Noname manuscript No. (will be inserted by the editor) Distributed Algorithms for Covering, Packing and Maximum Weighted Matching Christos Koufogiannakis · Neal E. Young Received: date / Accepted: date Abstract This paper gives poly-logarithmic-round, dis- 1 Background and results tributed δ-approximation algorithms for covering prob- lems with submodular cost and monotone covering con- Many distributed systems are composed of components straints (Submodular-cost Covering). The approx- that can only communicate locally, yet need to achieve a imation ratio δ is the maximum number of variables in global (system-wide) goal involving many components. any constraint. Special cases include Covering Mixed The general possibilities and limitations of such sys- Integer Linear Programs (CMIP), and Weighted tems are widely studied [42,49,54,38,39,12]. It is of Vertex Cover (with δ = 2). specific interest to see which fundamental combinato- Via duality, the paper also gives poly-logarithmic- rial optimization problems admit efficient distributed round, distributed δ-approximation algorithms for Frac- algorithms that achieve approximation guarantees that tional Packing linear programs (where δ is the max- are as good as those of the best centralized algorithms. imum number of constraints in which any variable oc- Research in this spirit includes works on Set Cover curs), and for Max Weighted c-Matching in hy- (Dominating Set) [26,40,41], capacitated domi- pergraphs (where δ is the maximum size of any of the nating set [37], Capacitated Vertex Cover [16, hyperedges; for graphs δ = 2). 17], and many other problems. This paper presents dis- The paper also gives parallel (RNC) 2-approximation tributed approximation algorithms for some fundamen- algorithms for CMIP with two variables per constraint tal covering and packing problems. and Weighted Vertex Cover. The algorithms use the standard synchronous com- The algorithms are randomized. All of the approx- munication model: in each round, nodes can exchange imation ratios exactly match those of comparable cen- a constant number of messages with neighbors and per- tralized algorithms.1 form local computation [54]. There is no restriction on message size or local computation. The algorithms are Keywords Approximation algorithms · Integer linear efficient | they finish in a number of rounds that is programming · Packing and covering · Vertex cover · poly-logarithmic in the network size [42]. Matching C. Koufogiannakis 1.1 Covering Problems Department of Computer Science and Engineering University of California, Riverside Consider optimization problems of the following form: given a non-decreasing, continuous, and submodular2 N. E. Young cost function c : IRn ! IR , and a set of constraints C Department of Computer Science and Engineering + + 3 University of California, Riverside where each constraint S 2 C is closed upwards , 1 n Preliminary versions appeared in [34,35]. Work by the find x 2 IR+ minimizing c(x) s.t. (8S 2 C) x 2 S. first author was partially supported by the Greek State Scholarship Foundation (IKY). Work by the second author 2 Formally, c(x) + c(y) ≥ c(x ^ y) + c(x _ y) where ^ and _ was partially supported by NSF awards CNS-0626912, CCF- are component-wise minimum and maximum, respectively. 0729071. 3 If x 2 S and y ≥ x, then y 2 S. 2 Christos Koufogiannakis, Neal E. Young Submodular-cost Covering includes all problems vertex cover) can be computed in an expected O(log n) of this form [36]. The (approximation) parameter δ is rounds via the algorithm of Israeli and Itai [25]. the maximum number of elements of x on which any Maximal Matching is also in NC [9,47,30] and in constraint x 2 S depends. RNC | parallel poly-log time with polynomially many n In the above formulation, x ranges over IR+, but, be- randomized processors [25] | hence so is 2-approximat- cause the constraints can be non-convex, one can handle ing Unweighted Vertex Cover. arbitrarily restricted variable domains by incorporating For Weighted Vertex Cover, previous to this them into the constraints [36]. For example, Weighted work, no efficient distributed 2-approximation algorithm P Vertex Cover is equivalent to minimize v cvxv sub- was known. Similarly, no parallel NC or RNC 2-ap- n ject to x 2 IR+ and proximation algorithm was known. in 1994, Khuller, Vishkin and Young gave 2(1 + ")-approximation algo- xu ≥ 1 or xw ≥ 1 (8(u; w) 2 E): rithms: a distributed algorithm taking O(log n log 1=") (Given any 2-approximate solution x to this formula- rounds and a parallel algorithm, in NC for any fixed tion | which allows x 2 IR | rounding each x down u + u " [31].4 Given integer vertex weights, their result gives to its floor gives a 2-approximate integer solution.) distributed 2-approximation