Distributed Fractional Packing and Maximum Weighted b-matching via Tail-Recursive Duality

Transcription

1 Ditributed Fractional Packing and Maximum Weighted b-matching via Tail-Recurive Duality Chrito Koufogiannaki, Neal E. Young Department of Computer Science, Univerity of California, Riveride {ckou, Abtract. We preent efficient ditributed δ-approximation algorithm for fractional packing and maximum weighted b-matching in hypergraph, where δ i the maximum number of packing contraint in which a variable appear (for maximum weighted b-matching δ i the maximum edge degree for graph δ = 2). (a) For δ = 2 the algorithm run in O(log m) round in expectation and with high probability. (b) For general δ, the algorithm run in O(log 2 m) round in expectation and with high probability. 1 Background and reult Given a weight vector w IR m +, a coefficient matrix A IR n m + and a vector b IR n +, the fractional packing problem i to compute a vector x IR m + to maximize m j=1 w jx j and at the ame time meet all the contraint m j=1 A ijx j b i ( i = 1... n). We ue δ to denote the maximum number of packing contraint in which a variable appear, that i, δ = max j {i A ij 0}. In the centralized etting, fractional packing can be olved optimally in polynomial time uing linear programming. Alternatively, one can ue a fater approximation algorithm (i.e. [11]). maximum weighted b-matching on a (hyper)graph i the variant where each A ij {0, 1} and the olution x mut take integer value (without lo of generality each vertex capacity i alo integer). An intance i defined by a given hypergraph H(V, E) and b Z V + ; a olution i given by a vector x Z E + maximizing e E w ex e and meeting all the vertex capacity contraint e E(u) x e b u ( u V ), where E(u) i the et of edge incident to vertex u. For thi problem, n = V, m = E and δ i the maximum (hyper)edge degree (for graph δ = 2). maximum weighted b-matching i a cornertone optimization problem in graph theory and Computer Science. A a pecial cae it include the ordinary maximum weighted matching problem (b u = 1 for all u V ). In the centralized etting, maximum weighted b-matching on graph belong to the well-olved cla of integer linear program in the ene that it can be olved in polynomial time [5, 6, 19]. Moreover, getting a 2-approximate 1 olution for maximum weighted matching i relatively eay, ince the obviou greedy algorithm, which elect the heaviet edge that i not conflicting with already elected edge, give a 2-approximation. For hypergraph the problem i NP-hard, ince it generalize et packing, one of Karp 21 NP-complete problem [10]. Our reult. In thi work we preent efficient ditributed δ-approximation algorithm for the above problem. If the input i a maximum weighted b-matching intance, the algorithm produce integral olution. The method we ue i of particular interet in the ditributed etting, where it i the firt primal-dual extenion of a non-tandard local-ratio technique [13, 2]. For fractional packing where each variable appear in at mot two contraint (δ = 2), we how a ditributed 2-approximation algorithm running in O(log m) round in expectation and with high probability. Thi i the firt 2-approximation algorithm requiring only O(log m) round. Thi improve the approximation ratio over the previouly bet known algorithm [14]. (For a ummary of known reult ee Figure 1.) Partially upported by NSF award CNS , CCF Since it i a maximization problem it i alo referred to a a 1/2-approximation

2 problem approx. ratio running time where when O( ) O(1) [22] O(log 2 n) [23] O(m) [7] 2004 O(1)(> 2) O(log n) [14] 2006 max weighted matching on graph (4 + ε) O(ε 1 log ε 1 log n) [18] 2007 (2 + ε) O(log ε 1 log n) [17] 2008 (1 + ε) O(ε 4 log 2 n) [17] 2008 (1 + ε) O(ε 2 + ε 1 log(ε 1 n) log n) [20] O(log 2 n) [17, 20] (ε = 1) O(log n) here 2009 fractional packing with δ = 2 max weighted matching on hypergraph fractional packing with general δ O(1)(> 2) O(log m) [14] O(log m) here 2009 O(δ) > δ O(log m) [14] 2006 δ O(log 2 m) here 2009 O(1) > 12 O(log m) [14] 2006 δ O(log 2 m) here 2009 Fig. 1. Ditributed algorithm for fractional packing and maximum weighted matching. For fractional packing where each variable appear in at mot δ contraint, we give a ditributed δ-approximation algorithm running in O(log 2 m) round in expectation and with high probability, where m i the number of variable. For mall δ, thi improve over the bet previouly known contant factor approximation [14], but the running time i lower by a logarithmic-factor. For maximum weighted b-matching on graph we give a ditributed 2-approximation algorithm running in O(log n) round in expectation and with high probability. maximum weighted b-matching generalize the well tudied maximum weighted matching problem. For a 2-approximation, our algorithm i fater by at leat a logarithmic factor than any previou algorithm. Specifically, in O(log n) round, our algorithm give the bet known approximation ratio. The bet previouly known algorithm compute a (1 + ε)- approximation in O(ε 4 log 2 n) round [17] or in O(ε 2 + ε 1 log(ε 1 n) log n) round [20]. For a 2- approximation both thee algorithm need O(log 2 n) round. For maximum weighted b-matching on hypergraph with maximum hyperedge degree δ we give a ditributed δ-approximation algorithm running in O(log 2 m) round in expectation and with high probability, where m i the number of hyperedge. Our reult improve over the bet previouly known O(δ)-approximation ratio by [14], but it i lower by a logarithmic factor. Related work for Maximum Weighted Matching. There are everal work conidering ditributed maximum weighted matching on edge-weighted graph. Uehara and Chen preent a contant time O( )- approximation algorithm [22], where i the maximum