DOCSLIB.ORG

Efficient Theoretic and Practical Algorithms for Linear Matroid

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Efficient Theoretic and Practical Algorithms for Linear Matroid Journal of Computer and System Sciences 1273 journal of computer and system sciences 53, 129147 (1996) article no. 0054 CORE Efficient Theoretic and Practical AlgorithmsMetadata, for citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Linear Matroid Intersection Problems Harold N. Gabow* and Ying Xu Department of Computer Science, University of Colorado at Boulder, Boulder, Colorado 80309 Received February 1, 1989 This paper presents efficient algorithms for a number Efficient algorithms for the matroid intersection problem, both car- of matroid intersection problems. The main algorithm is dinality and weighted versions, are presented. The algorithm for for weighted matroid intersection. It is based on cost weighted intersection works by scaling the weights. The cardinality scaling, generalizing recent work in matching [GaT87], algorithm is a special case, but takes advantage of greater structure. Efficiency of the algorithms is illustrated by several implementations on and network flow [GoT]. A special case is of this problem linear matroids. Consider a linear matroid with m elements and rank n. is cardinality matroid intersection, and several improve- Assume all element weights are integers of magnitude at most N. Our ments are given for this case. The efficiency of our fastest algorithms use time O(mn1.77 log(nN)) and O(mn1.62) for intersection algorithms is illustrated by implementa- weighted and unweighted intersection, respectively; this improves the tions on linear matroids. This is the broadest class of previous best bounds, O(mn2.4) and O(mn2 log n), respectively. Corresponding improvements are given for several applications of matroids with efficient representations. (Implementations of matroid intersection to numerical computation and dynamic systems. our general matroid algorithm on other matroids are given ] 1996 Academic Press, Inc. in [GX].) To state the specific results, assume that a linear matroid is given consisting of m elements with dimension n. For 1. INTRODUCTION weighted intersection assume that all weights are integers at most N in magnitude. We distinguish two types of The matroid intersection problem is interesting from a algorithms: those that use only naive procedures for theoretic point of view and because of its rich set of applica- manipulating matrices (e.g., the simple O(n3) algorithm for tions. Concerning theory, matroid intersection generalizes matrix multiplication) and those that take advantage bipartite graph matching; powerful duality results (e.g., the of sophisticated procedures (e.g., the O(n2.4) multiplica- KonigEgervary minimax theorem) generalize to matroids tion algorithm of [CW]). Algorithms of the first type do (e.g., Edmonds' matroid intersection theorem) [L]. Concer- not have as good an asymptotic time bound. However, they ning applications, matroid intersectioncanbeusedto analyze can be more practical than algorithms of the second type, continuous systems arising in electrical networks, elastic which have large constants hidden in the asymptotic structures, and chemical processing plants. The analysis is bounds. attractive because problems that might naturally be solved by The main previous contribution for linear matroids is the numerical computation, subject to rounding error and inac- cardinality intersection algorithm of Cunningham [Cu]. It curacies in the data, are replaced by discrete calculations, finds a maximum cardinality matching on a linear matroid which are more efficient and relatively insensitive to these in time O(mn2 log n), using naive matrix manipulation. Our numerical difficulties. Examples of such applications of naive algorithm for cardinality intersection is essentially the matroid intersection include solvability and dynamic degrees same as Cunningham's. Using fast matrix multiplication our of freedom of an electrical network [I] and their generaliza- algorithm runs in time O(mn1.62). tiontoarbitrary dynamicsystemsgovernedby lineardifferen- For the weighted problem, our naive algorithm finds a max- tial equations [M87a]. (For electrical networks the matroids imum weight perfect matching in time O(mn2 log n log(nN)). tend to be graphic; for general systems they are linear. This bound is just a factor log(nN) more than Cunningham's Applications are discussed further in Section 5.) bound for the simpler problem without weights. Using fact 1.77 * Research supported in part by NSF Grant Nos. MCS-8302648, matrix multiplication our algorithm runs in time O(mn DCR-851191. log(nN)). The previous best bound for this problem is 129 0022-0000Â96 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. File: 571J 127301 . By:BV . Date:29:08:96 . Time:11:51 LOP8M. V8.0. Page 01:01 