Tight Approximation Algorithms for Maximum Separable Assignment Problems

Transcription

1 MATHEMATICS OF OPERATIONS RESEARCH Vo. 36, No. 3, August 011, pp issn X eissn /moor INFORMS Tight Approximation Agorithms for Maximum Separabe Assignment Probems Lisa Feischer Department of Computer Science, Dartmouth Coege, Hanover, New Hampshire 03755, Miche X. Goemans Department of Mathematics, Massachusetts Institute of Technoogy, Cambridge, Massachusetts 0139, Vahab S. Mirrokni Googe Research, NewYork, New York 10011, Maxim Sviridenko IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, A separabe assignment probem (SAP) is defined by a set of bins and a set of items to pack in each bin; a vaue, f ij, for assigning item j to bin i; and a separate packing constraint for each bin i.e., for each bin, a famiy of subsets of items that fit in to that bin. The goa is to pack items into bins to maximize the aggregate vaue. This cass of probems incudes the maximum generaized assignment probem (GAP) 1 and a distributed caching probem (DCP) described in this paper. Given a -approximation agorithm for finding the highest vaue packing of a singe bin, we give (i) A poynomia-time LP-rounding based 1 1/e -approximation agorithm. (ii) A simpe poynomia-time oca search / + 1 -approximation agorithm, for any > 0. Therefore, for a exampes of SAP that admit an approximation scheme for the singe-bin probem, we obtain an LP-based agorithm with 1 1/e -approximation and a oca search agorithm with 1 -approximation guarantee. Furthermore, for cases in which the subprobem admits a fuy poynomia approximation scheme (such as for GAP), the LP-based agorithm anaysis can be strengthened to give a guarantee of 1 1/e. The best previousy known approximation agorithm for GAP is a 1 -approximation by Shmoys and Tardos and Chekuri and Khanna. Our LP agorithm is based on rounding a new inear programming reaxation, with a provaby better integraity gap. To compement these resuts, we show that SAP and DCP cannot be approximated within a factor better than 1 1/e uness NP DTIME n O og og n, even if there exists a poynomia-time exact agorithm for the singe-bin probem. We extend the 1 1/e -approximation agorithm to a constant-factor approximation agorithms for a nonseparabe assignment probem with appications in maximizing revenue for budget-constrained combinatoria auctions and the AdWords assignment probem. We generaize the oca search agorithm to yied a 1 approximation agorithm for the maximum k-median probem with hard capacities. Key words: approximation agorithms; assignment probems; knapsack probem; inear programming; hardness resuts MSC000 subject cassification: Primary: 68W5; secondary: 90C7 OR/MS subject cassification: Primary: 007; secondary: 645 History: Received August 3, 008; revised January 1, 010, May 6, 010, August 17, 010, and January 1, Introduction. In this paper, we study a genera cass of maximizing assignment probems with packing constraints. In particuar, we study the separabe assignment probems (SAP): we are given a set U of n bins and a set H of m items, and a vaue, f ij, for assigning item j to bin i. We are aso given a separate packing constraint for each bin i; packing here refers to the assumption that if a subset is feasibe for a bin, then any subset of it is aso feasibe. The goa is to find an assignment of items to bins with the maximum aggregate vaue. For each bin, we define the singe-bin subprobem as the optimization probem over feasibe sets associated with the packing constraint for the bin. This genera cass of probems incudes severa more specific probems as specia cases, some we-studied and others newer and motivating our work here. A these probems are NP-compete because for a of them, the knapsack probem is a specia case. Maximum Generaized Assignment Probem (GAP): Given a set of bins with capacity constraint and a set of items that have a possiby different size and vaue for each bin, pack a maximum-vaued subset of items into the bins. This probem has severa appications in inventory panning. Distributed Caching Probem (DCP): This probem motivated our study of SAP. There is a set of n cache ocations and m requests. Each cache ocation has a storage capacity and a bandwidth imit. Each request is of a particuar request type and a particuar bandwidth. 1 GAP is as foows: given a set of bins and a set of items that have a different size and vaue for each bin, pack a maximum-vaued subset of items into the bins. 416