vertex degree. Wattenhofer and Wattenhofer improve thi reult, howing a randomized 5-approximation algorithm taking O(log 2 n) round [23]. Hoepman how a determinitic 2-approximation algorithm taking O(m) round [7]. Lotker, Patt-Shamir and Roén give a randomized (4+ε)-approximation algorithm running in O(ε 1 log ε 1 log n) round [18]. Lotker, Patt-Shamir and Pettie improve thi reult to a randomized (2 + ε)-approximation algorithm taking O(log ε 1 log n) round [17]. Their algorithm ue a a black box any ditributed contant-factor approximation algorithm for maximum weighted matching which take O(log n) round (i.e. [18]). Moreover, they mention (without detail) that there i a ditributed (1+ε)-approximation algorithm taking O(ε 4 log 2 n) round, baed on the parallel algorithm by Hougardy and Vinkemeier [8]. Nieberg preent a (1 + ε)-approximation algorithm in O(ε 2 +ε 1 log(ε 1 n) log n) round [20]. The latter two reult give randomized 2-approximation algorithm for maximum weighted matching in O(log 2 n) round. Related work for Fractional Packing. Kuhn, Mocibroda and Wattenhofer how efficient ditributed approximation algorithm for fractional packing [14]. They firt how a (1+ε)-approximation algorithm for fractional packing with logarithmic meage ize, but the running time depend on the input coefficient. For unbounded meage ize they how a contant-factor approximation algorithm for fractional packing which take O(log m) round. If an integer olution i deired, then ditributed randomized rounding ([15])

3 can be ued. Thi give an O(δ)-approximation for maximum weighted b-matching on (hyper)graph with high probability in O(log m) round, where δ i the maximum hyperedge degree (for graph δ = 2). (The hidden contant factor in the big-o notation of the approximation ratio can be relative large compared to a mall δ, ay δ = 2). Lower bound. The bet lower bound known for ditributed packing and matching are given by Kuhn, Mocibroda and Wattenhofer [14]. They prove that to achieve a contant or even a poly-logarithmic approximation ratio for fractional maximum matching, any algorithm require at leat Ω( log n/ log log n) round and Ω(log / log log ), where i the maximum vertex degree. Other related work. For unweighted maximum matching on graph, Iraeli and Itai give a randomized ditributed 2-approximation algorithm running in O(log n) round [9]. Lotker, Patt-Shamir and Pettie improve thi reult giving a randomized (1 + ε)-approximation algorithm taking O(ε 3 log n) round [17]. Czygrinow, Hańćkowiak, and Szymańka how a determinitic 3/2-approximation algorithm which take O(log 4 n) round [4]. A (1 + ε)-approximation for maximum weighted matching on graph i in NC [8]. The ret of the paper i organized a follow. In Section 2 we decribe a non-tandard primal-dual technique to get a δ-approximation algorithm for fractional packing and maximum weighted b-matching. In Section 3 we preent the ditributed implementation for δ = 2. Then in Section 4 we how the ditributed δ-approximation algorithm for general δ. We conclude in Section 5. 2 Covering and packing Koufogiannaki and Young how equential and ditributed δ-approximation algorithm for general covering problem [13, 12], where δ i the maximum number of covering variable on which a covering contraint depend. A a pecial cae their algorithm compute δ-approximate olution for fractional covering problem of the form min{ n i=1 b iy i : n i=1 A ijy i w j ( j = 1..m), y IR n +}. The linear programming dual of uch a problem i the following fractional packing problem: max{ m j=1 w jx j : m j=1 A ijx j b i ( i = 1... n), x IR m + }. For packing, δ i the maximum number of packing contraint in which a packing variable appear, δ = max j {i A ij 0}. Here we extend the ditributed approximation algorithm for fractional covering by [12] to compute δ-approximate olution for fractional packing uing a non-tandard primal-dual approach. Notation. Let C j denote the j-th covering contraint ( n i=1 A ijy i w j ) and P i denote the i-th packing contraint ( m j=1 A ijx j b i ). Let Var(S) denote the et of (covering or packing) variable indexe that appear in (covering or packing) contraint S. Let Con(z) denote the et of (covering or packing) contraint indexe in which (covering or packing) variable z appear. Let N(x ) denote the et of packing variable that appear in the packing contraint in which x appear, that i, N(x ) = {x j j Var(P i ) for ome i Con(x )} = Var(Con(x )). Fractional Covering. Firt we give a brief decription of the δ-approximation algorithm for fractional covering by [13, 12] 2. The algorithm perform tep to cover non-yet-atified covering contraint. Let y t be the olution after the firt t tep have been performed. (Initially y 0 = 0.) Given y t, let wj t = w j n i=1 A ijyi t be the lack of C j after the firt t tep. (Initially w 0 = w.) The algorithm i given by Alg. 1. There may be covering contraint for which the algorithm never perform a tep becaue they are covered by tep done for other contraint with which they hare variable. Alo note that increaing y i for all i Var(C ), decreae the lack of all contraint which depend on y i. Our general approach. [13] how that the above algorithm i a δ-approximation for covering, but they don t how any reult for matching or other packing problem. Our general approach i to recat their analyi a a primal-dual analyi, howing that the algorithm (Alg. 1) implicitly compute a olution to the dual packing problem of interet here. To do thi we ue the tail-recurive approach implicit in previou local-ratio analye [3]. 2 The algorithm i equivalent to local-ratio when A {0, 1} n m and y {0, 1} n [1, 2]. See [13] for a more general algorithm and a dicuion on the relation between thi algorithm and local ratio.