Codes: 6104 Signs: 4603 . Length: 60 pic 11 pts, 257 mm 130 GABOW AND XU achieved by Frank's algorithm [F] (and a number of similar because of repeated matrix multiplications and inversions. algorithms). This gives time O(mn3) using naive multiplica- Hence the bounds either assume we are working over a tion and O(mn2.4) using fast multiplication. Frank's algo- finite field, or the bounds count the number of arithmetic rithm has the theoretic advantage that the time bound is operations. strongly polynomial. (Advantages and disadvantages of I denotes the identity matrix. The dimension of this strong polynomial algorithms compared to scaling algo- matrix will be clear from context. rithms are discussed in [G, GoT].) The reader is assumed familiar with the basic notions of It is interesting to compare our results with recent scaling matroids (see, e.g., [A, L, W]). Let M be a matroid over a algorithms for minimum cost network flow [GoT], the set of elements E. The span of a set of elements S is denoted assignment problem [GaT87], minimum weight matching sp(S). Suppose e is an element and B is a base. The notation [GaT88], and others. These papers present scaling algo- C(e, B) denotes the fundamental circuit of e in B if e  B,or rithms with a time bound for the weighted problem that is the fundamental cocircuit of e for B if e # B. (It will be clear essentially within a factor log(nN) of the best-known bound from context whether or not e # B.) On occasion we write for the (simpler) cardinality problem. As mentioned, this is C(e, B, M) if the matroid M is not clear. If S and T are sets also the case for our weighted matroid intersection algo- of elements then SÂT denotes the matroid restricted to S rithm using naive multiplication. However, using fast multi- with T contracted.IfMand N are matroids, their direct sum plication there is a much larger gap between our bounds for is denoted M+N. weighted and cardinality problems, about n0.15. We leave Let the element set E be partitioned into blocks of size open the problem of closing this gap, or making other two called pairs. Thus each element e has a mate, denoted eÄ , improvements to our algorithms. such that [e, eÄ ] is a pair. (We usually drop brackets and The rest of this section gives some terminology and denote a pair as e, eÄ .) A matching M is an independent set notation; then it sketches Frank's algorithm for weighted of pairs, i.e., e # M if and only if eÄ # M .Amaximum car- matroid intersection [F]. Our algorithm uses several dinality matching has the greatest number of pairs possible. ideas of this algorithm. Section 2 presents our algorithm A matching is perfect if it is a base of M.Ifw:EÄRis a for weighted matroid intersection, on general matroids. function that assigns a real-valued weight to each element, Section 3 gives the algorithm for cardinality matching, the weight of a set S of elements is defined as w(S)= plus other extensions of the weighted algorithm. Section 4 [w(e)|e#S].Amaximum perfect matching is a perfect gives implementations of the matching algorithms matching that has the largest weight possible. Maximum on linear matroids. Section 5 sketches the extensions weight maximum cardinality matching is defined similarly. A to related problems such as independent assignment maximum weight matching is a matching with the largest and linking (definitions of these problems are in weight possible. Section 5). It concludes by giving a number of applica- The above problems generalize the problem of matching tions to problems in numerical computation and dynamic on graphs. This paper is concerned with the ``bipartite'' systems. versions of these matching problems. Specifically, through- If S is a set and e an element, S+e denotes S _ [e] and out this paper the M is a direct sum of matroids, M=M0+ S&e denotes S&[e]. The symmetric difference of sets S M1 , where Mi is a matroid on elements Ei , i=0, 1 (so and T is denoted SÄT. We use interval notation for sets of E=E0 _E1) and every pair contains one element from each integers: for integers i and j,[i..j]=[k | k is an integer, matroid. The cardinality matroid intersection problem is to ik j],[i..j)=[k | k is an integer, ik< j], etc. The log find a maximum cardinality matching on such a matroid M. function denotes logarithm base two. We use the following The weighted matroid intersection problem is to find convention to sum the values of a function: If f is a real- either a maximum perfect matching, maximum weight max- valued function defined on elements and S is a set of imum cardinality matching, or maximum weight matching, elements, then f(S)=[f(s)|s#S]. on M. The algorithms for linear matroids do matrix multiplica- The following notation is useful in the context of intersec- tion. We could state the time bounds in terms of M(n), the tion problems. For any set of elements S and i # [0, 1], Si time to multiply two n_n matrices.