2 Mathematics of Operations Research 36(3), pp , 011 INFORMS 417 Given a set of requests with request types where each request type has a size, the service provider decides (1) which request types to store at each cache ocation, subject to storage capacity; and () which subset of requests to answer, subject to the request type seection and avaiabe bandwidth at the cache. The storage space for a request type in a cache ocation is independent of the number of requests for that type, the bandwidth required to serve requests of the same type is the sum of bandwidth requirements for the individua requests. For each potentia assignment of request to cache, there is an associated profit that depends on the connection cost for the request to the cache. The goa is to maximize the profit of providing requests. Our resuts. Our resuts depend on an agorithm to sove the singe-bin subprobem in SAP. In an instance of the singe-bin subprobem, we are given a bin i, a set of items with vaue v j for each item j, and a packing constraint for bin i, i.e., a ower-idea famiy, I i of feasibe subsets of items that can be packed in bin i. A -approximation agorithm for 1 for the singe-bin subprobem is an agorithm that outputs a subset of items S I i such that for any other subset S I i of items, j S v j j S v j. Given a -approximation agorithm for the singe-bin subprobem, we give (i) A poynomia-time LP-rounding based 1 1/e -approximation agorithm. (ii) A simpe, poynomia-time oca search / + 1 -approximation agorithm. 3 For a the specific probems mentioned above, there is an approximation scheme for the singe-bin subprobem so that we obtain an LP-based agorithm with 1 1/e -approximation and a oca search agorithm with 1 -approximation guarantee. Furthermore, if the singe-bin subprobem admits an FPTAS, a more carefu anaysis gives a 1 1/e-approximation agorithm for SAP. To compement these resuts, we show that SAP and DCP cannot be approximated within a factor better than 1 1/e uness NP DTIME n O og og n, even if there exists a poynomia-time exact agorithm for the singe-bin subprobem. We extend the 1 1/e-approximation agorithm to a nonseparabe assignment probem with appications in maximizing revenue for the budget-constrained combinatoria auctions and the AdWords assignment probem. We generaize the oca search agorithm to yied a 1 approximation agorithm for the maximum k-median probem with hard capacities. Reated work. For the specia case of GAP, Shmoys and Tardos [7] present an LP-rounding -approximation agorithm for the minimization probem. Chekuri and Khanna [10] observed that a 1 -approximation for GAP is impicit in Shmoys and Tardos [7]. Their method is based on an LP formuation with the integraity gap of at east. (Thus the LP we introduce here is provaby stronger.) Chekuri and Khanna [10] deveop PTASs for a specia case of this probem caed the mutipe knapsack probem. In this probem, each item has the same size and the same profit for a bins. They aso cassify the APX-hard specia cases of GAP. Fisher et a. [15] consider the probem of maximizing a submoduar function subject to the set being the independent set of a matroid. They describe both a simpe greedy agorithm and a oca search agorithm that give an approximation guarantee of 1. Chekuri [9] shows that SAP can be modeed as such a probem; his reduction is described in 3. Therefore, the proofs of Fisher et a. impy that if the singe-bin subprobem is soved optimay, their agorithms yied 1 approximation for SAP. In this paper, we describe a simiar oca search agorithm and show that it aso yieds a good guarantee for SAP, even when the singe-bin subprobem is soved approximatey. Sviridenko shows that the greedy agorithm gives a 1 1/e-approximation for maximizing a submoduar function subject to one knapsack constraint (Sviridenko [8]), but does not consider assignment type probems with sets of packing constraints. Indeed the simpe greedy agorithm does worse than 1 1/e even for the mutipe knapsack probem. Using LP techniques (Ageev and Sviridenko [1], Sviridenko [8]), approximation agorithms with the guarantee of 1 1/e for some maximum coverage probems are known. These techniques are different from ours and cannot hande SAP as the packing constraints in SAP are more genera. Gomes et a. [0] use a set packing LP and a rounding scheme simiar to the one we use here but to obtain a 1 1/e-approximation agorithm to sove the partia Latin square extension probem. In particuar, their LP does not capture knapsack packing constraints. Independent of our work, Nutov et a. [5] give a 1 1/e-approximation agorithm for a specia case of GAP with fixed profits. They use a different LP formuation that does not capture the genera GAP. A famiy I H of subsets of a set H is ower-idea if for each two subsets S and R such that R S, if S I, then R I. 3 For the oca search agorithm, we can set to be exponentiay sma, i.e., = 1/ cn for any constant c > 0.