4 greedy δ-approximation algorithm for fractional covering [13, 12] alg Initialize y 0 0, w 0 w, t While there exit an unatified covering contraint C do a tep for C : 3. Set t = t Let β w t 1 min i Var(C) b i/a i.... OPT cot to atify C given the current olution 5. For each i Var(C ): 6. Set yi t = y t 1 i + β /b i.... increae y i inverely proportional to it cot 7. For each j Con(y i) update wj t = w t 1 j A ijβ /b i.... new lack 8. Return y = y t. After the t-th tep of the algorithm, define the reidual covering problem to be min{ n i=1 b iy i : n i=1 A ijy i wj t ( j = 1..m), y IR n +} and the reidual packing problem to be it dual, max{ m j=1 wt j x j : m j=1 A ijx j b i ( i = 1... n), x IR m + }. The algorithm will compute δ-approximate primal and dual pair (x t, y T t ) for the reidual problem for each t. A hown in what follow, the algorithm increment the covering olution x in a forward way, and the packing olution y in a tail-recurive manner. Standard Primal-Dual approach doe not work. For even imple intance, generating a δ-approximate primal-dual pair for the above greedy algorithm require a non-tandard approach. For example, conider min{y 1 +y 2 +y 3 : y 1 +y 2 1, y 1 +y 3 5, y 1, y 2 0}. If the greedy algorithm (Alg. 1) doe the contraint in either order and chooe β maximally, it give a olution of cot 10. In the dual max{x x 13 : x 12 + x 13 1, x 12, x 13 0}, the only way to generate a olution of cot 5 i to et x 13 = 1 and x 12 = 0. A tandard primaldual approach would raie the dual variable for each covering contraint when that contraint i proceed (eentially allowing a dual olution to be generated in an online fahion, contraint by contraint). That can t work here. For example, if the contraint y 1 + y 2 1 i covered firt by etting y 1 = y 2 = 1, then the dual variable x 12 would be increaed, thu preventing x 13 from reaching 1. Intead, auming the tep to cover y 1 + y 2 1 i done firt, the algorithm hould not increae any packing variable until a olution to the reidual dual problem i computed. After thi tep the reidual primal problem i min{y 1 + y 2 + y 3 : y 1 + y 2 1, y 1 + y 3 4, y 1, y 2 0}, and the reidual dual problem i max{ x x 13 : x 12 + x 13 1, x 12, x 13 0}. Once a olution x to the reidual dual problem i computed (either recurively or a hown later in thi ection) then the dual variable x 12 for the current covering contraint hould be raied maximally, giving the dual olution x for the current problem. In detail, the reidual dual olution x i x 12 = 0 and x 13 = 1 and the cot of the reidual dual olution i 4. Then the variable x 12 i raied maximally to give x 12. However, ince x 13 = 1, x 12 cannot be increaed, thu x = x. Although neither dual coordinate i increaed at thi tep, the dual cot i increaed from 4 to 5, becaue the weight of x 13 i increaed from w 13 = 4 to w 13 = 5. (See Figure 2 in the appendix.) In what follow we preent thi formally. Fractional Packing. We how that the greedy algorithm for covering create an ordering of the covering contraint for which it perform tep, which we can then ue to raie the correponding packing variable. Let t j denote the time 3 at which a tep to cover C j wa performed. Let t j = 0 if no tep wa performed for C j. We define the relation C j C j on two covering contraint C j and C j which hare a variable and for which the algorithm performed tep to indicate that contraint C j wa done firt by the algorithm. Definition 1. Let C j C j if Var(C j ) Var(C j ) and 0 < t j < t j. Note that the relation i not defined for covering contraint for which a tep wa never performed by the algorithm. Then let D be the partially ordered et (poet) of all covering contraint for which the algorithm performed a tep, ordered according to. D i partially ordered becaue i not defined for covering contraint that do not hare a variable. In addition, ince for each covering contraint C j we have a correponding dual packing variable x j, abuing notation we write x j x j if C j C j. Therefore, D i alo a poet of packing variable. 3 In general by time we mean ome reaonable way to ditinguih in which order tep were performed to atify covering contraint. For now, the time at which a tep wa performed can be thought a the tep number (line 3 at Alg. 1). It will be lightly different in the ditributed etting.