Load more

Recommended publications
  • Parity Systems and the Delta-Matroid Intersection Problem
    Parity Systems and the Delta-Matroid Intersection Problem Andr´eBouchet ∗ and Bill Jackson † Submitted: February 16, 1998; Accepted: September 3, 1999. Abstract We consider the problem of determining when two delta-matroids on the same ground-set have a common base. Our approach is to adapt the theory of matchings in 2-polymatroids developed by Lov´asz to a new abstract system, which we call a parity system. Examples of parity systems may be obtained by combining either, two delta- matroids, or two orthogonal 2-polymatroids, on the same ground-sets. We show that many of the results of Lov´aszconcerning ‘double flowers’ and ‘projections’ carry over to parity systems. 1 Introduction: the delta-matroid intersec- tion problem A delta-matroid is a pair (V, ) with a finite set V and a nonempty collection of subsets of V , called theBfeasible sets or bases, satisfying the following axiom:B ∗D´epartement d’informatique, Universit´edu Maine, 72017 Le Mans Cedex, France. [email protected] †Department of Mathematical and Computing Sciences, Goldsmiths’ College, London SE14 6NW, England. [email protected] 1 the electronic journal of combinatorics 7 (2000), #R14 2 1.1 For B1 and B2 in and v1 in B1∆B2, there is v2 in B1∆B2 such that B B1∆ v1, v2 belongs to . { } B Here P ∆Q = (P Q) (Q P ) is the symmetric difference of two subsets P and Q of V . If X\ is a∪ subset\ of V and if we set ∆X = B∆X : B , then we note that (V, ∆X) is a new delta-matroid.B The{ transformation∈ B} (V, ) (V, ∆X) is calledB a twisting.
    [Show full text]
  • CS675: Convex and Combinatorial Optimization Spring 2018 Introduction to Matroid Theory
    CS675: Convex and Combinatorial Optimization Spring 2018 Introduction to Matroid Theory Instructor: Shaddin Dughmi Set system: Pair (X ; I) where X is a finite ground set and I ⊆ 2X are the feasible sets Objective: often “linear”, referred to as modular Analogues of concave and convex: submodular and supermodular (in no particular order!) Today, we will look only at optimizing modular objectives over an extremely prolific family of set systems Related, directly or indirectly, to a large fraction of optimization problems in P Also pops up in submodular/supermodular optimization problems Optimization over Sets Most combinatorial optimization problems can be thought of as choosing the best set from a family of allowable sets Shortest paths Max-weight matching Independent set ... 1/30 Objective: often “linear”, referred to as modular Analogues of concave and convex: submodular and supermodular (in no particular order!) Today, we will look only at optimizing modular objectives over an extremely prolific family of set systems Related, directly or indirectly, to a large fraction of optimization problems in P Also pops up in submodular/supermodular optimization problems Optimization over Sets Most combinatorial optimization problems can be thought of as choosing the best set from a family of allowable sets Shortest paths Max-weight matching Independent set ... Set system: Pair (X ; I) where X is a finite ground set and I ⊆ 2X are the feasible sets 1/30 Analogues of concave and convex: submodular and supermodular (in no particular order!) Today, we will look only at optimizing modular objectives over an extremely prolific family of set systems Related, directly or indirectly, to a large fraction of optimization problems in P Also pops up in submodular/supermodular optimization problems Optimization over Sets Most combinatorial optimization problems can be thought of as choosing the best set from a family of allowable sets Shortest paths Max-weight matching Independent set ..