3 418 Mathematics of Operations Research 36(3), pp , 011 INFORMS Baev and Rajaraman [3] study a probem of data pacement in networks. They formaize a minimization version of our probem in which they need to pace objects in caches to minimize the tota connection costs. They give a constant-factor approximation for this probem, which is improved to factor 10 by Swamy [4]. In the concusion of Baev and Rajaraman [3], they suggest considering the probem with bandwidth as an important extension. After the conference version of this paper Feischer et a.[16], Feige and Vondrak [14] improve the approximation factor for GAP from 1 1/e to 1 1/e + for a sma constant. In fact, they use the same LP formuation and proved that the integraity gap of this LP is better than 1 1/e for GAP. Recenty, Vondrak [9] and Cainescu et a. [6] give a more genera 1 1/e-approximation agorithm for a cass of submoduar maximization probems that incudes GAP. This agorithm aso impies a 1 1/e -approximation agorithm for SAP. The distributed caching probem (DCP). Here, we formay define the distributed caching probems that are specia cases of SAP. The genera distributed caching probem that we formaize here is denoted by CapIBDC: CapIBDC: Let U be a set of n cache ocations with given avaiabe capacities A i and given avaiabe bandwidths B i for each cache ocation i. There are k request types; 4 each request type t has a size a t (1 t k). Let H be a set of m requests with a reward R j, a required bandwidth b j, a request type t j for each request j, and a cost c ij for connecting each cache ocation i to each request j. The profit of providing request j by cache ocation i is f ij = R j c ij. A cache ocation i can service a set of requests S i if it satisfies the bandwidth constraint: j S i b j B i, and the capacity constraint: t t j j S i a t A i (this means that the sum of the sizes of the request types of the requests in cache ocation i shoud be ess than or equa to the avaiabe capacity of cache ocation i). We say that a set S i of requests is feasibe for cache ocation i if it satisfies both bandwidth and capacity constraints for bin i. The goa of the CapIBDC probem is to find a feasibe assignment of requests to cache ocations to maximize the tota profit; i.e., the tota reward of requests that are provided minus the connection costs of these requests. We aso consider the foowing variants and specia cases of the CapIBDC probem: Probem without bandwidth constraints (CapDC). The CapDC probem is a specia case of CapIBDC probem without bandwidth constraints. Probem without capacity constraints (IBDC). The IBDC probem is a specia case of CapIBDC probem without capacity constraints. From this definition, one can see that IBDC is a specia case of CapDC where there exists exacty one request of each request type, the avaiabe capacity of cache ocations in CapDC is equa to the avaiabe bandwidth of cache ocations in IBDC, and the size of request types in CapDC corresponds to the bandwidth requirement of requests in IBDC. IBDC is a specia case of GAP. Uniform probems. The uniform CapDC probem is a specia case of the CapDC probem where the size of a request types is the same, i.e., a t = a for a 1 t k. The uniform IBDC probem is a specia case of the IBDC probem where the bandwidth requirement of a requests is the same, i.e., b j = b for a j H. We refer to a variants of the distributed caching probems as DCP. A fractiona variant (CapFBDC). In CapFBDC probem, a cache ocation might satisfy a request fractionay. In other words, a the constraints in CapFBDC are the same except the fact that the assignment of a request to cache ocations can be done fractionay, and the bandwidth constraint hod for the sum of fractiona assignment of requests to a cache ocation, e.g., if haf of a request is assigned to a cache ocation, ony haf of bandwidth consumption of the request is charged to that cache ocation, and thus, ony haf of the bandwidth consumption of the request appears in the bandwidth constraint. However, note that as ong as any fraction of a request is assigned to a cache ocation, the cache ocation needs to carry a copy of that request type, and the capacity of the request type appears competey in the storage constraint. The main reason to consider this setting is that a request may capture the requests for the same fie in a region. In this case, some of these fies might get service from one cache ocation and others from another cache ocation. The profit obtained for servicing a fraction p 0 1 of the request j by the cache ocation i is f ij p. Note that we get this profit even if the tota sum of a fractions served for request j is ess than one. An aternative a-or-nothing variant of the probem, where we get the profit for a request j ony if it is competey served, is not considered in this paper.. LP-based approximation agorithms. In this section, we give a 1 1/e -approximation for SAP and its variants where is the approximation factor of the agorithm for the singe-bin subprobem. The genera approach is to formuate an (exponentia-size) integer program, sove the inear program reaxation approximatey, round the soution to the inear program, and prove that the rounded soution has this guarantee. There 4 Request type can be thought as different fies that shoud be deivered to cients.

