Multi-budgeted Matchings and Intersection via Dependent Rounding

Chandra Chekuri∗ Jan Vondrak´ † Rico Zenklusen‡

Abstract ing: given a fractional point x in a polytope P ⊂ Rn, ran- Motivated by multi-budgeted optimization and other applications, domly round x to a solution R corresponding to a we consider the problem of randomly rounding a fractional solution of P . Here P captures the deterministic constraints that we x in the (non-) and wish the rounding to satisfy, and it is natural to assume that polytopes. We show that for any fixed δ > 0, a given point P is an integer polytope (typically a {0, 1} polytope). Of course the important issue is what properties we need R to x can be rounded to a random solution R such that E[1R] = (1 − δ)x and any linear function of x satisfies dimension-free satisfy, and this is dictated by the application at hand. A Chernoff-Hoeffding concentration bounds (the bounds depend on property that is useful in several applications is that R sat- δ and the expectation µ). We build on and adapt the swap isfies concentration properties for linear functions of x: that n rounding scheme in our recent work [9] to achieve this result. is, for any vector a ∈ [0, 1] , we want the linear function P 1 Our main contribution is a non-trivial martingale based analysis a(R) = i∈R ai to be concentrated around its expectation . framework to prove the desired concentration bounds. In this Ideally, we would like to have E[1R] = x, which would P paper we describe two applications. We give a randomized PTAS mean E[a(R)] = i aixi. If this is not feasible, we would for matroid intersection and matchings with any fixed number of like E[1R] to be approximately equal to x. budget constraints. We also give a deterministic PTAS for the case A natural question is the following. For which polytopes of matchings. The concentration bounds also yield related results P can we implement such a rounding procedure? A strong when the number of budget constraints is not fixed. As a second property that implies concentration for linear functions is 2 application we obtain an algorithm to compute in polynomial that of negative correlation , as shown by Panconesi and time an ε-approximate Pareto-optimal set for the multi-objective Srinivasan [25]. In our recent work [9] we described a variants of these problems, when the number of objectives is a rounding scheme called randomized swap rounding, to unify fixed constant. We rely on a result of Papadimitriou and Yannakakis and generalize some existing results. In particular, we [26]. showed that if P is a matroid polytope (the convex hull of independent sets of the matroid), then one can round 1 Introduction x ∈ P to a random independent set R such that E[1R] = x and the coordinates of 1 satisfy negative correlation. Randomized rounding of a fractional solution into an integral R We also showed negative correlation properties for certain solution is a powerful and ubiquitious technique in approxi- restricted subsets of variables when P is the intersection of mation algorithm design. Following the influential work of two ; this restricted setting nevertheless captures the Raghavan and Thompson [27] for routing, packing and cov- important and useful special case of the assignment polytope ering problems, several different randomized rounding meth- where one obtains negative correlation for the edge variables ods have been developed over the years. Dependent ran- incident to each vertex of the underlying bipartite graph domized rounding methods have recently found many ap- [14]. There are, however, applications where one requires plications [1, 31, 14, 11, 19, 3, 2, 29, 6]. The term depen- concentration properties for arbitrary linear functions of the dent refers to the property of the rounding scheme that en- variables. If one wants to use the property of negative sures that the rounded solution satisfies some additional con- correlation, then matroids are essentially the only structures straints in a deterministic fashion — this implies that the co- where this can be done: We have recently shown that a {0, 1} ordinates of the fractional solution cannot be independently polytope P has the property that any x ∈ P can be rounded rounded. An abstract view of such a scheme is the follow- to a vertex R of P such that E[1R] = x and the coordinates of 1R are negatively correlated, if and only if P is an axis- ∗ Department of Computer Science, University of Illinois, Urbana, IL parallel projection of a matroid base polytope [10]. 61801. Supported in part by NSF grants CCF-0728782 and CCF-1016684. E-mail: [email protected]. †IBM Almaden Research Center, San Jose, CA 95120. E-mail: 1In this paper we focus on non-negative vectors a ∈ [0, 1]n although [email protected]. there are applications for vectors a ∈ [−1, 1]n (see [1]). ‡ 2 Dept. of Mathematics, MIT, Cambridge, MA 02139. E-mail: We call {0, 1}-random variables X1,...,Xr negatively correlated, Q Q Q Q [email protected]. Supported by the Swiss National Science Foun- if E[ i∈T Xi] ≤ i∈T E[Xi] and E[ i∈T (1 − Xi)] ≤ i∈T (1 − dation, grant number: PBEZP2-129524. E[Xi]) for all subsets T ⊆ [r]. Given that negative correlation is too strong a property This can be seen from a well-known pseudo-polynomial time to hold beyond matroid polytopes, it is natural to consider reduction of b-matchings to regular matchings (see [30]), and whether the property of concentration for linear functions the fact that our bounds are dimension free. Obtaining an can still be obtained via different means for other polytopes. efficient (that is, polynomial instead of pseudo-polynomial) An important special case to consider is the assignment poly- rounding procedure requires more work, and we defer the tope (equivalently the bipartite simple b-matching polytope). details. Arora, Frieze and Kaplan [1], motivated by applications to Our methods also yield related bounds that depend on approximation schemes for dense-graph optimization prob- n, and that work for negative coefficients. Our main applica- lems, considered this problem in a certain restricted setting tions in this paper, however, require dimension-free bounds and guaranteed concentration bounds in an√ additive sense and we defer further discussion to a later version of the (with standard deviation on the order of n · poly(log n) paper. Our framework extends to polytopes/combinatorial and poly(log n), respectively, depending on whether the lin- structures that have certain exchange properties; Section 4 ear function under consideration has arbitrary coefficients or outlines this framework. non-negative coefficients). We discuss more details of their result and its relationship to our work later in the paper. It 1.1 Applications In this paper, we present two applica- is easy to show that even in the setting of the assignment tions, each of which applies to non-bipartite matchings and polytope, one cannot obtain concentration bounds while pre- matroid intersection. serving the expectation exactly. Multi-budgeted Optimization: There has been substantial In this paper we give a rounding scheme for two recent interest in the problem of optimizing linear (and also (classes of) polytopes, namely the polytope corresponding submodular functions) over a combinatorial structure with to the intersection of two matroids, and the non-bipartite additional linear/budget/packing constraints [5, 16, 22, 18, graph matching polytope. The rounding procedure for 2, 9]. Given a polynomial-time solvable problem, such as non-bipartite graph matching polytopes can easily be ap- maximum matching, can we optimize (in particular maxi- plied to simple b-matchings by a standard reduction from b- mize) over feasible solutions with an additional budget con- matchings to matchings [30]. Note that the assignment poly- straint of the form aT x ≤ b where a is a non-negative weight tope is a special case of both the intersection of two matroids vector? It is easy to see that the problem becomes NP- and simple b-matchings. Our main result is the following. hard via a reduction from knapsack. Randomized and deter- THEOREM 1.1. Let P be either a matroid intersection poly- ministic polynomial-time approximation schemes (PTASes) tope or a (non-bipartite graph) matching polytope. For any have been developed for important combinatorial optimiza- 1 tion problems subject to a single budget constraint: these fixed 0 < γ ≤ 2 , there is an efficient randomized rounding procedure, such that given a point x ∈ P , it outputs a ran- include spanning trees [28], matchings and independent sets dom feasible solution R corresponding to a (integer) vertex in two matroids [5]. Two approaches have been used. of P such that E[1R] = (1 − γ)x. In addition, for any lin- Algebraic methods give a randomized pseudo- P polynomial time algorithm for the exact version of some ear function a(R) = i∈R ai with ai ∈ [0, 1], and for any ε ∈ [0, 1], we have: of these problems [8]; that is, given an integer B and an integer weight vector w, is there a solution S such that −µγε2/20 • If µ ≤ E[a(R)], Pr[a(R) ≤ (1 − ε)µ] ≤ e . w(S) = B? Such algorithms can be used to obtain a 2 randomized PTAS for any fixed number of budgets [16]. In • If µ ≥ E[a(R)], Pr[a(R) ≥ (1 + ε)µ] ≤ e−µγε /20. the case of matroids and matroid intersection, this approach For any t ≥ 2, is limited to representable (i.e. linear) matroids [8]. A different, deterministic approach is via the use of La- • If µ ≥ E[a(R)], Pr[a(R) ≥ tµ] ≤ e−µγ(2t−3)/20. grangian relaxation combined with various technical proper- ties of the underlying combinatorial structure. This approach We emphasize that the concentration bounds above de- was first used to develop a PTAS for spanning trees [28], and pend only on µ, γ, ε and are independent of n, the dimen- recently, matchings and matroid intersection with one budget sion of the polytope. Moreover, there is no restriction on [5]. These results have been also extended to matchings sub- the linear functions a(R) for which we prove concentration ject to two budget constraints and matroid independent sets bounds, other than the normalization condition a ∈ [0, 1]. i with multiple budgets [16]. We note that a randomized PTAS Roughly speaking, a(R) for the rounded solution is concen- √ for multi-budgeted matroid independent sets is also given in trated in a window of size O( µ) around µ = E[a(R)], even [9] via the randomized swap rounding scheme. if µ  n. In this paper we obtain the following theorem as a The rounding scheme and concentration bounds in the relatively straightforward consequence of Theorem 1.1. above theorem also hold for non-bipartite graph b-matchings. THEOREM 1.2. For any fixed number of budget constraints, The above was not previously known for general ma- there is a randomized PTAS for multi-budgeted matroid in- troids. We remark that if the matroids are representable, then tersection and multi-budgeted non-bipartite graph matching. the results in [26] combined with the known exact algorithm for linear matroid intersection [8] imply a randomized fully Our techniques offer several advantages over previous polynomial-time approximation scheme (FPTAS). The same methods. First, we obtain a PTAS for matroid intersec- can be done for multi-objective matchings as well, using the tion without any restriction on representability; this was not randomized exact algorithm presented in [24]. We note that known prior to our work. Second, the concentration bounds our techniques can be used to obtain a deterministic algo- naturally allow for a good approximation even when the rithm for multi-objective matchings, similar to Theorem 1.3. number of budget constraints is not fixed as long as the con- We omit the details here. straints are sufficiently “loose”. More precisely, if the size 1 of each element is O( log m )B where B is minimum bud- Further extensions. Our framework allows for a common get size, then we can handle m budget constraints simul- generalization of the multi-budgeted and multi-objective set- taneously. We can also handle the case when all except a ting, with any fixed number of budgets and a fixed number small fixed number of constraints are loose. Alternatively, if of objectives. More precisely, we can find an ε-approximate we can tolerate an overflow of the budget constraints by an Pareto-optimal set in the presence of a constant number of O(log m) factor, we can handle m budget constraints with- budget constraints. This is dicussed further in Section 5. out any assumption on element sizes. These results arise nat- Other applications may arise in settings that involve a match- urally from our concentration results. ing or matroid intersection problem with additional con- Finally, our approach can potentially be derandomized straints. with the method of pessimistic estimators to obtain deter- ministic PTASes. (Unlike techniques relying on randomized 1.2 Technical contribution and relation to prior work exact-weight algorithms, whose derandomization is a major Our rounding builds on and modifies the swap-rounding open problem.) In the case of matroid intersection, deran- scheme from our recent work [9] that, given a point x domization of our approach seems non-trivial but possible in a n-dimensional polytope P , produces a random vertex in principle. In this work, we give a deterministic PTAS for corresponding to a solution R as follows: the case of multi-budgeted matchings; a deterministic PTAS • Express x as a convex combination of vertices of P , Pn was previously known only for two budgets [16]. Our algo- that is, x = αivi where each vi is a vertex of P P i=1 rithm can be viewed (indirectly) as a derandomization of our and i αi = 1. . • Let w1 = v1, β1 = α1, and in each stage, merge two vertices wi, vi+1 into a new vertex wi+1 with THEOREM 1.3. For any fixed number of budgets, there is a coefficient βi+1 = βi + αi+1 in such a way that deterministic PTAS for multi-budgeted matching. E[βi+1wi+1] = βiwi + αi+1vi+1 (the merge oper- ation). Multi-objective Optimization: In multi-objective opti- • After n − 1 stages, obtain a vertex wn = 1R such that Pn mization we have several different objective functions E[1R] = i=1 αivi = x. f1, . . . , fk and the goal is to simultaneously optimize them over a given polytope P . We restrict our attention to lin- Clearly, the crucial step which we have not described here is the merge operation. For matroids, the strong ear objective functions given by weight vectors w1, . . . , wk. Here, one is interested in the so called Pareto-optimal so- base-exchange property is useful in merging two bases by lutions; a solution S is Pareto-optimal if no other solution swapping two elements at a time and this leads to negative S0 dominates S for each objective. Papadimitriou and Yan- correlation. However, more complicated structures such nakakis [26] defined the notion of a succinct ε-approximate as matchings and independent sets in the intersection of Pareto-optimal set which consists of a polynomial number two matroids cannot be merged with a bounded number of elements being swapped at a time. For example, if M and of solutions S for given objectives w1, . . . , wk such that ev- 1 ery solution S0 is approximately dominated by some point M2 are two matchings, their symmetric difference M1∆M2 0 consists of a collection of disjoint cycles and paths. If one S ∈ S; that is wi(S) ≥ (1 − ε)wi(S ) for all 1 ≤ i ≤ k. We use our rounding scheme combined with a result in of these cycles/paths C is long, merging the matchings to [26] to obtain the following. preserve expectation requires one to pick all edges in C ∩M1 or C ∩ M2, leading to positive correlation among many THEOREM 1.4. There is a randomized polynomial-time al- variables. gorithm that for any fixed ε > 0 computes a ε-approximate In order to avoid the above, we modify the merge Pareto-optimal set for matroid intersection with a fixed num- process so that the cycles and paths used in the merge ber of objectives. operation are short. We do this by breaking long cycles and paths, effectively deleting some edges. (The idea of can be quite small when compared to n. Moreover, unlike breaking long cycles/paths is natural and not new; it has the simpler setting of bipartite matchings, the merge process been used in related contexts, for instance in [1] and more for matroid intersection is complex and hard to analyze in a recently in [15].) Therefore, instead of E[1R] = x, we can direct fashion as in [1]. We remark that our martingale setup only ensure that E[1R] is close to x. The main technical can be used to derive results similar to those in [1] also for difficulty is to analyze the above process. Although the the more general setting of matroid intersection. above is intuitive for matchings, the process is considerably We also mention the work in [15] on multi-budgeted more complicated for matroid intersection. Despite this, it matroid bases and bipartite matchings, when the number of was shown in [9] that one can define a merge process that is budgets is a fixed constant. Unlike our work, they allow based on covering the symmetric difference by a distribution the budget constraints to be violated by a (1 + ε)-factor; for over cycles; this decomposition was originally developed matroid bases the weight of the solution obtained is optimal, in [23] for the analysis of a local-search algorithm. One can while for bipartite matchings the weight of the solution is adapt the idea of breaking cycles even to this more general at least (1 − ε) times the optimum. The main technique of setting. [15] is iterated rounding, with additional ideas for bipartite The first difficulty in proving concentration bounds for matchings. our rounding scheme is that negative correlation is no longer Organization: In Section 2, we present our rounding proce- available. Therefore, we resort to a martingale-based analy- dure for matchings. In Section 3, we present our rounding sis. Our goal is to prove dimension-free bounds, and there- procedure for matroid intersection. In Section 4, we prove fore direct applications of Azuma’s inequality are insuffi- the main concentration results. Finally, we give more details cient. Our approach is to bound the total variance of the P on our applications in Section 5. We describe our determin- rounding process as function of the expectation µ = aixi, istic PTAS for multibudgeted matchings in Appendix A. in a way inspired by the lower-tail bound for submodular All our algorithms can be implemented in polynomial functions in [9]. However, further difficulties arise because time via well-known and standard assumptions/algorithms of the fact that swap operations are randomly distributed over for matchings and matroids. Due to space constraints, we large sets of elements. Hence it is not obvious how to esti- do not discuss the details in this version of the paper. mate the variance of a sequence of swap operations. This leads to a fairly technical analysis which we present in Sec- 2 Rounding in the matching polytope tion 4. The swap-rounding process is crucial for the analysis since an element that participates in a swap step effectively In this section, we describe how our rounding approach disappears after the step. We set up a somewhat generic anal- can be applied to a point in the (non-bipartite) matching polytope. Let G = (V,E) be an undirected graph. We ysis framework that is applicable to both of our problems. It E could also be useful for related problems. denote by M ⊆ 2 the set of matchings in G, and by Apart from our recent work [9], the work that is most P = conv({1M | M ∈ M}) the matching polytope for related to this paper is that of Arora, Frieze and Kaplan G. Let x ∈ P be a point in the matching polytope that is an [1]. They considered the problem of rounding a fractional input to our rounding procedure. x in a bipartite graph into a near-perfect Our rounding procedure builds on the swap-rounding matching such that concentration properties hold for any framework from [9]. First, a convex decomposition x = Pn α 1 with I ∈ M is obtained for x. Assume linear function of the edge variables. They developed an `=1 ` I` ` algorithm that is similar in spirit to swap-rounding with that the coefficients are ordered so that α1 ≥ · · · ≥ αn. the idea of breaking long cycles when merging two cycles. (Even though the ordering of the coefficients is not crucial, However, there are crucial differences in both the results it simplifies the analysis.) Let J1 := I1. We proceed in and the techniques. First, they obtain concentration bounds stages, where in the k-th stage, a new matching Jk+1 ∈ M is produced by “merging” Jk and Ik+1. The formal rounding that allow an additive√ deviation depending polynomially in n (essentially as O˜( n)), where n is the size of the procedure SwapRound is described in the figure below. graph. This is necessary for them since they consider negative coefficients, and these bounds were sufficient for Pn Algorithm SwapRound(x = `=1 α`1I` ): their applications. Further, these large additive terms imply Reorder so that α1 ≥ α2 ≥ ... ≥ αn; that the rounding scheme and the analysis could be coarse J1 = I1; — after some preliminary processing to reduce the number For (k = 1 to n − 1) do of merge steps to be poly-logarithmic in n, the analysis uses Pk Jk+1 =MergeMatchings(αk+1,Ik+1, `=1 α`,Jk); standard Chernoff-Hoeffding bounds for each merge, and a EndFor union bound over the merge steps. However, the bounds P Output Jn. we seek are relative to the expectation µ = aixi which It remains to specify the MergeMatchings procedure. with respect to α, β, and (ii) assuming δ = 0, the step en- Given two matchings I,J ∈ M with corresponding coeffi- sures that each element of I∆J is equally likely to be re- cients α, β, we want to choose MergeMatchings(α, I, β, J) moved as added. However, since δ > 0, the important point such that a (random) matching M is returned that is in expec- here (in contrast to the work in [9]) is that the process creates tation close to the given linear combination of I and J, i.e., a slight bias towards the elements we are removing. This bias E[(α + β)1M ] ≈ α1I + β1J , and such that the correlation allows the use of short swap sets which in turn helps prove between different edges of M is weak enough for concen- concentration bounds. The bias also means that the process tration bounds to hold. The approach in [9] is to consider is no longer a martingale. Nevertheless, we show in Sec- the symmetric difference I∆J which consists of alternating tion 4 that the process can be analyzed by defining a related paths and cycles S1,...,Sm; a merged matching M is ob- martingale. tained by including all edges I ∩J in M, and for each set Si, with probability α all edges of S ∩ I are included in M, α+β i Algorithm MergeMatchings(α, I, β, J): otherwise all edges of Si ∩J are included in M. This scheme While (I 6= J) do satisfies the property that E[(α+β)1M ] = α1I +β1J , how- Generate a collection of alternating paths/cycles {Pi} ever, it creates positive correlations between many variables in I∆J of length ≤ 2p such that inside the sets Si. As mentioned earlier, such strong positive P P ρi1Pi = (1 + δ)ρ1I\J + ρ1J\I , ρi = 1, ρi ≥ 0, correlations are unavoidable if x is to be preserved in expec- 0 and a collection of alternating paths/cycles {Pi } tation. Our approach returns a merged matching M whose in I∆J of length ≤ 2p such that expectation is slightly below the target value α1I + β1J , in P σ P σ 1 0 = 1 + σ1 , σ = 1, σ ≥ 0. i Pi 1+δ I\J J\I i i return we obtain concentration results for any linear func- βσρi With probability αρ+βσ for each i, let I := I∆Pi. tion f(M). Intuitively, we would like to break the Si into else with probability αρσi for each i, let J := J∆P 0. smaller segments by removing some edges. The lemma be- αρ+βσ i low describes the properties we desire for the segments. The EndWhile choice of the parameter p will be discussed later. Output I.