    [Show full text]
  • Matroid Intersection
    EECS 495: Combinatorial Optimization Lecture 8 Matroid Intersection Reading: Schrijver, Chapter 41 • colorful spanning forests: for graph G with edges partitioned into color classes fE1;:::;Ekg, colorful spanning forest is Matroid Intersection a forest with edges of different colors. Define M1;M2 on ground set E with: Problem: { I1 = fF ⊂ E : F is acyclicg { I = fF ⊂ E : 8i; jF \ E j ≤ 1g • Given: matroids M1 = (S; I1) and M2 = 2 i (S; I ) on same ground set S 2 Then • Find: max weight (cardinality) common { these are matroids (graphic, parti- independent set J ⊆ I \I 1 2 tion) { common independent sets = color- Applications ful spanning forests Generalizes: Note: In second two examples, (S; I1 \I2) not a matroid, so more general than matroid • max weight independent set of M (take optimization. M = M1 = M2) Claim: Matroid intersection of three ma- troids NP-hard. • matching in bipartite graphs Proof: Reduction from directed Hamilto- For bipartite graph G = ((V1;V2);E), let nian path: given digraph D = (V; E) and Mi = (E; Ii) where vertices s; t, is there a path from s to t that goes through each vertex exactly once. Ii = fJ : 8v 2 Vi; v incident to ≤ one e 2 Jg • M1 graphic matroid of underlying undi- Then rected graph { these are matroids (partition ma- • M2 partition matroid in which F ⊆ E troids) indep if each v has at most one incoming edge in F , except s which has none { common independent sets = match- ings of G • M3 partition :::, except t which has none 1 Intersection is set of vertex-disjoint directed = r1(E) +r2(E)− min(r1(F ) +r2(E nF ) paths with one starting at s and one ending F ⊆E at t, so Hamiltonian path iff max cardinality which is number of vertices minus min intersection has size n − 1.
    [Show full text]
  • Packing of Arborescences with Matroid Constraints Via Matroid Intersection
    Mathematical Programming (2020) 181:85–117 https://doi.org/10.1007/s10107-019-01377-0 FULL LENGTH PAPER Series A Packing of arborescences with matroid constraints via matroid intersection Csaba Király1 · Zoltán Szigeti2 · Shin-ichi Tanigawa3 Received: 29 May 2018 / Accepted: 8 February 2019 / Published online: 2 April 2019 © The Author(s) 2019 Abstract Edmonds’ arborescence packing theorem characterizes directed graphs that have arc- disjoint spanning arborescences in terms of connectivity. Later he also observed a characterization in terms of matroid intersection. Since these fundamental results, intensive research has been done for understanding and extending these results. In this paper we shall extend the second characterization to the setting of reachability-based packing of arborescences. The reachability-based packing problem was introduced by Cs. Király as a common generalization of two different extensions of the spanning arborescence packing problem, one is due to Kamiyama, Katoh, and Takizawa, and the other is due to Durand de Gevigney, Nguyen, and Szigeti. Our new characterization of the arc sets of reachability-based packing in terms of matroid intersection gives an efficient algorithm for the minimum weight reachability-based packing problem, and it also enables us to unify further arborescence packing theorems and Edmonds’ matroid intersection theorem. For the proof, we also show how a new class of matroids can be defined by extending an earlier construction of matroids from intersecting submodular functions to bi-set functions based on an idea of Frank. Mathematics Subject Classification 05C05 · 05C20 · 05C70 · 05B35 · 05C22 B Csaba Király [email protected] Zoltán Szigeti [email protected] Shin-ichi Tanigawa [email protected] 1 Department of Operations Research, ELTE Eötvös Loránd University and the MTA-ELTE Egerváry Research Group on Combinatorial Optimization, Budapest Pázmány Péter sétány 1/C, 1117, Hungary 2 Univ.