4 Mathematics of Operations Research 36(3), pp , 011 INFORMS 419 are two main issues here: proving the goodness of the rounded soution, and obtaining a good soution to the arge inear program in poynomia time. We first discuss the approach in the context of SAP and then discuss some extensions..1. Separabe assignment probems Formuation. We give an exponentia-size integer programming formuation for SAP. Let I i for i U be the set of a feasibe assignments of items to bin i; these are the feasibe soutions to the singe-bin subprobem for bin i. For a set S I i, et Xi S be the indicator variabe that indicates if we choose S as the subset of items for bin i. The first constraint is that we cannot assign more than one set to a bin i, thus for a i U, S I i Xi S = 1. Moreover, we cannot assign each item to more than one bin: i S I i j S Xi S 1. Our objective is to find an assignment of items to bins to maximize the sum of profits, i.e., i S I i fi S Xi S where fi S = j S f ij. By reaxing the 0-1 variabes to nonnegative rea variabes, we obtain the foowing inear programming reaxation: max f S i X S i (1) i U S I i s.t. X S i 1 j H i U S I i j S S I i X S i = 1 X S i 0 Let LP(SAP) denote the objective function vaue of this LP. i U i U S I i Rounding the fractiona soution. Given a soution to the inear program (1), independenty for each bin i, assign set S to i with probabiity Xi S. In the resuting soution, some item j coud be contained in more than one of the sets assigned to the bins. In this case, item j is assigned to the bin among these bins with the maximum f ij -vaue. Note that the resuting assignment after this step is feasibe because the famiy I i for each bin i is ower-idea. Theorem.1. The expected vaue of the rounded soution is at east 1 1 1/n n LP(SAP). Proof. For item j, sort the bins i for which Y i = S I i j S Xi S is nonzero in the nonincreasing order of f ij. Without oss of generaity, assume that these bins are 1 and f 1j f j f j 0. With probabiity Y 1 the set that is assigned to bin 1 contains j, thus item j is assigned to bin 1. In this case, the vaue of item j is f 1j. With probabiity 1 Y 1 Y, bin 1 does not have item j in its subset and bin has item j in its set. In this case, the vaue of item j is f j. Proceeding simiary, we obtain that the expected vaue for request j in the rounded soution is f 1j Y 1 + f j 1 Y 1 Y + + f j 1 i=1 1 Y i Y. The contribution of item j in the vaue of the fractiona soution is 1 i f ij Y i. This in conjunction with Lemma.1 beow yieds the resut. The foowing emma is fokore, and we give a simpe proof of it for competeness. Lemma.1. f 1 Y 1 + f 1 Y 1 Y + + f 1 i=1 1 Y i Y 1 1 1/ 1 i f i Y i whenever Y i 0 for a i and i Y i 1 and f 1 f f 0. Proof. From the arithmetic/geometric mean inequaity, one can derive (see Lemma 3.1 in Goemans and Wiiamson [19]): ( k 1 ) ( k ) Y Y 1 Y Y i Y k = 1 1 Y i i=1 i=1 ( 1 ( 1 1 k ) k ) k for any k. Mutipying this inequaity by f k f k+1 0 where f +1 = 0 and using the monotonicity of 1 1 1/k k, we get ( k 1 ( k 1 ) f k Y 1 + f k 1 Y 1 Y + + f k 1 Y i )Y k f k+1 Y 1 f k+1 1 Y 1 Y f k+1 1 Y i Y k i=1 i=1 Y i i=1 ( ( ) ) k f k Y i f k+1 Y i i=1

5 40 Mathematics of Operations Research 36(3), pp , 011 INFORMS Summing these inequaities for 1 k where f +1 = 0, the eft-hand side of the inequaity becomes f 1 Y 1 + f 1 Y 1 Y + + f 1 i=1 1 Y i Y and the right-hand side of the inequaity becomes 1 1 1/ i=1 f iy i. This proves the desired inequaity of the emma. Soving the LP. The number of variabes in the inear program (1) is exponentia. To sove this LP, we first sove its dua (), given beow: min q i + j () i U j H s.t. q i + j f S i j S j 0 j H i U S I i The dua inear program () has a poynomia number of variabes but exponentiay many constraints. We rewrite it as a fractiona covering probem as foows: min q i + j (3) i U j H s.t. q i i j 0 j H i U Here, i is the poytope defined by constraints of the form q i j S f ij j for a S I i. To