5 Definition 2. A revere order of poet D i an order C j1, C j2,..., C jk (or equivalently x j1, x j2,..., x jk ) uch that for l > i either we have C jl C ji or the relation i not defined for contraint C ji and C jl (becaue they do not hare a variable). Then the following figure (Alg. 2) how the equential δ-approximation algorithm for fractional packing. greedy δ-approximation algorithm for fractional packing alg Run Alg. 1, recording the poet D. 2. Let T be the number of tep performed by Alg Initialize x T 0, t T.... note that t will be decreaing from T to 0 4. Let Π be ome revere order of D.... any revere order of D work, ee Lemma 1 5. For each variable x D in the order given by Π do: 6. Set x t 1 = x t. 7. Raie x t 1 until a packing contraint that depend on x t 1 m j=1 Aijxt 1 j ). 8. Set t = t Return x = x 0. i tight, that i, et x t 1 = max i Con(xj )(b i The algorithm imply conider the packing variable correponding to covering contraint that Alg. 1 did tep for, and raie each variable maximally without violating the packing contraint. The order in which the variable are conidered matter: the variable hould be conidered in the revere of the order in which tep were done for the correponding contraint, or an order which i equivalent (ee Lemma 1). (Thi flexibility i neceary for the ditributed etting.) The olution x i feaible at all time ince a packing variable i increaed only until a packing contraint get tight. Lemma 1. Alg. 2 return the ame olution x uing (at line 4) any revere order of D. Proof. Let Π = x j1, x j2,..., x jk and Π = x j1, x j2,..., x jk be two different revere order of D. Let x Π,1...m be the olution computed o far by Alg. 2 after raiing the firt m packing variable of order Π. We prove that x Π,1...k = x Π,1...k. Aume that Π and Π have the ame order for their firt q variable, that i j i = j i for all i q. Then, x Π,1...q = x Π,1...q. The firt variable in which the two order diagree i the (q + 1)-th one, that i, j q+1 j q+1. Let = j q+1. Then x hould appear in ome poition l in Π uch that q + 1 < l k. The value of x depend only on the value of variable in N(x ) at the time when x i et. We prove that for each x j N(x ) we have x Π,1...q j = x Π,1...l j, thu x Π,1...q each packing variable only once thi implie x Π,1...k = x Π,1...q = x Π,1...l. Moreover ince the algorithm conider = x Π,1...l = x Π,1...k. (a) For each x j N(x ) with x x j, the variable x j hould have already been et in the firt q tep, otherwie Π would not be a valid revere order of D. Moreover each packing variable can be increaed only once, o once it i et it maintain the ame value till the end. Thu, for each x j uch that x x e, we have x Π,1...q j = x Π,1...q j = x Π,1...l j. (b) For each x j N(x ) with x j x, j cannot be in the interval [j q+1,..., j l 1 ) of Π, otherwie Π would not be a valid revere order of D. Thu, for each x j uch that x j x, we have x Π,1...q j x Π,1...l j = 0. So in any cae, for each x j N(x ), we have x Π,1...q j = x Π,1...l j The lemma follow by induction on the number of edge. and thu x Π,1...q = x Π,1...l. = x Π,1...q j = The following lemma and weak duality prove that the olution x returned by Alg. 2 i δ-approximate. Lemma 2. For the olution y and x returned by Alg. 1 and Alg. 2 repectively, m j=1 w jx j 1/δ n i=1 b iy i. Proof. Lemma 1 how that any revere order of D produce the ame olution, o w.l.o.g. here we aume that the revere order Π ued by Alg. 2 i the revere of the order in which tep to atify covering contraint were performed by Alg. 1.

6 When Alg. 1 doe a tep to atify the covering contraint C (by increaing y i by β /b i for all i Var(C )), the cot of the covering olution i b iy i increae by at mot δβ, ince C depend on at mot δ variable ( Var(C ) δ). Thu the final cot of the cover y i at mot D δβ. Define Ψ t = j wt j xt j to be the cot of the packing xt. Recall that x T = 0 o Ψ T = 0, and that the final packing olution i given by vector x 0, o the the cot of the final packing olution i Ψ 0. To prove the theorem we have to how that Ψ 0 D β. We have that Ψ 0 = Ψ 0 Ψ T = T t=1 Ψ t 1 Ψ t o it i enough to how that Ψ t 1 Ψ t β where C i the covering contraint done at the t-th tep of Alg. 1. Then, Ψ t 1 Ψ t i j = w t 1 = w t 1 = β x t 1 β 1 b i = β w t 1 j x t 1 j wjx t t j (1) x t 1 x t j (w t 1 j w t j)x t 1 j (2) i Con(x ) j {Var(P j) } A i max + i Con(x ) b i m j=1 i Con(x ) j {Var(P j) } A ij β b i x t 1 j (3) A ij β b i x t 1 j (4) A ij x t 1 j (for i.t. contraint P i become tight after raiing x ) (5) In equation (2) we ue the fact that x t = 0 and x t 1 j = x t j for all j. For equation (3), we ue the fact that the reidual weight of packing variable in N(x ) are increaed. If x j > 0 for j, then x j wa increaed before x (x x j ) o at the current tep w t 1 j > wj t > 0, and wt 1 j wj t = i Con(x A ) ij β b i. For equation (4), by the definition of β we have w t 1 = β max Ai i Con(x) b i. In inequality (5) we keep only the term that appear in the contraint P i that get tight by raiing x. The lat equality hold becaue P i i tight, that i, m j=1 A ijx j = b i. The following lemma how that Alg. 2 return integral olution if the coefficient A ij are 0/1 and the b i are integer, thu giving a δ-approximation algorithm for maximum weighted b-matching. Lemma 3. If A {0, 1} n m and b Z n + then the returned packing olution x i integral, that i, x Z m +. Proof. Since all non-zero coefficient are 1, the packing contraint are of the form j Var(P x i) j b i ( i). We prove by induction that x Z m +. The bae cae i trivial ince the algorithm tart with a zero olution. Aume that at ome point we have x t Z m +. Let x D, be the next packing variable to be raied by the algorithm. We how that x t 1 Z + and thu the reulting olution remain integral. The algorithm et x t 1 = min i Con(x){b i m j=1 xt 1 j } = min i Con(x){b i m j=1 xt j } 0. By the induction hypothei, each x t j Z +, and ince b Z n +, then x t 1 i alo a non-negative integer. 