    [Show full text]
  • Distributed Throughput Maximization in Wireless Mesh Networks Via Pre-Partitioning
    1 Distributed Throughput Maximization in Wireless Mesh Networks via Pre-Partitioning Andrew Brzezinski, Gil Zussman Senior Member, IEEE, and Eytan Modiano Senior Member, IEEE Abstract— This paper considers the interaction between chan- and scheduling but unlike most previous works, we do not nel assignment and distributed scheduling in multi-channel multi- assume that the traffic statistics are known. Alternatively, radio Wireless Mesh Networks (WMNs). Recently, a number we assume a stochastic arrival process and present a novel of distributed scheduling algorithms for wireless networks have emerged. Due to their distributed operation, these algorithms partitioning approach that enables throughput maximization can achieve only a fraction of the maximum possible throughput. in each partition by distributed scheduling algorithms. As an alternative to increasing the throughput fraction by Joint scheduling and routing in a slotted multihop wireless designing new algorithms, we present a novel approach that network with a stochastic packet arrival process was con- takes advantage of the inherent multi-radio capability of WMNs. sidered in the seminal paper by Tassiulas and Ephremides We show that this capability can enable partitioning of the network into subnetworks in which simple distributed scheduling [23]. In that paper they presented the first centralized policy algorithms can achieve 100% throughput. The partitioning is that is guaranteed to stabilize the network (i.e. provide 100% based on the notion of Local Pooling. Using this notion, we char- throughput) whenever the arrival rates are within the stability acterize topologies in which 100% throughput can be achieved region. The results of [23] have been extended to various distributedly. These topologies are used in order to develop a settings of wireless networks and input-queued switches (e.g.
    [Show full text]
  • Approximation by Lexicographically Maximal Solutions in Matching And
    Approximation by Lexicographically Maximal Solutions in Matching and Matroid Intersection Problems Krist´of B´erczia Tam´as Kir´alya Yutaro Yamaguchib Yu Yokoic Abstract We study how good a lexicographically maximal solution is in the weighted matching and matroid intersection problems. A solution is lexicographically maximal if it takes as many heaviest elements as possible, and subject to this, it takes as many second heaviest elements as possible, and so on. If the distinct weight values are sufficiently dispersed, e.g., the minimum ratio of two distinct weight values is at least the ground set size, then the lexicographical maximality and the usual weighted optimality are equivalent. We show that the threshold of the ratio for this equivalence to hold is exactly 2. Furthermore, we prove that if the ratio is less than 2, say α, then a lexicographically maximal solution achieves (α/2)-approximation, and this bound is tight. Keywords: Matching, Matroid intersection, Lexicographical maximality, Approximation ratio arXiv:2107.09897v1 [math.CO] 21 Jul 2021 aMTA-ELTE Momentum Matroid Optimization Research Group and MTA-ELTE Egerv´ary Research Group, Department of Operations Research, E¨otv¨os Lor´and University, Budapest, Hungary. Email: {kristof.berczi,tamas.kiraly}@ttk.elte.hu b Department of Informatics, Faculty of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan. Email: yutaro [email protected] cPrinciples of Informatics Research Division, National Institute of Informatics, Tokyo, Japan. Email: [email protected] 1 Introduction Matching in bipartite graphs is one of the fundamental topics in combinatorics and optimization. Due to its diverse applications, various optimality criteria of matchings have been proposed based on the number of edges, the total weight of edges, etc.
    [Show full text]
  • The Intersection of a Matroid and a Simplicial Complex 1
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 11, November 2006, Pages 4895–4917 S 0002-9947(06)03833-5 Article electronically published on June 19, 2006 THE INTERSECTION OF A MATROID AND A SIMPLICIAL COMPLEX RON AHARONI AND ELI BERGER Abstract. A classical theorem of Edmonds provides a min-max formula re- lating the maximal size of a set in the intersection of two matroids to a “cov- ering” parameter. We generalize this theorem, replacing one of the matroids by a general simplicial complex. One application is a solution of the case r = 3 of a matroidal version of Ryser’s conjecture. Another is an upper bound on the minimal number of sets belonging to the intersection of two matroids, needed to cover their common ground set. This, in turn, is used to derive a weakened version of a conjecture of Rota. Bounds are also found on the dual parameter—the maximal number of disjoint sets, all spanning in each of two given matroids. We study in detail the case in which the complex is the complex of independent sets of a graph, and prove generalizations of known results on “independent systems of representatives” (which are the special case in which the matroid is a partition matroid). In particular, we define a no- tion of k-matroidal colorability of a graph, and prove a fractional version of a conjecture, that every graph G is 2∆(G)-matroidally colorable. The methods used are mostly topological. 1. Introduction The point of departure of this paper is a notion which has been recently developed and studied, that of an independent system of representatives (ISR), which is a generalization of the notion of a system of distinct representatives (SDR).