LEMMA 2.1. Let I,J ∈ M, p ∈ Z+, and δ = 1/(p − 1). The choice of the parameter p will depend on the Then we can find in polynomial time a collection of sets parameter γ of Theorem 1.1 which controls the loss in P1,...,Pm ⊆ I∆J with coefficients ρi = 1/m, such that expectation. There is a trade-off in choosing p: the larger m ≤ p|I∆J|, and for ρ = p − 1: p is chosen, the smaller is the bias, i.e., the closer is the (i) For i ∈ {1, . . . , m}, I∆Pi ∈ M. matching M returned by SwapRound to the point x in (ii) For 1 ≤ i ≤ m, |I ∩ Pi| ≤ p, |J ∩ Pi| ≤ p and hence expectation. However, a larger value of p leads also to 0 |Pi| ≤ 2p. longer swap sets in the collections {Pi} and {Pi } used in Pm (iii) i=1 ρi1Pi = (1 + δ)ρ1I\J + ρ1J\I . MergeMatchings, which results in weaker concentration. The relation between p and γ together with the concentration We defer the proof of the lemma to Section 2.1, and describe results will be discussed in Section 4. our merging procedure MergeMatchings. It works in itera- tions, where in each iteration either I or J is modified such 2.1 Creating collections of swap sets of small size In that |I∆J| strictly decreases, until I = J. If we (randomly) this section we prove Lemma 2.1, the basis of which is the decide to alter I, we construct a collection of short alter- following lemma. nating paths/cycles {Pi}, according to Lemma 2.1. Then a path Pi is chosen out of {Pi} with probability ρi and I LEMMA 2.2. Let I,J ∈ M, p ∈ Z+, and let S ⊆ I∆J is replaced by I∆Pi. Similarly, when we (randomly) de- be a connected component of I∆J. In polynomial time, a 0 cide to change J, a collection of paths/cycles {Pi } of length collection of sets P1,...,Pm ⊆ S can be obtained such that at most 2p is constructed with probabilities {σi} such that m ≤ max{p, |S|} and: P p−1 σ 1 0 = σ1 + σ1 , for some constant σ > 0. i P1 p I\J J\I • For 1 ≤ i ≤ m, I∆Pi ∈ M. Again, such a family can be obtained through Lemma 2.1, • For 1 ≤ i ≤ m, |I ∩ Pi| ≤ p, |J ∩ Pi| ≤ p and hence by exchanging the roles of I and J, and by scaling ρ. Be- |Pi| ≤ 2p. cause of their use in modifying I and J, we call the al- • Each edge of S ∩ I appears in exactly p of the Pi and 0 ternating paths/cycles {Pi} and {Pi } also swap sets. The each edge of S ∩ J in exactly p − 1 of the Pi. MergeMatchings algorithm is summarized in the box where δ = 1/(p − 1). Proof. The set S is either an alternating path or cycle. The random step to choose which set I or J to alter is Assume first that S is an alternating path (e1, . . . , er) that based on α, β, σ, ρ in a natural fashion to ensure two proper- is not a cycle. The case of alternating cycles is very similar ties: (i) the merged matching proportionally represents I,J and we discuss it briefly at the end of the proof. If |S| ≤ 2p, then we choose m = p paths P1,...,Pm, where each path are defined over the same ground set N. It was proved by Pi consists of all edges in S. To make sure that each edge Edmonds [12] that this polytope is integral and hence any of S ∩ J is contained in exactly p − 1 sets, for each edge fractional solution can be expressed as a convex combination in J we select a unique path from the set of m = p paths of integer solutions. We assume that the matroids are given and remove it from that path. Since |S| ≤ 2p and S is an via membership oracles: that is, given a set S ⊆ N the alternating path, |S ∩ J| ≤ p, and hence this is possible to oracle returns whether S is independent in the matroid. Let do. One can easily check that the resulting sets fulfill the n = |N| denote the cardinality of the ground set. conditions of the lemma. Now consider the case |S| > 2p. Assume e1 ∈ 3.1 Decomposition into irreducible paths and cycles I (the case e1 ∈ J is analogous). For a fixed j ∈ The basis of our rounding procedure is a decomposition of {1, . . . , p}, we can break the path (e1, . . . , er) into short the symmetric difference between two independent sets into subpaths by removing the edges e2j, e2j+2p, e2j+4p,... . feasible exchanges. This is similar to the decomposition Let P1,P2,...,Pm ⊆ S be the collection of all subpaths developed in [23, 9]; however, with some modifications due obtained in this way for all choices of j ∈ {1, . . . , p}. Each to the fact that we want to break long exchange cycles into subpath has length at most 2p − 1. It is easy to verify that shorter exchange paths. We use the construction of [9] as a each edge e ∈ I ∩ S is contained in exactly p of the subpaths black box and describe the important differences here. P1,...,Pm, each edge e ∈ J ∩ S is contained in exactly Again, we use the standard constructs of matroid inter- p − 1 subpaths, and m ≤ |S|. It remains to observe that section; see [30]. For I ∈ I1 ∩ I2, we define two digraphs