3 Ditributed Fractional Packing with δ = Ditributed model for δ = 2 We aume the network in which the ditributed computation take place ha vertice for covering variable (packing contraint) and edge for covering contraint (packing variable). So, the network ha a node u i for every covering variable y i. An edge e j connect vertice u i and u i if y i and y i belong to the ame covering contraint C j, that i, there exit a contraint A ij y i + A i jy i w j (δ = 2 o there can be at mot 2 variable in each covering contraint). We aume the tandard ynchronou communication model, where in each round, node can exchange meage with neighbor, and perform ome local computation [21]. We alo aume no retriction on meage ize and local computation. (Note that a ynchronou model algorithm can be tranformed into an aynchronou algorithm with the ame time complexity [21].) (6)

7 3.2 Ditributed algorithm for δ = 2 Koufogiannaki and Young how a ditributed implementation of Alg. 1, for (fractional) covering with δ = 2 that run in O(log n) round in expectation and with high probability [12]. In thi ection we augment their algorithm to ditributively compute 2-approximate olution to the dual fractional packing problem without increaing the time complexity. The high level idea i imilar to that in the previou ection: run the ditributed algorithm for covering to get a partial order of the covering contraint for which tep were performed, then conider the correponding dual packing variable in ome revere order raiing them maximally. The challenge here i that the ditributed algorithm for covering can perform tep for many covering contraint in parallel. Moreover, each covering contraint, ha jut a local view of the ordering, that i, it only know it relative order among the covering contraint with which it hare variable. Ditributed Fractional Covering with δ = 2. Here i a hort decription of the ditributed 2-approximation algorithm for fractional covering (Alg. 5 in appendix from [12]). In each round, the algorithm doe tep on a large ubet of remaining edge (covering contraint), a follow. Each vertex (covering variable) randomly chooe to be a leaf or a root. A not-yet-atified edge e j = (u i, u r ) between a leaf u i and a root u r with b i /A ij b r /A rj i active for the round. Each leaf u i chooe a random tar edge (u i, u r ) from it active edge. Thee tar edge form tar rooted at root. Each root u r then perform tep (of Alg. 1) on it tar edge (in any order) until they are all atified. Note that in a round the algorithm perform tep in parallel for edge not belonging to the ame tar. For edge belonging to the ame tar, their root perform tep for ome of them one by one. There are edge for which the algorithm never perform tep becaue they are covered by tep done for adjacent edge. In the ditributed etting we define the time at which a tep to atify C j i done a a pair (t R j, ts j ), where t R j denote the round in which the tep wa performed and t S j denote that within the tar thi tep i the t S j -th one. Let tr j = 0 if no tep wa performed for C j. Overloading Definition 1, we redefine a follow. Definition 3. Let C j C j (or equivalently x j x j ) if Var(C j ) Var(C j ) (j and j are adjacent edge in the ditributed network) and ([0 < t R j < tr j ] or [tr j = tr j and t S j < ts j ]). The pair (t R j, ts j ) i enough to ditinguih which of two adjacent edge had a tep to atify it covering contraint performed firt. Adjacent edge can have their covering contraint done in the ame round only if they belong to the ame tar (they have a common root), thu they differ in t S j. Otherwie they are done in different round, o they differ in t R j. Thu the pair (tr j, ts j ) and relation define a partially ordered et D of all edge done by the ditributed algorithm for covering. Lemma 4. ([12]) Alg. 5 (for fractional covering with δ = 2) finihe in T = O(log m) round in expectation and with high probability. Simultaneouly, Alg. 5 et (t R j, ts j ) for each edge e j for which it perform a tep (0 < t R j T ), thu defining a poet of edge D, ordered by. Ditributed Fractional Packing with δ = 2. Alg. 3 implement Alg. 2 in a ditributed fahion. Firt, it run Alg. 1 uing the ditributed implementation by [12] (Alg. 5) and recording D. Meanwhile, a it dicover the partial order D, it begin the econd phae of Alg. 2, raiing each packing variable a oon a it can. Specifically it wait to et a given x j D until after it know that (a) x j i in D, (b) for each x j N(x j ) whether x j x j, and (c) each uch x j i et. In other word, (a) a tep ha been done for the covering contraint C j, (b) each adjacent covering contraint C j i atified and (c) for each adjacent C j for which a tep wa done after C j, the variable x j ha been et. Subject to thee contraint it et x j a oon a poible. Note that ome node will be executing the econd phae of the algorithm (packing) while ome other node are till executing the firt phae (covering). Thi i neceary becaue a given node cannot know when ditant node are done with the firt phae. All x j will be determined in 2T round by the following argument. After round T, D i determined. Then by a traightforward induction on t, within T + t round, every contraint C j for which a tep wa done at round T t of the firt phae, will have it variable x j et. Theorem 1. For fractional packing where each variable appear in at mot two contraint there i a ditributed 2-approximation algorithm running in O(log m) round in expectation and with high probability, where m i the number of packing variable.