    [Show full text]
  • On the Graphic Matroid Parity Problem
    On the graphic matroid parity problem Zolt´an Szigeti∗ Abstract A relatively simple proof is presented for the min-max theorem of Lov´asz on the graphic matroid parity problem. 1 Introduction The graph matching problem and the matroid intersection problem are two well-solved problems in Combi- natorial Theory in the sense of min-max theorems [2], [3] and polynomial algorithms [4], [3] for finding an optimal solution. The matroid parity problem, a common generalization of them, turned out to be much more difficult. For the general problem there does not exist polynomial algorithm [6], [8]. Moreover, it contains NP-complete problems. On the other hand, for linear matroids Lov´asz provided a min-max formula in [7] and a polynomial algorithm in [8]. There are several earlier results which can be derived from Lov´asz’ theorem, e.g. Tutte’s result on f-factors [15], a result of Mader on openly disjoint A-paths [11] (see [9]), a result of Nebesky concerning maximum genus of graphs [12] (see [5]), and the problem of Lov´asz on cacti [9]. This latter one is a special case of the graphic matroid parity problem. Our aim is to provide a simple proof for the min-max formula on this problem, i. e. on the graphic matroid parity problem. In an earlier paper [14] of the present author the special case of cacti was considered. We remark that we shall apply the matroid intersection theorem of Edmonds [4]. We refer the reader to [13] for basic concepts on matroids. For a given graph G, the cycle matroid G is defined on the edge set of G in such a way that the independent sets are exactly the edge sets of the forests of G.
    [Show full text]
  • Exact and Approximation Algorithms for Weighted Matroid Intersection
    Noname manuscript No. (will be inserted by the editor) Exact and Approximation Algorithms for Weighted Matroid Intersection Chien-Chung Huang · Naonori Kakimura · Naoyuki Kamiyama Received: date / Accepted: date Abstract In this paper, we propose new exact and approximation algorithms for the weighted matroid intersection problem. Our exact algorithm is faster than previous algorithms when the largest weight is relatively small. Our approximation algorithm delivers a (1 − ϵ)-approximate solution with a running time significantly faster than most known exact algorithms. The core of our algorithms is a decomposition technique: we decompose an in- stance of the weighted matroid intersection problem into a set of instances of the unweighted matroid intersection problem. The computational advantage of this ap- proach is that we can make use of fast unweighted matroid intersection algorithms as a black box for designing algorithms. Precisely speaking, we prove that we can solve the weighted matroid intersection problem via solving W instances of the unweighted matroid intersection problem, where W is the largest given weight. Furthermore, we can find a (1 − ϵ)-approximate solution via solving O(ϵ−1 log r) instances of the un- weighted matroid intersection problem, where r is the smallest rank of the given two matroids. Our algorithms are simple and flexible: they can be adapted to special cases of the weighted matroid intersection problem, using unweighted matroid intersection algorithms. In this paper, we will show the following results. A preliminary version of this paper appears in the proceeding of ACM-SIAM Symposium on Discrete Algorithms (SODA), 2016. Chien-Chung Huang CNRS, Ecole´ Normale Superieure,´ Paris, France.