I∆Pi ∈ M for i ∈ {1, . . . , m}, i.e., that dI∆Pi (v) ≤ 1 for DM1 (I) and DM2 (I) as follows. v ∈ V . We have dI∆Pi (v) = dI (v) = 1 for all vertices • For each i ∈ I, j ∈ N \ I with I + j − i ∈ I1, we have v ∈ V that are not endpoints of Pi. If v is an endpoint of Pi, an arc (i, j) ∈ DM1 (I); then by the way how the Pi are constructed, the edge e ∈ Pi • For each i ∈ I, j ∈ N \ I with I + j − i ∈ I2, we have that is adjacent to v is either an edge in I, in which case we an arc (j, i) ∈ DM2 (I). have dI∆Pi (v) = 0, or if e ∈ J then v is also an endpoint of When we refer to a matching in DMl (I) for l = 1, 2 we the path S, and hence dI∆Pi (v) ≤ dS(v) = 1. mean a matching in an undirected graph where the arcs

The argument above can be easily extended to S being of DMl (I) are treated as undirected edges. We define a an alternating cycle. If |S| ≤ 2p then again m = p paths digraph DM1,M2 (I) as the union of DM1 (I) and DM2 (I).

P1,...,Pm can be chosen where each path consists of all A directed cycle in DM1,M2 (I) corresponds to a chain of edges in S except up to one edge of J, to fulfill the condition feasible swaps. It is not necessarily the case that the entire that each edge in S ∩ J is contained in exactly p − 1 of the cycle gives a valid exchange in both matroids. Nonetheless, paths. If |S| > 2p, then a collection of paths P1,...,Pm it is known that if a cycle decomposes into two matchings fulfilling the claims of the lemma is obtained by taking all which are unique on their set of vertices, respectively in paths of length 2p − 1 starting with an edge in I and going DM1 (I) and DM2 (I), then the cycle corresponds to a along the cycle in a fixed direction. One can check that this feasible swap. This motivates the following definition. collection of paths satisfies the conditions of the lemma.

DEFINITION 3.1. We call a directed cycle C in DM1,M2 (I)

Finally, applying Lemma 2.2 to each component of I∆J irreducible if C∩DM1 (I) is the unique matching in DM1 (I) separately, a collection of short paths/cycles P1,...,Pm and C∩DM2 (I) is the unique matching in DM2 (I) covering is obtained that covers all elements of I∆J, and implies exactly the vertex set V (C). Otherwise, we call C reducible. Lemma 2.1. Lemma 2.2 shows that in fact all probabilities {ρi} Let us assume for simplicity that we have two sets in Lemma 2.1 can be chosen to be equal, and the same I,J ∈ I1 ∩I2 and |I| = |J|. We also assume that I ∩J = ∅; holds for the probabilities {σi}. Still, we presented the if not, we can formally replace elements in I ∩ J by parallel algorithm in the slightly more general setting to present a copies which appear in I,J respectively that can always be unified framework for matchings and matroid intersection. exchanged without affecting independence. The following For matroid intersection the decomposition into swap sets lemma, building on previous work [23], was proved in [9]. requires non-uniform probabilities {ρi}.

LEMMA 3.1. Let M` = (N, I`), ` = 1, 2, be matroids on 3 Rounding for matroid intersection ground set N. Suppose that I,J ∈ I1 ∩ I2 and |I| = |J|. In this section, we describe our new procedure to round Then we can find in polynomial time a collection of irre- a fractional solution in the matroid intersection polytope, ducible cycles {C1,...,Cm}, m ≤ |I∆J|, in DM1,M2 (I), Pm i.e. the intersection of two matroid polytopes P (M1) ∩ with coefficients γi ≥ 0, i=1 γi = 1, such that for some Pm P (M2) where both M1 = (N, I1) and M2 = (N, I2) γ > 0, i=1 γi1V (Ci) = γ1I∆J . In this work, we need a modification of this decompo- of J \ I with a coefficient of γ. Finally, we normalize the P sition lemma which uses irreducible paths in addition to cy- coefficients so that ρi = 1, which gives the statement of cles. We define an irreducible path as follows. the lemma for some value ρ > 0.

EFINITION D 3.2. We call a directed path P in DM1,M2 (I) Observe that switching the roles of I and J, we can work 0 irreducible if both its endpoints are in I, P ∩ DM1 (I) is the with DM1,M2 (J) and obtain a collection of paths/cycles Pi unique matching on its vertices in D (I), and P ∩D (I) Pm M1 M2 with coefficients σi such that σi1V (P 0) = σ1I\J + is the unique matching on its vertices in D (I). i=1 i M2 (1 + δ)σ1J\I . Equivalently, we can scale σ so that Pm σ σ 1 0 = 1 + σ1 . i=1 i V (Pi ) 1+δ I\J J\I LEMMA 3.2. For any irreducible path P in DM1,M2 (I), the set I∆V (P ) is independent in I ∩ I . 1 2 3.2 The rounding procedure The rounding procedure Proof. Observe that the edges of P alternate between follows the same framework as the rounding algorithm for D (I) and D (I), the vertices of P alternate between I matchings (Section 2). Again, we start with a convex combi- M1 M2 Pn and N \ I, and since both endpoints are in I, the length of P nation x = i=1 αi1Ii where Ii ∈ I1 ∩ I2. Any fractional solution x ∈ P (M1) ∩ P (M2) can be efficiently decom- is even. Consider the matching P ∩DM1 (I) and its vertex set posed in this manner, see [30]. Assume also that the coeffi- W1 = V (P ) \{w2}, where w2 is the endpoint of P incident cients are ordered so that α1 ≥ ... ≥ αn. Let J1 := I1. As to an edge of DM2 (I). By assumption, this is the unique in the matchings case we proceed in stages, where in the k- matching in DM1 (I) covering W1, and so I∆W1 ∈ I1. By removing another element, we cannot violate independence, th stage, a new independent set Jk+1 ∈ I1 ∩ I2 is produced from J and I . and hence we also have I∆V (P ) = (I∆W1) \{w2} ∈ I1. k k+1 Similarly, we prove that I∆V (P ) ∈ I2.