sove the LP, we wi need a separation agorithm for i. We define an -approximate separation agorithm for poytope i to be an agorithm that given a point q i j j H either returns a vioated constraint or guarantees that q i / j j H is feasibe for i. We et LP(Dua SAP) denote the objective function vaue of the inear program (). In the foowing emma we show that given an approximate separation orace for the dua of the packing program, we can get an approximate soution for the prima program using the eipsoid agorithm. This fact for a cass of packing-covering inear programs is aso observed by Carr and Vempaa [7] and by Jain et a. [1]. For competeness, we give our proof of this fact. Lemma.. For any > 0, given a poynomia-time -approximate separation agorithm for i, we can design a poynomia-time -approximation agorithm to sove the inear program () and hence the inear program (1). Proof. We run the eipsoid agorithm on the inear program () using a -approximate separation agorithm. More precisey, we move the objective into the set of constraints by adding the constraint i U qi + j H j v to the current inear program. For a given v, we use the eipsoid agorithm to determine if this LP is feasibe and use binary search to find the smaest feasibe vaue v. Using the eipsoid agorithm in this binary search framework with a -approximate separation agorithm, suppose that the process of the agorithm terminates with a soution qi j j H such that v = i U qi + j H j. Thus, we know that the inear program () with the new constraint is infeasibe for v where depends on the precision of the binary search. 5 Thus, the optima soution to the LP () is at east v. Because we use a -approximate separation agorithm, we are not guaranteed that this soution is feasibe. However, we know that qi / j j H is feasibe. Thus, the optima soution to the LP () is at most v / ; thus the optima soution to () is between v and v /. In the execution of the eipsoid agorithm for v, we check a poynomia number of constraints. This set of constraints is enough to show that the vaue of the dua is greater than v. The dua of this restricted LP is equivaent to the inear program (1) restricted to the variabes corresponding to this set of constraints (by setting a other variabes to zero). By LP-duaity, the cost of the soution to this program is at east v. Thus, the soution to this poynomia-sized LP has vaue at east v. By setting sufficienty sma, this is an -approximation agorithm for the prima inear program because LP(SAP) LP(Dua SAP) v /. An approximate soution to the inear programs (1) and () can aso be obtained via Lagrangian LP agorithms (Garg and Könemann [17], Potkin et a. [5], Young [30, 31]). We can use a -approximation agorithm for the singe-bin subprobem for bin i to design a -approximate separation agorithm for i. The -approximate separation agorithm for i asks, given q i j j H, find a 5 We can set to be 1/ cn for any constant c > 0.

6 Mathematics of Operations Research 36(3), pp , 011 INFORMS 41 set S I i such that q i < j S f ij j. It is sufficient to find the set S I i that maximizes j S f ij j. Because I i is ower-idea, we know that if q i < 0, then the set S = vioates the above inequaity. Moreover, we can consider ony items j for which f ij j is positive. In fact, we can set max 0 f ij j as the vaue of item j in the singe-bin subprobem and use a -approximation agorithm for the singe-bin subprobem, to find a subset S I i with vaue qi such that for any set S I i, qi = j S f ij j j S f ij j. We know that either qi > q i in which case we find a vioated constraint, or qi q i. In the atter case, we know that for any subset S I i, j S f ij j qi / q i/. Therefore, in this case q i / j j U is feasibe for i. Hence, a -approximation agorithm for the singe-bin subprobem is a -approximate separation agorithm for i. The above and previous discussion yied a guarantee of 1 1 1/n n. Because can be chosen to be exponentiay sma (it determines the parameters of a binary search) and we have that 1 1 1/n n 1 1/e + 1/ 3n, we can remove the dependence on and obtain the foowing genera resut. 