    [Show full text]
  • Matroid Optimization and Algorithms Robert E. Bixby and William H
    Matroid Optimization and Algorithms Robert E. Bixby and William H. Cunningham June, 1990 TR90-15 MATROID OPTIMIZATION AND ALGORITHMS by Robert E. Bixby Rice University and William H. Cunningham Carleton University Contents 1. Introduction 2. Matroid Optimization 3. Applications of Matroid Intersection 4. Submodular Functions and Polymatroids 5. Submodular Flows and other General Models 6. Matroid Connectivity Algorithms 7. Recognition of Representability 8. Matroid Flows and Linear Programming 1 1. INTRODUCTION This chapter considers matroid theory from a constructive and algorithmic viewpoint. A substantial part of the developments in this direction have been motivated by opti­ mization. Matroid theory has led to a unification of fundamental ideas of combinatorial optimization as well as to the solution of significant open problems in the subject. In addition to its influence on this larger subject, matroid optimization is itself a beautiful part of matroid theory. The most basic optimizational property of matroids is that for any subset every max­ imal independent set contained in it is maximum. Alternatively, a trivial algorithm max­ imizes any {O, 1 }-valued weight function over the independent sets. Most of matroid op­ timization consists of attempts to solve successive generalizations of this problem. In one direction it is generalized to the problem of finding a largest common independent set of two matroids: the matroid intersection problem. This problem includes the matching problem for bipartite graphs, and several other combinatorial problems. In Edmonds' solu­ tion of it and the equivalent matroid partition problem, he introduced the notions of good characterization (intimately related to the NP class of problems) and matroid (oracle) algorithm.
    [Show full text]
  • Faster Matroid Intersection
    2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) Faster Matroid Intersection Deeparnab Chakrabarty Yin Tat Lee Aaron Sidford Dartmouth College University of Washington Stanford University [email protected] Microsoft Research [email protected] [email protected] Sahil Singla Sam Chiu-wai Wong Princeton University Microsoft Research Institute for Advanced Study [email protected] [email protected] Abstract—In this paper we consider the classic matroid I. Matroids are fundamental objects in combinatorics, and intersection problem: given two matroids M1 =(V,I1) and the abstract definition above generalizes a wide range of M V,I V 2 =( 2) defined over a common ground set , compute concepts ranging from acyclic graphs to linearly independent a set S ∈I1 ∩I2 of largest possible cardinality, denoted by r. We consider this problem both in the setting where each matrices. M V,I M V,I Mi is accessed through an independence oracle, i.e. a routine Given two matroids 1 =( 1) and 2 =( 2) which returns whether or not a set S ∈Ii in Tind time, and over the same ground set, the matroid intersection problem is Mi the setting where each is accessed through a rank oracle, to find a set I ∈I1 ∩I2 with the largest cardinality |I|. This i.e. a routine which returns the size of the largest independent problem generalizes many important combinatorial opti- subset of S in Mi in Trank time. mization problems such as bipartite matching, arborescences In each setting we provide faster exact and approximate in digraphs, and packing spanning trees. Unsurprisingly, algorithms.
    [Show full text]
  • CS675: Convex and Combinatorial Optimization Fall 2019 Introduction to Matroid Theory
    CS675: Convex and Combinatorial Optimization Fall 2019 Introduction to Matroid Theory Instructor: Shaddin Dughmi Set system: Pair (X ; I) where X is a finite ground set and I ⊆ 2X are the feasible sets Objective: often “linear”, referred to as modular Analogues of concave and convex: submodular and supermodular (in no particular order!) Today, we will look only at optimizing modular objectives over an extremely prolific family of set systems Related, directly or indirectly, to a large fraction of optimization problems in P Also pops up in submodular/supermodular optimization problems Optimization over Sets Most combinatorial optimization problems can be thought of as choosing the best set from a family of allowable sets Shortest paths Max-weight matching Independent set ... 1/30 Objective: often “linear”, referred to as modular Analogues of concave and convex: submodular and supermodular (in no particular order!) Today, we will look only at optimizing modular objectives over an extremely prolific family of set systems Related, directly or indirectly, to a large fraction of optimization problems in P Also pops up in submodular/supermodular optimization problems Optimization over Sets Most combinatorial optimization problems can be thought of as choosing the best set from a family of allowable sets Shortest paths Max-weight matching Independent set ... Set system: Pair (X ; I) where X is a finite ground set and I ⊆ 2X are the feasible sets 1/30 Analogues of concave and convex: submodular and supermodular (in no particular order!) Today, we will look only at optimizing modular objectives over an extremely prolific family of set systems Related, directly or indirectly, to a large fraction of optimization problems in P Also pops up in submodular/supermodular optimization problems Optimization over Sets Most combinatorial optimization problems can be thought of as choosing the best set from a family of allowable sets Shortest paths Max-weight matching Independent set ..
    [Show full text]