in O(log n log nC^)) rounds, Submodular-cost Covering includes the follow- where C^ is the average vertex weight. The required ing problems as special cases: number of rounds is reduced to (expected) O(log nC^) CIP, covering integer programs with variable upper by Grandoni, K¨onemannand Panconesi [18,15]. m×n m n bounds: given A 2 IR+ , w 2 IR+ , u 2 IR+, As noted in [38], neither of these algorithms is effi- n cient (taking a number of rounds that is polylogarith- minimize c · x subject to x 2 ZZ+, Ax ≥ w, and x ≤ u. mic in the number of vertices). CMIP, covering mixed integer linear programs: CIP Kuhn, Moscibroda and Wattenhofer describe dis- with both integer and fractional variables. tributed approximation algorithms for fractional cov- Facility Location, Weighted Set Cover (that is, ering and packing linear programs [39]. They show an CIP with Aij 2 f0; 1g; wi = 1), Weighted Ver- O(1)-approximation with high probability (w.h.p.) in tex Cover (that is, Weighted Set Cover with O(log m) rounds (m is the number of covering con- δ = 2), and probabilistic (two-stage stochastic) vari- straints). The approximation ratio is greater than 2 for ants of these problems. Fractional Weighted Vertex Cover. For (inte- In the centralized (non-distributed) setting there are ger) Weighted Vertex Cover and Weighted Set two well-studied classes of polynomial-time approxima- Cover (where each Aij 2 f0; 1g) combining their al- tion algorithms for covering problems: gorithms with randomized rounding gives O(log ∆)-ap- proximate integer solutions in O(log n) rounds, where (i) O(log ∆)-approximation algorithms where ∆ is the ∆ is the maximum number of constraints in which any maximum number of constraints in which any vari- variable occurs. able occurs (e.g. [27,46,10,56,57,32]), and Distributed lower bounds. The best lower bounds (ii) O(δ)-approximation algorithms where δ is the max- known for Weighted Vertex Cover are by Kuhn, imum number of variables in any constraint (e.g. [4, Moscibroda and Wattenhofer: to achieve even a poly- 22,21,5,19,7,8,55,2,36]), including most famously logarithmic approximation ratio requires in the worst 2-approximation algorithms for Weighted Ver- case Ω(plog n= log log n) rounds. In graphs of constant tex Cover. degree ∆, Ω(log ∆/ log log ∆) rounds are required [38]. The algorithms here match those in class (ii). New results for covering problems. This paper Related work. For Unweighted Vertex Cover, gives the following results for covering problems. it is well known that a 2-approximate solution can be { Section 2 describes the first efficient distributed 2- found by computing any maximal matching, then tak- approximation algorithm for Weighted Vertex ing the cover to contain the endpoints of the edges in the Cover. The algorithm runs in O(log n) rounds in matching. A maximal matching (and hence a 2-approx- expectation and with high probability. imate vertex cover) can be computed deterministically in O(log4 n) rounds using the algorithm of Ha´n´ckowiak, { Section 3, generalizing the above result, describes Karonski and Panconesi [20] or in O(∆+log∗ n) rounds the first efficient distributed 2-approximation algo- using the algorithm of Panconesi and Rizzi [51], where 4 Their result extends to δ(1 + ")-approximating Weighted ∆ is the maximum vertex degree. Or, using randomiza- Set Cover: a distributed algorithm taking O(δ log n log 1=") tion, a maximal matching (and hence a 2-approximate rounds, and a parallel algorithm (in NC for fixed " and δ). Distributed Algorithms for Covering, Packing and Maximum Weighted Matching 3 problem approx. ratio # rounds where when 2 O(log n) (random) [25] 1986 Unweighted Vertex Cover 2 O(log4 n) [20] 2001 2 + "O(log "−1 log n) [31] 1994 Weighted Vertex Cover 2 O(log n log nC^) [31] 1994 2 O(log nC^) (random) [18,15] 2005 2 O(log n) (random) here Weighted Set Cover O(log ∆) O(log m) (random) [39] 2006 δ O(log m) (random) here CMIP (with δ = 2) 2 O(log m) (random) here CMIP δ O(log2 m) (random) here Table 1 Comparison of distributed algorithms for covering problems. δ is the maximum number of variables in any constraint. ∆ is the maximum number of constraints in which any variable occurs. rithm for CMIP (covering mixed integer linear pro- of generality each vertex capacity cj is also an integer.) grams with variable upper bounds) restricted to in- An instance is defined by a given hypergraph H(V; E) jV j stances where each constraint has at most two vari- and cost vector c 2 ZZ+ ; a solution is given by a vector ables (δ = 2). The algorithm runs in O(log m) rounds jEj P y 2 ZZ+ maximizing e2E weye and meeting all the in expectation and with high probability, where m P vertex capacity constraints e2E(u) ye ≤ cu (8u 2 V ), is the number of constraints.