The next lemma is an adaptation of Lemma 3.1 for our purposes. Pn Algorithm SwapRound(x = `=1 α`1I` ): Reorder so that α1 ≥ α2 ≥ ... ≥ αn LEMMA 3.3. Let M` = (N, I1), ` = 1, 2, be matroids on J1 = I1; ground set N, and let δ = 1/(p − 1), p ∈ Z. Suppose that For (k = 1 to n − 1) do I,J ∈ I1 ∩I2 and |I| = |J|. Then we can find in polynomial Pk Jk+1 =MergeIntersectionSets(αk+1,Ik+1, `=1 α`,Jk); time a collection of irreducible cycles/paths {P1,...,Pm} of 2 EndFor length at most 2p − 1, m ≤ |I∆J| , in DM1,M2 (I), with Pm Output Jn. coefficients ρi ≥ 0, i=1 ρi = 1, such that for some ρ > 0, Pm i=1 ρi1V (Pi) = (1 + δ)ρ1I\J + ρ1J\I . The merge operation is performed in much the same Proof. Let {C1,...,Cm0 } be the collection of irreducible way as in [9], the difference being that we use the swap cycles on I∆J provided by Lemma 3.1. Consider a cycle Ci path/cycle structure provided by Lemma 3.3. Suppose we with coefficient γi, which has length 2`i. Assume for now are merging two sets I,J which are disjoint. If not, we that `i > p. For each vertex v ∈ V (Ci) ∩ I, let Piv be a formally add copies of the shared elements, and we consider directed path of length 2p − 1 starting at v, going along the trivial exchanges between copies of the same element. This cycle. The number of such paths is `i ≤ |I∆J|. We assign does not affect the actual rounding procedure, but helps in a coefficient ρi = γi/(p − 1) = δγi to each of these paths. the analysis in the sense that each element of I ∪ J will be Since every vertex in V (C) ∩ I is contained in p such paths, processed exactly once in the merging stage. and every vertex in V (C)\I is contained in p−1 such paths, We apply Lemma 3.3 twice, to obtain (1) a con- P these paths contribute (1 + δ)γi to each vertex of V (C) ∩ I vex combination of irreducible paths/cycles ρi1V (Pi) and γi to each vertex of V (C) \ I. in DM1,M2 (I), representing feasible swaps from I to J, If `i ≤ p, we can keep the cycle Ci in our collection and (2) a convex combination of irreducible paths/cycles P with ρ = (1 + δ)γ . This cycle would contribute (1 + δ)γ σ 1 0 in D (J), representing feasible swaps i i i i V (Pi ) M1,M2 to all its vertices. To make the contributions consistent with from J to I. These swap sets cover the symmetric differ- the case above, we replace Ci with suitable coefficients by ence I∆J almost uniformly, but as in the rounding algorithm paths where some vertex of V (Ci) ∩ J is removed. It is easy for matchings, there is a slight bias towards the elements P to see that in this way we can decrease the contribution to we are removing. For example, we have ρi1V (Pi) = each vertex of V (Ci) ∩ J to γi. (1 + δ)ρ1I\J + ρ1J\I ; i.e., elements of I are covered more We repeat this for every cycle Ci, to produce a collection often than the elements of J, when we work with I. This of at most |I∆J|2 paths/cycles. These cover each vertex of means elements are removed from I more often than added I \ J with a coefficient of exactly (1 + δ)γ and each vertex to I. Algorithm MergeIntersectionSets(α, I, β, J): While (I 6= J) do Generate a collection of irreducible paths/cycles {Pi} in DM ,M (I) each containing at most 2p vertices P P 1 2 such that ρi1V (Pi) = (1 + δ)ρ1I\J + ρ1J\I , ρi = 1, ρi ≥ 0, 0 and a collection of irreducible paths/cycles {Pi } in DM1,M2 (J) each containing at most 2p vertices P σ P such that σ 1 0 = 1 + σ1 , σ = 1, σ ≥ 0. i V (Pi ) 1+δ I\J J\I i i βσρi With probability αρ+βσ for each i, let I := I∆V (Pi). αρσi 0 else with probability αρ+βσ for each i, let J := J∆V (Pi ). EndWhile Output I.

We proceed in a sequence of random swaps, where each E[1D] = (1 + δ)ρ1I\J + ρ1J\I , swap is chosen randomly from a distribution corresponding to this path/cycle structure, as can be seen from a concise αρ 0 • with probability αρ+βσ , we replace J by J∆D where description of the merge operation in the box. Every time σ E[1D0 ] = 1I\J + σ1J\I . we perform a swap, the size of I∆J shrinks. We remark that 1+δ it is necessary to recompute the entire path/cycle structure Our goal is to prove the following. after each swap. We repeat this procedure, until I and J become identical. n P THEOREM 4.1. Let a ∈ [0, 1] and let a(S) = i∈S ai be the associated linear function. Let x = Pn α 1 ∈ P 4 Dimension-free concentration for swap-based i=1 i Ii and let µ = E[a(R)], where R = J ∈ F is a random set processes n obtained by the randomized swap rounding procedure using In this section, we prove concentration bounds for a general swaps satisfying |D∩Ik+1|, |D∩Jk| ≤ p as explained above. random process which encapsulates both our rounding pro- 1 2 Then E[1R] ≥ (1 − p ) x and cedures for matroid intersection and matching. The rounding 2 i) For ε ∈ [0, 1], Pr[a(R) ≤ (1 − ε)µ] ≤ e−µε /8p, procedures from the previous two sections can be described −µε2/8p at a high level as follows. ii) For ε ∈ [0, 1], Pr[a(R) ≥ (1 + ε)µ] ≤ e . iii) For t ≥ 2, Pr[a(R) ≥ tµ] ≤ e−µ(2t−3)/8p. 4.1 The swap-based random process Let P ⊆ [0, 1]N be an integer polytope and let F ⊆ 2N be the family of the Given this theorem, we can derive Theorem 1.1 as 1 subsets of N that correspond to vertices of P . Given is a follows. Given γ ∈ (0, 2 ] in Theorem 1.1, we pick the Pn smallest integer p such that 2 ≤ γ. (Note that p ≥ 4 and convex combination x = i=1 αi1Ii ∈ P , where αi ≥ 0, p Pn 1 1 1 2 i=1 αi = 1 and Ii ∈ F. We further assume for technical δ = p−1 ≤ 3 .) Then we have E[1R] ≥ (1 − p ) x ≥ reasons that the indices are ordered so that α1 ≥ ... ≥ 2 (1 − p )x ≥ (1 − γ)x. As we show below, we can in αn. Assuming this ordering slightly simplifies our analysis. fact scale down the initial coefficients in such a way that However, with some modifications, the proof technique E[1R] = (1 − γ)x, as required by Theorem 1.1. Since we employ could also be used without the assumption of 2 we picked the smallest integer p satisfying p ≤ γ, we have α the i being ordered, leading to slightly weaker constants p ≤ 2 +1 ≤ 2.5 . This is why 1/8p in the exponent becomes in the exponents of the concentration bounds presented in γ γ γ/20 in Theorem 1.1. Theorem 1.1. Let us point out some differences between this work We let J := I . In the k-th merge operation, we process 1 1 and [9]. In [9], the random process in terms of the fractional J and I to produce a new set J . The merge operation k k+1 k+1 solution throughout the algorithm forms a martingale; i.e., proceeds in a sequence of swap operations between J and k the expectation is preserved in each step. This is not I , where each swap set D is chosen from a certain k+1 true here, since the structure of exchange cycles and paths distribution so that D contains at most p elements from each provided by Lemmas 2.1 and 3.3 is biased, in the sense of J ,I . The exact structure of the random swaps is given k k+1 that elements are more often removed than added. In by Lemma 2.1 for matchings and by Lemma 3.3 for matroid order to facilitate the analysis, we first define a modified intersection. In summary, if the current sets are I,J with random process which is related to the evolution of the coefficients α, β > 0, there are coefficients ρ, σ > 0 such fractional solution and in fact forms a martingale. This that allows us to apply our martingale analysis and finally prove βσ • with probability αρ+βσ , we replace I by I∆D where the concentration results that we claimed. 4.2 A modified martingale process We define β` = Proof. Note that the elements that have not been processed P` i=1 αi, and consider a related linear combination x˜ = yet are counted in x˜k,t with coefficients α˜i which are smaller Pn i=1 α˜i1Ii , with than αi - this accounts for the fact that they will be kicked out more often than added. At the beginning of the process, αi + βi−1 α˜ = α , we have J1 = I1 and formally J1 ∩ I1 = ∅; therefore, i α + (1 + δ)2β i i i−1 n n X X where δ = 1/(p − 1) is the parameter used in the rounding x˜1,0 = β11J1 +α ˜21I2 + α˜`1I` = α˜`1I` procedure. The modified process starts from the point x˜ = `=3 `=1 Pn i=1 α˜i1Ii . As we merge the first k sets, we produce a because β1 = α1 =α ˜1. new set Jk, and the coefficient assigned to this set will be To prove the martingale property, consider a fractional Pk βk = i=1 αi. I.e., at this point the linear combination solution x˜k,t. The procedure operates on the sets Jk and Pn becomes βk1Jk + `=k+1 α˜`1I` . Eventually, we obtain a Ik+1. If it decides to modify Jk, some coordinates on Jk lose Pn set Jn with coefficient βn = i=1 αi = 1. βk (when they are removed from Jk) and some coordinates I β + α − α˜ The new random process defines a martingale, as we on k+1 gain k k+1 k+1 (when they move from I \ J I ∩ J state more precisely in Lemma 4.1. Thus if R = J is the k+1 k to k+1 k). If the procedure decides to modify n I J α final outcome of the rounding procedure, we will show that k+1, some coordinates on k gain k+1 (when they move to Pn Jk ∩ Ik+1), and some coordinates on Ik+1 lose α˜k+1 (when E[1R] = x˜ = i=1 α˜i1Ii . Note that each coefficient α˜i is 2 1 2 1 2 they are removed from Ik+1). at least αi/(1 + δ) = αi/(1 + p−1 ) = (1 − p ) αi. Hence 1 2 Now let us compute the expected change for a fixed E[1R] ≥ (1 − ) x, as required by Theorem 4.1. p element j ∈ Jk. To simplify notation, let us use α = We remark that in order to achieve the condition α , β = β and α˜ =α ˜ . The probability that we E[1 ] = (1 − γ)x for some γ ≥ 2 , as required by Theo- k+1 k k+1 R p remove j from J is the probability that we modify J and j rem 1.1, we can proceed as follows. Assuming that the initial k k 0 Pn 0 participates in the chosen swap path, i.e. linear combination is x = i=1 αi1Ii , we can scale the co- Pn X αρσi αρσ efficients down in a suitable way to obtain x = i=1 αi1Ii Pr[j is removed from Jk] = = . Pn αρ + βσ αρ + βσ (formally adding an empty set with coefficient 1− i=1 αi), i:j∈V (Pi) 0 so that α˜i = (1 − γ)αi. We omit the details, as this would further encumber the notation. Similarly, the probability that an element j ∈ Jk gains by Next, we refine the definition of the modified random being added to Ik+1 is process and describe what we mean by x˜ in the middle of the X βσρi βσρ Pr[j is added to I ] = = . merge operation. The following lemma describes the process k+1 αρ + βσ αρ + βσ i:i∈V (P 0) and proves that it is a martingale. i In the first case, the coordinate Xj loses β, while in the LEMMA 4.1. Let F denote the family of feasible solutions second case, it gains α. Therefore, if X0 denotes the (either matchings or sets in matroid intersection). Let j Pn coordinate value after this step, x = i=1 αi1Ii , where Ii ∈ F. Let J1 = I1, and for k ∈ {1, . . . , n − 1} let Jk+1 be the set produced by 0 αρσ βσρ E[Xj | Xj] = Xj − β + α = Xj. merging Jk,Ik+1 as in the procedure MergeMatchings or αρ + βσ αρ + βσ Pk MergeIntersectionSets. Define βk = i=1 αi and x˜ = It is similar but slightly more involved to analyze the change Pn i=1 α˜i1Ii , where in coordinates on Ik+1. The probability that we remove j from Ik+1 is the probability that Ik+1 is modified and j αi + βi−1 α˜i = 2 αi. happens to be on the chosen swap path: αi + (1 + δ) βi−1 X βσρi After t steps of merging Jk with Ik+1, assuming the elements Pr[j is removed from Ik+1] = 0 αρ + βσ already processed are in Jk ∩ Ik+1, let us define i:j∈V (Pi ) βσρ x˜ = (β + α )1 + β 1 = (1 + δ) . k,t k k+1 Jk∩Ik+1 k Jk\Ik+1 αρ + βσ n X The probability that j is added to Jk is the following: +α ˜k+11Ik+1\Jk + α˜`1I` . `=k+2 X αρσi Pr[j is added to Jk] = Pn αρ + βσ Then we have x˜ = i=1 α˜i1Ii = x˜1,0 and the process i:i∈V (Pi) (x˜1,0, x˜1,1,..., x˜2,0, x˜2,1,...) forms a martingale. In par- 1 αρσ = . ticular, if R is the final rounded solution, E[1R] = x˜ = x˜1,0. 1 + δ αρ + βσ Note the asymmetry here, due to the factors (1 + δ). Recall iv) We have a(D ∩ I), a(D ∩ J) ≤ 1 with probability 1. that in the first case, Xj loses α˜, while in the second case, (α+β)α Let us now analyze how a linear function a · x˜ behaves Xj gains α + β − α˜. A computation using α˜ = α+(1+δ)2β under such swap operations. Let Z be the change of a · x˜ reveals that due to the first swap, i.e., Z = a · (x˜0 − x˜). Furthermore, βσρ let W be the random variable given by a(D), where D is the E[X0 | X ] = X − (1 + δ) α˜ j j j αρ + βσ swap set used in the first swap iteration. We have |Z| ≤ W and W = a(D ∩ I) + a(D ∩ J) ≤ 2. First, the following 1 αρσ + (α + β − α˜) elementary inequality. 1 + δ αρ + βσ βσρ α + β LEMMA 4.2. Let Y = αAX − β(1 − A)X, where α, β ∈ = X − (1 + δ) α j αρ + βσ α + (1 + δ)2β [0, 1], A, X are (possibly correlated) random variables with 1 αρσ (α + β)(1 + δ)2β A ∈ {0, 1},X ∈ [0, 1], and E[Y ] = 0. Then + 1 + δ αρ + βσ α + (1 + δ)2β E[Y 2] ≤ αβE[X]. = Xj. Proof. Since A ∈ {0, 1}, Y 2 = α2AX2 + β2(1 − A)X2. Therefore,