8 Ditributed 2-approximation Fractional Packing with δ = 2 alg. 3 input: Graph G = (V, E) repreenting a fractional packing problem intance with δ = 2. output: Feaible x, 2-approximately minimizing w x. 1. Each edge e j E initialize x j Each edge e j E initialize done j fale.... thi indicate if x j ha been et to it final value 3. Until each edge e j ha et it variable x j (done j == true), perform a round: 4. Perform a round of Alg covering with δ = 2 augmented to compute (t R j,t S j ) 5. For each node u r that wa a root (in Alg. 5) at any previou round, conider locally at u r all tar S t r that were rooted by u r at any previou round t. For each tar S t r perform IncreaeStar(S t r). IncreaeStar(tar S t r): 6. For each edge e j S t r in decreaing order of t S j : 7. If IncreaePackingVar(e j) == UNDONE then BREAK (top the for loop). IncreaePackingVar(edge e j = (u i, u r)): 8. If e j or any of it adjacent edge ha a non-yet-atified covering contraint return UNDONE. 9. If t R j == 0 then: 10. Set x j = 0 and done j = true. 11. Return DONE. 12. If done j == fale for any edge e j uch that x j x j then return UNDONE. 13. Set x j = min { (b i j A ij x j )/A ij, (b r j A rj x j )/A rj } and donej = true. 14. Return DONE. Proof. By Lemma 4, Alg. 5 compute a covering olution y in T = O(log m) round in expectation and with high probability. At the ame time, the algorithm et (t R j, ts j ) for each edge e j for which it perform a tep to cover C j, and thu defining a poet D of edge. In the ditributed etting the algorithm doe not define a linear order becaue there can be edge with the ame (t R j, ts j ), that i, edge that are covered by tep done in parallel. However, ince thee edge mut be non-adjacent, we can till think that the algorithm give a linear order (a in the equential etting), where tie between edge with the ame (t R j, ts j ) are broken arbitrarily (without changing D). Similarly, we can analyze Alg. 3 a if it conider the packing variable in a revere order of D. Then, by Lemma 1 and Lemma 2 the returned olution x i 2-approximate. We prove that the x can be computed in at mot T extra round after the initial T round to compute y. Firt note that within a tar, even though it edge are ordered according to t S j they can all et their packing variable in a ingle round if none of them wait for ome adjacent edge packing variable that belong to a different tar. So in the ret of the proof we only conider the cae were edge are waiting for adjacent edge that belong to different tar. Note that 1 t R j T for each x j D. Then, at round T, each x j with t R j = T can be et in thi round becaue it doe not have to wait for any other packing variable to be et. At the next round, round T + 1, each x j with t R j = T 1 can be et; they are dependent only on variable x j with t R j = T which have been already et. In general, packing variable with tr j = t can be et once all adjacent x j with t R j t + 1 have been et. Thu by induction on t = 0, 1,... a contraint C j for which a tep wa done at round T t may have to wait until at mot round T + t until it packing variable x j i et. Therefore, the total number of round until olution x i computed i 2T = O(log m) in expectation and with high probability. The following theorem i a direct reult of Lemma 3 and Thm 1 and the fact that for thi problem m = O(n 2 ). Theorem 2. For maximum weighted b-matching on graph there i a ditributed 2-approximation algorithm running in O(log n) round in expectation and with high probability.

9 4 Ditributed Fractional Packing with general δ 4.1 Ditributed model for general δ Here we aume that the ditributed network ha a node v j for each covering contraint C j (packing variable x j ), with edge from v j to each node v j if C j and C j hare a covering variable y i 4. The total number of node in the network i m. Note that in thi model the role of node and edge i revered a compared to the model ued in Section 3. We aume the tandard ynchronou model with unbounded meage ize. 4.2 Ditributed algorithm Koufogiannaki and Young [12] how a ditributed δ-approximation algorithm for (fractional) covering problem with at mot δ variable per covering contraint that run in O(log 2 m) round in expectation and with high probability. Similar to the δ = 2 cae, here we ue thi algorithm to get a poet of packing variable which we then conider in a revere order, raiing them maximally. Ditributed covering with general δ. Here i a brief decription of the ditributed δ-approximation algorithm for (fractional) covering from [12]. To tart each phae, the algorithm find large independent ubet of covering contraint by running one phae of Linial and Sak (LS) decompoition algorithm, with any k uch that k Θ(ln m) 5 [16]. The LS algorithm, for a given k, take O(k) round and produce a random ubet R {v j j = 1... m} of the covering contraint, and for each covering contraint v j R a leader l(v j ) R, with the following propertie: Each v j R i within ditance k of it leader: ( v j R) d(v j, v l(j) ) k. Component do not hare covering variable (edge do not cro component): ( v j, v j R) v l(j) v l(j ) Var(v j ) Var(v j ) =. Each covering contraint node ha a chance to be in R: ( j = 1... m) Pr[v j R] 1/cm 1/k for ome c > 1. Next, each node v j R end it information (the contraint and it variable value) to it leader v l(j). Thi take O(k) round becaue v l(j) i at ditance O(k) from v j. Each leader then contruct (locally) the ubproblem induced by the covering contraint that contacted it and the variable of thoe contraint, with their current value. Uing thi local copy, the leader doe tep until all covering contraint that contacted it are atified. (Ditinct leader ubproblem don t hare covering variable, o they can proceed imultaneouly.) To end the phae, each leader u l return the updated variable information to the contraint that contacted v l. Each covering contraint node in R i atified in the phae. To extend the algorithm to compute a olution to the dual packing problem the idea i imilar to the δ = 2 cae, ubtituting the role of tar by component and the role of root by leader. With each tep done to atify the covering contraint C j, the algorithm record (t R j, ts j ), where tr j i the round and t S j i the within-the-component iteration in which the tep wa performed. Thi define a poet D of covering contraint for which it perform tep. Lemma 5. ([12]) The ditributed δ-approximation algorithm for fractional covering finihe in T = O(log 2 m) round in expectation and with high probability, where m i the number of covering contraint (packing variable). Simultaneouly, it et (t R j, ts j ) for each covering contraint C j for which it perform a tep (0 < t R j T ), thu defining a poet of covering contraint (packing variable) D, ordered by. Ditributed packing with general δ. (ketch) The algorithm (Alg. 4) i very imilar to the cae δ = 2. Firt it run the ditributed algorithm for covering, recording (t R j, ts j ) for each covering contraint C j for which it perform a tep. Meanwhile, a it dicover the partial order D, it begin computing the packing olution, raiing each packing variable a oon a it can. Specifically it wait to et a given x j D until after it know that (a) x j i in D, (b) for each x j N(x j ) whether x j x j, and (c) each uch x j i et. In other word, (a) a tep ha been done for the covering contraint C j, (b) each adjacent covering contraint 4 The computation can eaily be imulated on a network with node for covering variable or node for covering variable and covering contraint. 5 If node don t know a k Θ(ln m), a doubling technique can be ued a a work-around [12].