6 Theorem.. Given a poynomia-time -approximation agorithm for the singe-bin subprobem, there exists a poynomia-time 1 1/e -approximation agorithm for SAP. Soving the singe-bin subprobem. In this section, we show that the singe-bin probem for each probem cass in SAP discussed in the introduction has an approximation scheme. Thus, for a probem casses, this yieds poynomia-time 1 1/e -approximation agorithms. GAP: the singe-bin subprobem is a knapsack probem, for which an efficient FPTAS is we known. CapDC: the singe-bin subprobem is to pack request types into the bin (respecting the bin capacity) to maximize the vaue of the items that can then be assigned to the bin. An item j can be assigned ony if the corresponding type t j is assigned. This is a knapsack probem and thus has an FPTAS. CapIBDC: the singe-bin subprobem is the foowing two-dimensiona knapsack probem: bin i has A i avaiabe space and B i avaiabe bandwidth. Item j S has vaue v j, type t j, and bandwidth consumption r j. Each type t has size s t. A feasibe packing of items and types into bin i, satisfies tota bandwidth of items in bin i is at most B i, the tota size of request types is at most A i, and an item j is packed ony if type t j is aso packed. The goa is to to maximize the tota vaue of items packed in a bins. Shachnai and Tamir [6] describe a PTAS for this probem. In appendix, we design a simper PTAS which aso handes a fractiona variant, CapFBDC. Theorem.3. There exists a poynomia-time 1 1/e -approximation agorithm for GAP, CapDC, and CapIBDC for any > 0. Furthermore, if the subprobem admits an FPTAS as in GAP or CapDC, we can get rid of the by using the same argument as before Theorem. (and running the FPTAS with a guarantee of 1 1/ 3n ). This was brought to our attention by Raphae Yuster. Thus we have Theorem.4. Theorem.4. There exists a poynomia-time 1 1/e -approximation agorithm for GAP and CapDC... Extensions of SAP. The above genera framework can be extended very sighty to aso capture other casses of probems. Here we describe two extensions. The first adds a two-sided packing constraint: we are not ony packing items in bins, but the items are in casses, and there is a imit, in terms of a budget constraint, on the set of items that can be packed in each cass. The second extends DCP to fractiona probems. The issue in this ast case is how to sove the LP, because the number of feasibe packings is no onger finite, and thus the straightforward modification of (1) yieds a inear program with an infinite number of variabes. The AdWords Assignment Probem (AAP) is as foows: we are given a set U of n bidders with a imited budget B i for each bidder i. Each bidder i aso has an ad, i, to advertise. Ad i is a rectange of width w i and height i. We are aso given a set H of m AdWords. For each AdWord j, we can advertise the ads of a subset of bidders in a rectanguar area of width W j and height L j. The rectanges for ads may not overap and may not be rotated, thus a set T j of ads (or bidders) can be assigned to AdWord j iff the corresponding rectanges of ads in T j can be packed in the rectange area for AdWord j. For each AdWord j and each bidder i, we are given a vaue v ij B i that is the maximum vaue that bidder i is wiing to pay for AdWord j. This vaue might capture the cick-through-rate of ad i for AdWord j. For an assignment of AdWords to bidders (or ads), if we assign the set S i of AdWords to bidder i, the revenue from bidder i is equa to min B i j S i v ij. We need to find an assignment of AdWords to bidders, and a packing the rectange ads of those bidders assigned to AdWord j 6 The observation that one has enough room between 1 1/e and 1 1 1/n n was pointed out to us by Raphae Yuster (simiary to Proposition.1 in Nutov et a. [5]); see aso the improvement in the forthcoming Theorem.4.