2 2 2 2 2 4.3 Martingale analysis In the following, we work to- E[Y ] = α E[AX ] + β E[(1 − A)X ] wards the proof of Theorem 4.1. By scaling a by 1/p, let ≤ α2E[AX] + β2E[(1 − A)X] us assume that ai ∈ [0, 1/p], so that a(D ∩ I), a(D ∩ J) ≤ 1 for any swap set D. We prove Theorem 4.1 by proving using the fact that X ∈ [0, 1]. Since we assume E[Y ] = 2 α [AX] − β [(1 − A)X] = 0 Pr[a(R) ≤ (1 − ε)µ] ≤ e−µε /8, Pr[a(R) ≥ (1 + ε)µ] ≤ E E , we have 2 e−µε /8 and Pr[a(R) ≥ tµ] ≤ e−µ(2t−3)/8. 0 = (β − α)E[Y ] The core of the proof is to estimate the exponential mo- = (αβ − α2)E[AX] + (αβ − β2)E[(1 − A)X]. ment E[eλ(a(R)−µ)] and bound it by a factor depending only 2 on the expectation, of the form eO(λ µ). The difficulty in an- Adding this to the inequality above, we get alyzing this process is that the swap set in each step comes 2 from a probability distribution and hence it is not easy to E[Y ] ≤ αβE[AX] + αβE[(1 − A)X] = αβE[X]. charge the variance of the current step to the value of ele- ments that have been processed. We develop an inductive approach to this problem relying on two probabilistic lem- Next, we have the following inequality for a single swap. mas (Lemma 4.2 and 4.3) which appear to be new and some- 0 what different from traditional martingale analysis. LEMMA 4.3. Let x˜ be obtained from x˜ by a single swap We focus on the analysis of the modified random process step, using a random swap set D as above, where α ≤ β, 0 in terms of x˜. In the k-th stage, we are merging sets Jk and and let Z = a · (x˜ − x˜) and W = a(D). Then for any Pk λ ∈ [− 1 , 1 ], Ik+1, with coefficients βk = i=1 αk and α˜k+1. Define 4 4 ˜ ˜ λZ−λ2αβW also βk so that βk +α ˜k+1 = βk + αk+1. Recall that the E[e ] ≤ 1. coefficients are ordered in descending order of magnitude, Proof. Since λZ − λ2αβW ≤ λZ ≤ 1 , we use the so we always have α ≤ β . For simplicity, let us drop 4 k+1 k following elementary bound: ex ≤ 1 + x + 5 x2 for x ≤ 1 . the indices and denote the sets by I,J, and the coefficients 9 4 We obtain by α, β, α,˜ β˜. Since α˜ ≤ α and β˜ +α ˜ = β + α, we 2 have α˜ ≤ α ≤ β ≤ β˜. While I∆J is nonempty, E[eλZ−λ αβW ] ≤1 + E[λZ − λ2αβW ] we perform the following rounding step. Assume that 5 the current (modified) fractional solution is x˜. One swap + E[(λZ − λ2αβW )2]. 9 operation can be summarized as follows. We have E[Z] = a · E[x˜0 − x˜] = 0 due to condition (iii). Swap Operation. Therefore, i) With some probability, choose a random D ⊆ I∆J and 0 ˜0 ˜ 2 5 define I := I∆D, x := x − α˜1D∩I + α1D∩J . E[eλZ−λ αβW ] ≤ 1 − λ2αβE[W ] + λ2E[(Z − λαβW )2]. ii) Otherwise, choose a random D ⊆ I∆J (according to 9 a possibly different distribution), and let J := J∆D, The key is to estimate the last expectation. We claim that 0 ˜ x˜ := x˜ + β1D∩I − β1D∩J . iii) The distributions are chosen so that E[x˜0 | x˜] = x˜. (4.1) E[(Z − λαβW )2] ≤ αβ(1 + |λ|)2E[W ]. 5 2 2 If we prove this, we are done, because 9 (1 + |λ|) ≤ using the fact that E[W ] ≤ 2E[W ] since 0 ≤ W ≤ 2. 5 1 2 9 (1 + 4 ) < 1, and therefore Hence, 2 2 E[(Z − λαβW ) ] E[eλZ−λ αβW ] = E[Z2] − 2λαβE[ZW ] + λ2α2β2E[W 2] 5 ≤ 1 − λ2αβE[W ] + λ2αβ(1 + |λ|)2E[W ] ≤ 1. p 9 ≤ αβE[W ] + 2|λ|αβ 2αβE[W ] + 2λ2α2β2E[W ] Proof of (4.1). We introduce the following random variables: p = αβ(1 + |λ| 2αβ)2E[W ] • W = a(D ∩ I),W = a(D ∩ J). I J ≤ αβ(1 + |λ|)2E[W ] 0 0 • ZI = aI · (x˜ − x˜),ZJ = aJ · (x˜ − x˜), where aI , aJ using α + β ≤ 1 which implies that αβ ≤ 1/4. are the weight functions restricted to I,J respectively. Remark. The proof above is the only place where we • A is an indicator variable such that A = 1 if we are exploit the ordering of the exponents α1 ≤ · · · ≤ αn. modifying I (step i)), and A = 0 if we are modifying J One can show that even without assuming any particular (step ii)). ordering of the coefficients, α˜β˜ ≤ (1 + δ)2αβ holds. By slightly modifying the statement of this lemma and • I.e., W = WI + WJ and Z = ZI + ZJ , where the remaining part of the proof, this additional factor of (1 + δ)2 can be propagated in the exponent, still leading ˜ (4.2) ZI = −AαW˜ I + (1 − A)βWI to a concentration bound, however with a slightly weaker constant in the exponent. and Now we can add up the contributions to the exponential