10 Ditributed δ-approximation Fractional Packing with general δ alg. 4 input: Graph G = (V, E) repreenting a fractional packing problem intance. output: Feaible x, δ-approximately minimizing w x. 1. Initialize x For each j = 1... m initialize done j fale.... thi indicate if x j ha been et to it final value 3. Until each x j ha been et (done j == true) do: 4. Perform a phae of the δ-approximation algorithm for covering by [12], recording (t R j, t S j ). 5. For each node v K that wa a leader at any previou phae, conider locally at v K all component that choe v K a a leader at any previou phae. For each uch component K r perform IncreaeComponent(K r). IncreaeComponent(component K r): 6. For each j K r in decreaing order of t S j : 7. If IncreaePackingVar(j) == UNDONE then BREAK (top the for loop). IncreaePackingVar(j): 8. If C j or any C j that hare covering variable with C j i not yet atified return UNDONE. 9. If t R j == 0 then: 10. Set x j = 0 and done j = true. 11. Return DONE. 12. If done j == fale for any x j uch that x j x j then return UNDONE. 13. Set x j = min i Con(xj ) ( (bi j A ij x j )/A ij) ) and done j = true. 14. Return DONE. C j i atified and (c) for each adjacent C j for which a tep wa done after C j, the variable x j ha been et. Subject to thee contraint it et x j a oon a poible. To do o, the algorithm conider all component that have been done by leader in previou round. For each component, the leader conider the component packing variable x j in order of decreaing t S j. When conidering x j it check if each x j with x j x j i et, and if ye, then x j can be et and the algorithm continue with the next component packing variable (in order of decreaing t S j ). Otherwie the algorithm cannot yet decide about the remaining component packing variable. Theorem 3. For fractional packing where each variable appear in at mot δ contraint there i a ditributed δ-approximation algorithm running in O(log 2 m) round in expectation and with high probability, where m i the number of packing variable. The proof i omitted becaue it i imilar to the proof of Thm 1; the δ-approximation ratio i given by Lemma 1 and Lemma 2, and the running time ue T = O(log 2 m) by Lemma 5. The following theorem i a direct reult of Lemma 3 and Thm 3. Theorem 4. For maximum weighted b-matching on hypergraph, there i a ditributed δ-approximation algorithm running in O(log 2 m) round in expectation and with high probability, where δ i the maximum hyperedge degree and m i the number of hyperedge. 5 Concluion We how a new non-tandard primal-dual method, which extend the (local-ratio related) algorithm for fractional covering by [13, 12] to compute approximate olution to the dual fractional packing problem without increaing the time complexity (even in the ditributed etting). Uing thi new technique, we how a ditributed 2-approximation algorithm for fractional packing where each packing variable appear in at mot 2 contraint and a ditributed 2-approximation algorithm for maximum weighted b-matching on graph, both running in a logarithmic number of round. We alo preent a ditributed δ-approximation algorithm for fractional packing where each variable appear in at mot δ contraint and a ditributed δ-approximation algorithm for maximum weighted b-matching on hypergraph, both running in O(log 2 m) round, where m i the number of packing variable and hyperedge repectively.