(4.3) ZJ = AαWJ − (1 − A)βWJ . moment over an entire merge operation. Let us define Z1,Z2,Z3,... to be the changes of a · x˜ during successive swaps in merging I and J. Similarly, let us define W = The assumption that E[x˜0 | x˜] = 0 implies that E[Z ] = 1 I a(D ) to be the weight of the first swap, etc. Notice that E[Z ] = 0. Let us write E[(Z − λαβW )2] as follows: 1 J the number of swaps needed to merge I and J is a random variable since the swap sets are not necessarily of the same E[(Z−λαβW )2] = E[Z2]−2λαβE[WZ]+λ2α2β2E[W 2]. size. However, this does not impose any further difficulty in First, let us estimate E[Z2]. We have the analysis below. In particular, by adding dummy swaps at the end if necessary we can assume that the number of 2 2 swaps needed to merge I and J is always the same. Notice E[Z ] = E[(ZI + ZJ ) ] that Z ∈ [−1, 1] and E[Z | H ] = 0, where H is the 2 2 j j j−1 j−1 = E[ZI ] + 2E[ZI ZJ ] + E[ZJ ] history of the merging procedure up to swap iteration j − 1. 2 2 ≤ E[ZI ] + E[ZJ ] 1 1 LEMMA 4.4. Let λ ∈ [− 4 , 4 ] and k ∈ N be some fixed number of swap iterations, then because ZI and ZJ always have opposite signs. The variable Pk 2 ZJ when decomposed as given by (4.3) satisfies the assump- λ Zj λ αβa(I∆J) 2 E[e j=1 ] ≤ e . tions of Lemma 4.2 and hence E[ZJ ] ≤ αβE[WJ ]. Simi- larly, ZI as given by (4.2) satisfies the same assumption with Proof. We prove the result by induction on k. For k = 1 we ˜ 2 ˜ α,˜ β instead of α, β, and so E[ZI ] ≤ α˜βE[WI ]. Recall that have we ordered the coefficients, so that α ≤ β; this also means λZ λZ −λ2αβW λ2αβW that α˜ ≤ α ≤ β ≤ β˜, since α+β =α ˜+β˜. Hence, α˜β˜ ≤ αβ. E[e 1 ] = E[e 1 1 · e 1 ] 2 2 We can conclude that ≤ E[eλZ1−λ αβW1 ]eλ αβa(I∆J) 2 2 2 2 λ αβa(I∆J) E[Z ] ≤ E[ZI ]+E[ZJ ] ≤ αβE[WI ]+αβE[WJ ] = αβE[W ]. ≤ e ,

To estimate E[ZW ], we use the Cauchy-Schwartz in- where the first inequality follows from W1 ≤ a(I∆J) equality: with probability 1, and the second inequality follows from Lemma 4.3. |E[ZW ]| ≤ (E[Z2]E[W 2])1/2 ≤ (αβE[W ]E[W 2])1/2 Consider now the case k > 1. We have Pk Pk p λ Zj λZ1 λ Zj ≤ 2αβ E[W ], E[e j=1 ] = E[e E[e j=2 | H1]], where H1 encodes a particular outcome of the first swap and hence operation. Let D1 ⊆ I∆J be the swap set used in the first iteration. Let I0 and J 0 be the updated (random) sets obtained from I and J after the first swap using the swap set 0 0 (4.4) D1. Since a(I ∆J ) = a(I∆J) − W1, we can apply the Pn−1 λ(a(Jn)−µ) λ(βn−1a(Jn−1)− i=1 α˜ia(Ii)) inductive hypothesis to obtain E[e ] ≤ EHn−1 [e 2 Pk 2 λ αnβn−1(a(Jn−1)+a(In)) λ Zj λ αβ(a(I∆J)−W1) × e ] E[e j=2 | H1] ≤ e , h 2 Pn−1 i (λ+λ αn)(βn−1a(Jn−1)− α˜ia(Ii)) = EH e i=1 and hence n−1 2 Pn−1 λ αn(βn−1a(In)+ i=1 α˜ia(Ii)) λ Pk Z λ2αβa(I∆J) λZ −λ2αβW × e E[e j=1 j ] ≤ e E[e 1 1 ]  0 Pn−1 α˜i  λ a(Jn−1)− a(Ii) λ2αβa(I∆J) i=1 βn−1 ≤ e , = EHn−1 e

λ2α β a(I )+Pn−1 α˜ a(I ) where the last inequality follows from Lemma 4.3. × e n( n−1 n i=1 i i )

So far, we only considered one merge operation. In the following we show how to bound the exponential moment 0 0 for the whole swap rounding procedure. Given is a fractional where we set λ = λ(1 + λαn)βn−1. Notice that |λ | ≤ Pn |λ|(1 + |λ|α )β ≤ |λ|(1 + α )(1 − α ) ≤ |λ| ≤ 1 . solution x˜ = i=1 α˜i1Ii with Ii ∈ F for i ∈ [n]. Let n n−1 n n 4 0 1 1 J1 = I1, and let Jk for k ∈ {1, . . . , n} be the set obtained by Hence λ ∈ [− 4 , 4 ]. merging Jk−1 and Ik. After k merge operations, the current Next, we will apply the inductive hypothesis to the Pn H J fractional solution is given by βk+11Jk+1 + i=k+2 α˜i1Ii expectation over n−1. The idea is to observe that n−1 Pk+1 can be seen as a random set that corresponds to applying the where βk+1 = αi. The set returned by the algorithm i=1 Pn−1 αi swap rounding procedure to the point z = 1I , is R = Jn. i=1 βn−1 i which is a convex combination of n − 1 sets, and thus allows 1 1 LEMMA 4.5. Let λ ∈ [− 4 , 4 ], let Jn be obtained from x˜ us to apply the inductive hypothesis. The only difference, by randomized swap rounding as described above, and let when I1,...,In−1 are merged into Jn−1 while applying the µ = a · x˜ = E[a(Jn)]. Then rounding procedure to x compared to rounding z, is that the

2 coefficients of the terms in the convex decomposition of z E[eλ(a(Jn)−µ)] ≤ e2λ µ. are scaled by a factor of 1 compared to those of x. The βn−1 ˜ Proof. We proceed by induction on the number of terms n in same is true for all coefficients α˜i and βi used in the merging Pn procedure. However, this does not make any difference in the the convex decomposition of x˜ = i=1 α˜i1Ii . Notice that the base step of the inductive proof is trivial, since if n = 1, rounding procedure, since the rounding probabilities depend only on relative ratios of the coefficients. Thus, we obtain by then a(J1) = a(I1) = µ and the left-hand side is 1. So assume n ≥ 2. We have induction

E[eλ(a(Jn)−µ)]   λ[βn−1a(Jn−1)−(µ−α˜na(In))]  0 Pn−1 α˜i  λ a(Jn−1)− a(Ii) = EHn−1 [e i=1 βn−1 EHn−1 e λ[a(Jn)−(βn−1a(Jn−1)+˜αna(In))] × EHn [e | Hn−1]] 02 Pn−1 α˜i 2λ a(Ii) Pn−1 i=1 βn−1 λ[βn−1a(Jn−1)− i=1 α˜ia(Ii)] ≤ e = EHn−1 [e 2 2 Pn−1 2λ (1+λαn) βn−1 α˜ia(Ii) λ[a(Jn)−(βn−1a(Jn−1)+˜αna(In))] = e i=1 . × EHn [e | Hn−1]], where Hk denotes the history of the first k merge operations in the rounding process. The inner expectation corresponds Combining the above equation with (4.4), we get to the change in eλ(a·x˜) due to the last merge operation, merging Jn−1 and In. By Lemma 4.4 we have

λ[a(Jn)−(βn−1a(Jn−1)+˜αna(In))] 2 n−1 EHn [e | Hn−1] λ α β a(I )+P α˜ a(I ) [eλ(a(Jn)−µ)] ≤ e n( n−1 n i=1 i i ) 2 E λ αnβn−1a(Jn−1∆In) ≤ e 2λ2(1+λα )2β Pn−1 α˜ a(I ) ×e n n−1 i=1 i i 2 λ α β (a(J )+a(I )) n−1 n n−1 n−1 n 2 λ2(α +2(1+λα )2β ) P α˜ a(I ) ≤ e , = eλ αnβn−1a(In) ×e n n n−1 i=1 i i . Taking logs to simplify the writing, we obtain 5 Applications (4.5) In this section, we show how our concentration bounds imply the results on multi-budgeted and multi-objective optimiza- h i λ(a(Jn)−µ) 2 log E e ≤ λ αnβn−1a(In) tion. In fact, we prove a more general result, which implies | {z } both Theorem 1.2 and Theorem 1.4. First, let us define the ≤2˜αna(In) Pn−1 following. =2µ−2 i=1 α˜ia(Ii) n−1 X Multi-objective/multi-budget matching. Given a graph + λ2(α + 2β + (4λα + 2λ2α2 )β ) α˜ a(I ) n n−1 n n n−1 i i G = (V,E), k linear functions (“demands”) f1, . . . , fk : | {z } i=1 E =αn+2(1−αn) 2 → R+, and ` linear functions (“budgets”) g1, . . . , g` : =2−α E n 2 → R+, is there a matching M satisfying fi(M) ≥ Vi for n−1 ! all i ∈ [k] and g (M) ≤ B for all i ∈ [`]? 2 2 X i i ≤ λ 2µ + αn(−1 + (4λ + 2λ αn)βn−1) α˜ia(Ii) i=1 Multi-objective/multi-budget matroid intersection. Given two matroids M = (N, I ) and M = (N, I ), k linear where we used α ≤ (1 + δ)2α˜ ≤ 2˜α (recall that δ = 1 1 2 2 n n n functions (“demands”) f , . . . , f : 2N → , and ` linear 1 ≤ 1 ). The lemma will finally be proven by showing that 1 k R+ p−1 3 functions (“budgets”) g , . . . , g : 2N → , is there an 2 1 1 1 ` R+ (4λ+2λ αn)βn−1 ≤ 1 for λ ∈ [− , ]. Since the left-hand 4 4 independent set I ∈ I1 ∩ I2 satisfying fi(I) ≥ Di for all side is a convex function of λ, it suffices to check the two i ∈ [k] and g (I) ≤ B for all i ∈ [`]? 1 1 i i endpoints. For λ = − 4 , the LHS is (−1 + 8 αn)βn−1 ≤ 0. 1 1 For λ = 4 , we get (1+ 8 αn)βn−1 ≤ (1+αn)(1−αn) ≤ 1. We show that these problems can be solved up to a small By (4.5), log E eλ(a(Jn)−µ) ≤ 2λ2µ. relative error in the objective functions.

Now we can finish the proof of Theorem 4.1. THEOREM 5.1. For any ε > 0 and any constant number Proof. [Proof of Theorem 4.1] of demands and budgets k + `, there is a polynomial-time randomized algorithm which for any feasible instance of 1 i) For λ ∈ [− 4 , 0], we have multi-objective/multi-budget matching or matroid intersec- tion finds with high probability a feasible solution S such Pr[a(R) ≤ (1 − ε)µ] ≤ Pr[eλ(a(R)−µ) ≥ e−λεµ] that λ(a(R)−µ) E[e ] • Each linear budget constraint is satisfied: gi(S) ≤ Bi. ≤ −λεµ e • Each linear demand is nearly satisfied: fi(S) ≥ (1 − (2λ2+λε)µ ≤ e , ε)Di. If such a solution is not found, the algorithm returns a where the second inequality follows from Markov’s certificate that the instance is not feasible with demands Di inequality and the third one follows from Lemma 4.5. and budgets Bi. Notice that Jn as defined in Lemma 4.5 equals R. Choosing λ = −ε/4 proves the claim. Implications. First, let us consider instances with one 1 ii) For λ ∈ [0, 4 ] we have, objective k = 1 and a constant number of budgets `. Via λ(a(R)−µ) λεµ the algorithm in Theorem 5.1, we can perform binary search Pr[a(R) ≥ (1 + ε)µ] ≤ Pr[e ≥ e ] on the objective function and estimate within a factor of E[eλ(a(R)−µ)] 1 − ε the maximum value D such that there is a solution ≤ 1 eλεµ of value f1(S) ≥ (1 − ε)D1 and satisfying gi(S) ≤ Bi for 2 ≤ e(2λ −λε)µ all budgets. This gives a (randomized) PTAS for the multi- budgeted versions of matching and matroid intersection, where again the second inequality follows Markov’s hereby proving Theorem 1.2. inequality and the third one from Lemma 4.5. Choosing Now, let us consider the case of k objective functions λ = ε/4 proves the second part of Theorem 4.1. and no budget constraints (` = 0). Here, we can solve the iii) For the third part of Theorem 4.1, we consider ε = following promise problem in polynomial time: is there a 1 feasible solution S satisfying fi(S) ≥ (1 − ε)Di for all t − 1 ≥ 1 and we fix λ = 4 . As above, we obtain i, or no solution satisfies fi(S) ≥ Di for all i. Using the 2 Pr[a(R) ≥ tµ] ≤ e(2λ −λ(t−1))µ = e(3−2t)µ/8. multi-objective optimization framework of Papadimitriou and Yannakakis (see Theorem 2 in [26]), this is sufficient Recall that we scaled the coefficients ai and hence µ by 1/p, to find in polynomial time a ε-approximate Pareto set with so the theorem follows. respect to the ` objectives. This proves Theorem 1.4. Proof. [Sketch of proof of Theorem 5.1.] Fix ε > 0 and an integer polytope of interest such as matroid intersection let γ = ε/2. We guess a constant (depending on k, `, 1/ε) or matching in our case. If m is part of the input, then number of elements so that for each remaining element j, the maximum independent set problem is a special case 4 the value in each fi is at most ε Di and also its size with (even with only the packing constraints); unless P = NP 4 1−ε respect to each budget is at most ε Bi. In the following, we there is no O(n )-factor approximation for any fixed 4 4 just assume that fi(j) ≤ ε Di and gi(j) ≤ ε Bi for all i and ε > 0. However, if the packing constraints are loose or if j ∈ N. they can be violated, our rounding procedure yields a good Let P be either the matching polytope or the matroid approximation. We can obtain the following results easily intersection polytope, corresponding to the instance. We add via our concentration bounds, and hence omit a formal proof. linear constraints to P , to obtain • A (1 − ε)-approximation if constraints are “loose”, that is, max A ≤ c·ε/ log m for some sufficiently small 0 i,j ij P = {x ∈ P : ∀i ∈ [k]; fi(x) ≥ Di, ∀j ∈ [`]; gj(x) ≤ Bi}. but fixed constant c. • A (1 − ε, O( 1 log m)) bicriteria approximation where Since P has a polynomial-time separation oracle, we also ε the constraints are violated by an O( 1 log m) addi- have a polynomial-time separation oracle for P 0; therefore, ε tive/multiplicative factor. via the ellipsoid method, we can determine in polynomial time whether P 0 is empty or not. If P 0 is empty, we have a certificate that the problem infeasible. Otherwise, we can References find a (fractional) solution y ∈ P 0. We apply randomized swap rounding to y, to obtain a [1] S. Arora, A. Frieze and H. Kaplan. A new rounding procedure random solution R. 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With every function gi, a budget Bi ∈ Z+ is associated. For [24] K. Mulmuley, U. V. Vazirani, and V. V. Vazirani. Matching convenience, we often consider the weights w and lengths gi is as easy as matrix inversion. Combinatorica, 7(1):101–104, E as vectors in Z+, in which case we use the boldface notation 1987. w and g . Let M ⊆ 2E be the set of all matchings in G, and [25] A. Panconesi and A. Srinivasan. Randomized distributed i let P be the matching polytope, i.e., the convex hull of M. via an extension of the Chernoff-Hoeffding M bounds. SIAM Journal on Computing 26:350-368, 1997. We are interested in the following multi-budgeted maximum [26] C. H. Papadimitriou and M. Yannakakis. On the approxima- matching problem. bility of trade-offs and optimal access of web sources. Proc. (A.1) max{w(I) | I ∈ M, g (I) ≤ B ∀i ∈ [`]} of IEEE FOCS, 86–92, 2000. i i [27] P. Raghavan and C. D. Thompson. Randomized rounding: In the following, we present a PTAS for problem (A.1). a technique for provably good algorithms and algorithmic More precisely, we present a multi-criteria PTAS for prob- proofs. Combinatorica 7(4):365–374, 1987. lem (A.1), i.e., an algorithm running in polynomial time for [28] R. Ravi and M. Goemans. The constrained minimum span- a constant ε ∈ (0, 1] that returns a matching of weight at ning tree problem. Proc. of SWAT, 66–75, 1996. [29] B. Saha and A. Srinivasan. A new approximation technique least (1 − ε)OPT and violates each budget by a factor of at for resource-allocation problems. Proc. of ICS, 342–357, most 1 + ε. Such an algorithm can then be transformed into 2010. a PTAS for problem (A.1) using standard techniques (see for [30] A. Schrijver. Combinatorial optimization - polyhedra and example [16]). efficiency. Springer, 2003. In a first step we formulate an adapted version of our [31] A. Srinivasan. Distributions on level-sets with applications to randomized procedure which is easier to derandomize. For approximation algorithms, Proc. of IEEE FOCS, 588–597, the derandomization step we use the following standard 2001. Chernoff bound for Poisson trials. A proof can be found 3 1 ki 1 in [17] . Pi ,...,Pi are constructed iteratively: Pi = {e1, . . . , eγ1 }

consists of the largest sequence of edges e1, . . . , eγ1 , such THEOREM A.1. Let X1,...,Xn be independent random that ` (P 1) ≤ T , i.e., γ = max{h ∈ [|D|] | Pn i i i 1 variables taking values in [0, 1], let X = Xi and let Ph 2 i=1 gi(ej) ≤ Ti}. Analogously, P consists of the largest δ ∈ [0, 1]. Then j=1 i sequence of edges e , . . . e such that ` (P 2) ≤ T and −µδ2/3 γ1+1 γ2 i i i i) for µ ≥ E[X], Pr[X ≥ (1 + δ)µ] ≤ e , so on. −µδ2/2 ii) for µ ≤ E[X], Pr[X ≤ (1 − δ)µ] ≤ e . Next, we show that the number ki of sets in Pi satisfies ki ≤ 3N/α2. Notice that by construction of Pi, we have A.1 Merging two matchings In this section we show how j j+1 gi(Pi ∪ Pi ) > Ti for j ∈ [ki − 1]. Using this fact, one a convex combination x = α11I1 + α21I2 of two matchings can easily observe that the average length of the sets in Pi is I1,I2 ∈ M can be transformed into a matching J ∈ M with at least Ti/3. Hence, a similar objective value and such that a constant number of linear functions do not increase too much. We use ki X kiTi the following notion of an r-almost matching, which was (A.2) g (D) = g (P j) ≥ . i i i 3 introduced in a slightly more general form in [16]. A set j=1 I ⊆ E is called an r-almost matching for some r ∈ Z+, if I can be transformed into a matching by removing at most r T T Furthermore, we have gi x = gi (α11I1 + α21I2 ) and elements of I. T max max gi(D) = gi (1I1 + 1I2 ), and hence by non-negativity of Let gi = maxe∈I1∆I2 gi(e) for i ∈ [`], w = T 6 gi x NTi max w(e) N = ln(2`) gi and since α2 ≤ α1, we get gi(D) ≤ = . e∈I1∆I2 , and ε2 . The following α2 α2 theorem is the backbone of the PTAS to be presented. Its Combining this result with (A.2) we obtain ki ≤ 3N/α2 proof shows how to define swap paths such that at the same as claimed. time we have concentration for the objective function and For every i ∈ [`], let Pi be a partition of D as de- budgets, and efficient derandomization through exhaustive scribed above. Similarly, let Pw be a partition of D con- search is possible. structed analogously to the other ones with respect to the weight function w instead of a length function. Each parti-