11 Reference 1. R. Bar-Yehuda. One for the price of two: A unified approach for approximating covering problem. Algorithmica, 27(2): , R. Bar-Yehuda, K. Bendel, A. Freund, and D. Rawitz. Local ratio: a unified framework for approximation algorithm. ACM Computing Survey, 36(4): , R. Bar-Yehuda and D. Rawitz. On the equivalence between the primal-dual chema and the local-ratio technique. SIAM Journal on Dicrete Mathematic, 19(3): , A. Czygrinow, M. Hańćkowiak, and E. Szymańka. A fat ditributed algorithm for approximating the maximum matching. In the twelfth European Symoium on Algorithm, page , J. Edmond. Path, tree, and flower. Canadian Journal of Mathematic, 17: , J. Edmond and E. L. Johnon. Matching: A well-olved cla of integer linear program. Combinatorial Structure and Their Application, page 89 92, J.H. Hoepman. Simple ditributed weighted matching. Arxiv preprint c.dc/ , S. Hougardy and D.E. Vinkemeier. Approximating weighted matching in parallel. Information Proceing Letter, 99(3): , A. Iraeli and A. Itai. A fat and imple randomized parallel algorithm for maximal matching. Information Proceing Letter, 22:77 80, R. M. Karp. Reducibility among combinatorial problem. Complexity of Computer Computation, R. E. Miller and J. W. Thatcher, Ed., The IBM Reearch Sympoia Serie, New York, NY: Plenum Pre:85 103, C. Koufogiannaki and N.E. Young. Beating implex for fractional packing and covering linear program. In the forty-eighth IEEE ympoium on Foundation of Computer Science, page , C. Koufogiannaki and N.E. Young. Ditributed and parallel algorithm for weighted vertex cover and other covering problem. In the twenty-eighth ACM ympoium on Principle of Ditributed Computing, C. Koufogiannaki and N.E. Young. Greedy -approximation algorithm for covering with arbitrary contraint and ubmodular cot. In the thirty-ixth International Colloquium on Automata, Language and Programming, LNCS 5555: , See alo F. Kuhn, T. Mocibroda, and R. Wattenhofer. The price of being near-ighted. In the eventeenth ACM-SIAM Sympoium On Dicrete Algorithm, page , F. Kuhn and R. Wattenhofer. Contant-time ditributed dominating et approximation. In the twetwenty-econd ACM ympoium on Principle Of Ditributed Computing, page 25 32, N. Linial and M. Sak. Low diameter graph decompoition. Combinatorica, 13(4): , Z. Lotker, B. Patt-Shamir, and S. Pettie. Improved ditributed approximate matching. In the twelfth ACM Sympoium on Parallelim in Algorithm and Architecture, page , Z. Lotker, B. Patt-Shamir, and A. Roén. Ditributed approximate matching. In the twenty-ixth ACM ympoium on Principle Of Ditributed Computing, page , M. Müller-Hannemann and A. Schwartz. Implementing weighted b-matching algorithm: Toward a flexible oftware deign. In the Workhop on Algorithm Engineering and Experimentation (ALENEX), page 18 36, T. Nieberg. Local, ditributed weighted matching on general and wirele topologie. In the fifth ACM Joint Workhop on the Foundation of Mobile Computing, DIALM-POMC, page 87 92, D. Peleg. Ditributed computing: a locality-enitive approach. Society for Indutrial and Applied Mathematic, R. Uehara and Z. Chen. Parallel approximation algorithm for maximum weighted matching in general graph. Information Proceing Letter, 76(1-2):13 17, M. Wattenhofer and R. Wattenhofer. Ditributed weighted matching. In the eighteenth international ympoium on Ditributed Computing, page , Appendix Alg. 5 how the ditributed 2-approximation algorithm for fractional covering with δ = 2.

12 Execution tart here. Follow the arrow Final dual olution: x 0, x 1 dual cot = 5 min chooe y y y y y.t. y y y y tep y 1, y 1 primal cot += 2 y, y, y 0 max x 5x.t. x x 1 x x x, x 0 raie x 12 (it remain 0) maximally x 0, x 1 dual cot += 1 dual cot increae by 1 becaue w increae by 1 and x = min y ' y ' y '.t. y ' y ' y ' y ' y ', y ', y ' 0 max - x ' 4 x '.t. x ' x ' 1 x ', x ' 0 x ' 0, x ' 1 dual cot += 4 chooe y ' y ' tep y ' 4, y ' primal cot += 8 raie x ' maximally 13 (it become 1) min y '' y '' y ''.t. y '' y '' y '' y '' y '', y '', y '' 0 max -5 x '' 4 x ''.t. x '' x '' 1 x '', x '' 0 Bae cae y '' 1 y '' 2 y '' 3 0 x '' 12 x '' 13 0 dual cot = 0 Final primal olution: y 5, y 1, y 4 primal cot = 10 Fig. 2. Example of the execution of our greedy primal-dual algorithm (auming contraint y 1 + y 2 1 i choen firt).

13 Ditributed 2-approximation Fractional Covering with δ = 2 ([12]) alg. 5 input: Graph G = (V, E) repreenting a fractional covering problem intance with δ = 2. output: Feaible y, 2-approximately minimizing b y. 1. Each node u i V initialize y i Each edge e j E initialize t R j 0 and t S j auxiliary variable for Alg Until there i a vertex with unatified incident edge, perform a round: 4. Each node u i, randomly and independently chooe to be a leaf or a root for the round, each with probability 1/2. 5. Each leaf-to-root edge e j = (u i, u r) with unmet covering contraint i active at the tart of the round if u i i a leaf, u r i a root and b i/a ij b r/a rj. Each leaf u i chooe, among it active edge, a random one for the round. Communicate that choice to the neighbor. The choen edge form independent tar rooted tree of depth 1 whoe leave are leaf node and whoe root are root node. 6. For each root node u r, do: (a) Let Sr t contain the tar edge haring variable y r (at thi round t). (b) Until there exit an unatified edge (covering contraint) e j = (u i, u r) Sr, t perform Step(y, e j). Step(y, e j = (u i, u r)): 7. Let β j (w j A ijy i A rjy r) min{b i/a ij, b r/a rj}. 8. Set y i = y i + β j/b i and y r = y r + β j/b r. 9. Set t R j to the number of round performed o far. 10. Set t S j to the number of tep performed by root u r o far at thi round.