THEOREM A.2. Let x = α11I1 + α21I2 be a convex tion P1,..., P`, Pw represents a way of cutting the sequence T max combination of two matchings. If gi x ≥ Ngi for i ∈ [`] of edges (e1, . . . , e|D|) into subsequences. Let P be the par- T max and w x ≥ Nw , then a matching J can be obtained in tition corresponding to cutting the sequence (e1, . . . , e|D|) at 3(1+ε)(`+1)N time O(n ) such that I1 ∩ I2 ⊆ J ⊆ I1 ∪ I2, and all places where at least one of the partitions P1,..., P`, Pw T (i) gi(J) ≤ (1 + ε)gi x ∀i ∈ [`], cuts the sequence. Hence, P can be described as follows T max (ii) w(J) ≥ (1 − ε)w x − 6(1 + ε)(` + 1)Nwi . P = {P ∩ P ∩ · · · ∩ P | P ∈ P i ∈ [`],P ∈ P }. Proof. The theorem will be proven by presenting first a ran- w 1 ` i i for w w domized algorithm returning with high probability a 6(1 + ε)(` + 1)N-almost matching Je that can be transformed into Since every partition of P1,..., P`, Pw contains at most a matching J satisfying the claim of the theorem. The algo- 3N/q subsequences of (e1, . . . , e|D|), the partition P con- rithm will then be derandomized to get the result. Without tains at most 3N(` + 1)/q elements. loss of generality we assume α1 ≥ α2, and I1 ∩ I2 = ∅. We consider the following random process to create a Let D = I1∆I2 = I1 ∪ I2. We number the edges in merge Je of I1 and I2. We start with I1 and, independently P ∈ P P D = {e1, . . . , e|D|} such that edges belonging to the same for every set , we perform an edge-flip on the set cycle/path in D are numbered consecutively, and any two with probability α2. In the following we show that with high T edges ej, ej+1 for j ∈ [|D| − 1] are either adjacent in D probability, the random set Je satisfies i) gi(Je) ≤ gi x for or belong to different paths/cycles of D. This can easily be i ∈ [`], and ii) w(Je) ≥ (1 − ε)wT x. achieved by cutting each cycle of D at an arbitrary vertex Let i ∈ [`]. The length gi(Je) can be written as a sum of and appending the resulting set of paths one to the other by independent random variables gluing together the endpoints of the paths. T Consider a fixed i ∈ [`], and let Ti = gi x/N. We parti- X 1 ki (A.3) gi(Je) = XP , tion the edges of D into a collection Pi = {P ,...,P } i i P ∈P such that gi(P ) ≤ Ti ∀P ∈ P as follows. The sets with Pr[X = g (I ∩ P )] = α and Pr[X = g (I ∩ 3In [17], a proof is presented for Theorem A.1 for the case µ = E[X]. P i 1 1 P i 2 However, the same proof also shows the slightly more general statement P )] = α2. Notice that by construction of P, we have T given by Theorem A.1. XP ∈ [0, gi x/N]. Furthermore, E[gi(Je)] = α1gi(I1) + T α2gi(I2) = gi x. Hence, be used for edge-flips to obtain Je, we can guess these sets (A.4) in O(|P|3N(1+ε)(`+1)) time, and since trivially |P| ≤ n, " # n = |V | T gi(Je)N where , this computational complexity is bounded Pr[g (J) ≥ (1 + ε)g x] = Pr ≥ (1 + ε)N 3N(1+ε)(`+1) i e i T by O(n ). gi x By removing from Je all pending edges eγj , eγj+1 of the −Nε2/3 1 ≤ e = , sets P = {eγ , . . . , eγ } ∈ P that were used to perform 4`2 j j+1 edge-flips to obtain Je, a matching J is obtained. Notice that T max where the inequality follows by the Chernoff bound of w(J) ≥ (1−ε)w x−6(1+ε)(`+1)Nwi since w(Je) ≥ Theorem (A.1) point (i) with µ = N. Analogously, we (1−ε)wT x and at most 6(1+ε)(`+1)N edges of D∩Jehave can obtain the following bound for the weight function using to be removed from Je to obtain J. Furthermore, because all Theorem A.1 point (ii), i.e., T lengths are positive we have gi(J) ≤ gi(Je) ≤ (1 + ε)gi x.

T −Nε2/2 1 (A.5) Pr[wi(Je) ≤ (1 − ε)w x] ≤ e ≤ . 4`2 A.2 A PTAS for multi-criteria matching Let ε ∈ (0, 1] be the desired accuracy, and we set ε0 = ln(2)ε/2` and We now show that with high probability the set Je let N 0 = 6 ln(2`). In a first step of our algorithm, for is an 6(1 + ε)(` + 1)N-almost matching, i.e., it can be ε02 each i ∈ [`], we guess the N 0 longest edges E ⊆ E with transformed into a matching J by removing from J at most i e respect to g in an optimal solution. Furthermore, we guess 6(1+ε)(`+1)N edges. The number of edges that have to be i the 12(1+ 1 )`(`+1)N 0 heaviest edges E ⊆ E with respect removed from J to obtain a matching can easily be bounded ε0 w e to w. For a fixed ε, there is only a constant number of edges as follows. Let P0 ⊆ P be the paths that were used to obtain that have to be guessed, which can be done in polynomial J from I by flips, i.e., J = I ∆(∪ 0 P ). Clearly, J can e 1 e 1 P ∈P e time. be transformed into a matching by removing all edges at the ↑ Let E be the set of all edges e ∈ E \ Ei such that two ends of all paths in P0. Hence, Je is a 2|P0|-matching. i gi(e) > min{gi(f) | f ∈ Ei}. Since Ei should contain To show that Je is with high probability a 6(1 + ε)(` + 1) the longest edges of an optimal solution with respect to matching as claimed, we will show that with high probability g , no edges in E↑ will be added to the solution at a later 0 i i |P | ≤ 3(1 + ε)(` + 1). stage. Similarly we define E↑ = {e ∈ E \ E | w(e) > 0 w w Notice that |P | is the sum of independent 0-1 ran- ` min{w(f) | f ∈ E }}. Let F = E ∪ S E be the set dom variables Y for P ∈ P, with P [Y = 1] = α . w w i=1 i P P 2 of all guessed edges, and let F ↑ = E↑ ∪ S` E↑. Notice Furthermore, since the number of sets in P is bounded by w i=1 i F ∩ F ↑ 6= ∅ 3(` + 1)N/α , and each set in P is used to perform an edge- that the case corresponds to an inconsistent 2 guess. Hence, such a guess can be discarded. Problem (A.1) flip with probability α2, we have that the expected number of flips is bounded by 3(` + 1)N. Again, using the Chernoff can then be reduced accordingly, leading to the following bound presented in Theorem A.1 point (i), we obtain problem. (A.6) max{w(I) | I ∈ M,F ⊆ I,I ∩ F ↑ = ∅, X −3N(`+1)ε2/3 (A.7) Pr[ YP ≥ (1 + ε)3N(` + 1)] ≤ e gi(I) ≤ Bi ∀i ∈ [`]} P ∈P ∗ 2 1 Let x be an optimal vertex-solution of the following LP ≤ eNε /3 = . 4`2 relaxation of problem (A.7). (A.8) 1 0 T ↑ Hence with probability at least 1 − 4`2 , we have |P | ≤ max{w x | x ∈ PM, x(e) = 1 ∀e ∈ F, x(e) = 0 ∀e ∈ F , 3N(1 + ε)(` + 1). gT x ≤ B ∀i ∈ [`]} Using (A.4), (A.5) and (A.6) we can apply a union i i bound to obtain that the probability of Y satisfying simul- Notice that problem (A.8) corresponds to optimizing a linear taneously function on the face of the matching polytope PM defined T ↑ (i) gi(Je) ≤ (1 + ε)gi x ∀i ∈ [`], by x(e) = 1 ∀e ∈ F and x(e) = 0 ∀e ∈ F , with the ` ad- T T ∗ (ii) w(Je) ≥ (1 − ε)w x and ditional linear constraints gi x ≤ Bi for i ∈ [`]. Hence, x ∗ (iii) Je is obtained from I1 by at most 3N(1 + ε)(` + 1) lies on a face of PM of dimension at most `. Thus, x can be edge-flips, written as a convex combination of at most ` + 1 matchings. 1 1 Furthermore, such a decomposition can be found in polyno- is lower-bounded by 1 − (` 4`2 + 2 4`2 ) > 0. The algo- mial time using a constructive version of Caratheodory’s´ the- rithm to find a set Je satisfying the above conditions can x∗ = P`+1 α 1 I ∈ M, α ≥ easily be derandomized as follows. Since we know that orem. Hence, let j=1 j Ij , where j j P`+1 at most 3N(1 + ε)(` + 1) sets of the partition P have to 0 ∀j ∈ [` + 1] and j=1 αj = 1. The algorithm then itera- tively proceeds as our randomized scheme, where for merg- Since α 1 + α 1 ε = ε0 ing 1 I1 2 I2 we apply Theorem A.2 with to the   1 T ∗ max 1 0 max convex combination (α11I1 + α21I2 ). As in our ran- w x ≥ w(F ) ≥ |F |w = 12 1 + `(`+1)N w , α1+α2 0 Pj ε domized rounding scheme, let βj = s=1 αs for j ∈ [`+1]. 0 Furthermore, let J1 = I1, and for j ∈ {2, . . . , ` + 1} we de- we have 6(1 + ε0)`(` + 1)N 0wmax ≤ ε wT x∗ ≤ ε wT x∗, note by J the set obtained by merging β 1 + α 1 . 2 2 j j−1 Jj−1 j Ij and hence by (A.9) we obtain w(J) ≥ (1 − ε)wT x∗ as After ` steps, a matching J = J`+1 is obtained and returned desired. P`+1 by the algorithm. Let zj = βj1Jj + s=j+1 αs1Ij for j ∈ [` + 1] be the fractional matching after j − 1 merging steps. By Theorem A.2, the algorithm has polynomial running time for a fixed ε. Notice, that the conditions of Theo- rem A.2 are always satisfied because of the following. Ev- ery matching used throughout the algorithm contains the guessed edges F . Furthermore, the guessed edges satisfy 0 max 0 max gi(F ) ≥ N gi for i ∈ [`] and w(F ) ≥ N w , where max gi is the largest ith length among the non-fixed edges ↑ max E \ (F ∪ F ), and wi is the largest weight among the non-fixed edges E \ (F ∪ F ↑). It remains to show that J is a (1 ± ε) multi-criteria solution to problem (A.1). We first consider a length function gi for some fixed i ∈ [`], and show by induction on j that T 0 j−1 T ∗ gi zj ≤ (1 + ε ) gi x for j ∈ [` + 1]. For j = 1 the result trivially holds. For j ∈ {2, . . . , ` + 1} we have

`+1 T X gi zj = βjgi(Jj) + αsgi(Is) s=j+1 0 ≤ (1 + ε )(βj−1gi(Jj−1) + αjgi(Ij)) `+1 X + αsgi(Is) s=j+1

 `+1  0 X ≤ (1 + ε ) βj−1gi(Jj−1) + αsgi(Is)) s=j 0 j T ∗ ≤ (1 + ε ) gi x , where the first inequality follows by Theorem A.2 point (i) when applied to the merge operation that merged Jj−1 and Ij to obtain Jj, and the last inequality follows by the inductive hypothesis. Thus we obtain

 ln(2)ε` g (J) = gT z ≤ (1 + ε0)`gT x∗ = 1 + gT x∗ i i `+1 i 2` i ln(2) ε T ∗ ε T ∗  ε  T ∗ ≤ e 2 g x = 2 2 g x ≤ 1 + g x i i 2 i  ε  ≤ 1 + B . 2 i

Analogously, one can prove  ε  (A.9) w(J) ≥ 1 − wT x∗ −6(1+ε0)`(`+1)N 0wmax. 2