7 4 Mathematics of Operations Research 36(3), pp , 011 INFORMS inside rectange area of AdWord j. Our goa is to find an assignment that maximizes the tota revenue, i.e., the sum of revenue from a bidders. In reated work (Andeman and Mansour []), a 1 1/e -approximation agorithm is given for budgetconstrained auctions. Budget-constrained auctions (Garg et a. [18], Andeman and Mansour [], Chakrabarty and Goe [8]) are specia cases of AAP where we can assign each AdWord to at most one bidder. AAP cannot be formaized as a separabe assignment probem. However, it can be formuated as an exponentia-size program simiar to the inear program that we used for SAP. Let j be the famiy of the feasibe sets of bidders for the AdWord j. In the inear programing formuation, for each AdWord j and each subset T j, there exists a variabe Yj T representing the amount of set T of bidders that is assigned to an AdWord j; e.g., Yj T = 1 represents the assignment of set T of bidders to AdWord j. Because one subset of bidders is assigned to each AdWord j in the inear programming reaxation, the tota amount of subsets T assigned to any AdWord j is at most 1. In the inear program, for each AdWord j and each bidder i, there exists a variabe 0 z ij 1 representing the amount that bidder i is assigned to AdWord j. In a soution of AAP, z ij > 0 represents the assignment of AdWord j to bidder i. 7 Considering the budget constraints and feasibiity of the soution in the inear programming reaxation, we get the foowing are the prima and dua inear programming reaxation of the AAP probem. max i U j H s.t. z ij T j Y T v ij z ij (4) T j i T Y T j j = 1 j H v ij z ij B i j H i U j H i U Y T j 0 j H T j z ij 0 i U j H min B i q i + j (5) i U j H s.t. j y ij 0 i T v ij q i + y ij v ij y ij 0 q i 0 i U j H i U j H T j j H i U Simiar to soving inear program (1), and using a -approximation poynomia-time agorithm for the rectange packing probem for AdWord j, we can sove the prima inear program (4) within a factor in poynomia time. The separation orace of this probem is the rectange packing probem or the geometric two-dimensiona knapsack probem for which the best known agorithm is a 1/ -approximation agorithm by Jansen and Zhang []. Obtaining a PTAS for this probem is sti an open question. Given a fractiona soution, we round the soution as foows: for each AdWord j, we assign exacty one set of bidders to j and we assign a set T of bidders to j with probabiity Yj T. In the foowing, we prove that in the rounding process we do not ose more than a factor 1 1/e. Before stating the main emma of this section, we state a known emma whose proof is inside the proof of Theorem 4 of Andeman and Mansour []. In stating the foowing emma, we foow the notation from Andeman and Mansour []. Lemma.3 (Theorem 4 Andeman and Mansour []). Consider a set of numbers b i j B i for a j, and a set of probabiity vaues x ij for i I and j H. Aso consider a set of independent random variabes X ij where X ij is 1 with probabiity x ij and 0 otherwise. Let B i = j H x ij b ij, then assuming B i B i we have E min B i j H X ij b ij 1 1/e B i. 7 In a soution of AAP, for any bidder i, among AdWords j for which z ij > 0, z ij = 1 except for one fractiona variabe z ij which resuts from hitting the budget B i of bidder i. However, in the inear programming reaxation we reax this constraint to 0 z ij.

8 Mathematics of Operations Research 36(3), pp , 011 INFORMS 43 Lemma.4. Let B i be the revenue of bidder i in the fractiona soution and E i be the expected revenue from bidder i in the rounded soution. Then E i B i 1 1/e. Proof. As the objective function is i U j H v ij z ij = i U B i, we have B i = j H v ij z ij. Here, we descibe how to use Lemma.3 in our setting. Let X ij be the indicator random variabe that indicates if AdWord j is assigned to bidder i in the rounded soution. Based on our rounding method, the probabiity of assigning a set T of bidders to j with probabiity Yj T, and thus the probabiity of assigning bidder i to j is at east z ij, i.e., we get Pr X ij = 1 z ij. As a resut, the revenue from bidder i in the rounded soution is j H X ij v ij if this sum is ess than B i ; otherwise, the revenue from i is B i and so the expected revenue from bidder i is E i = E min B i j H X ij v ij. As a resut, by substituting z ij instead of x ij, v ij s instead of b ij s, B i instead of B i, and E i instead of E min B i j H X ij v ij, Lemma.3 impies that E i B i 1 1/e as desired. The above emma aong with the 1/ -approximation agorithm for the rectange packing probem (Jansen and Zhang []) give a 0 31-approximation agorithm for AAP. If the width of the rectanges of ads and the rectanguar area for advertisement (W j s) are the same, then the packing probem for each AdWord is a knapsack probem. Therefore, our agorithm is a 0 63-approximation agorithm for this specia case of AAP. Fractiona DCP. For the fractiona DCP (i.e., the CapFBDC probem as defined earier), we coud consider I i as the famiy of fractiona subsets of requests that are feasibe for cache ocation i and write simiar inear programs, but the number of variabes (or constrains) of these inear programs is infinite. Instead, we modify the LP formuation. Let i be the set of a vectors in 0 1 k such that for Z i i, 1 t k Z i t a t A i. For a vector Z i i, et Z i be a poytope defined as foows: Z i = y i R m j H y i j b j B i y i j Z i t j 0 y i j 1. Let S Z i be the set of extreme points of poytope Z i and et i = Z i i S Z i. For simpicity et i = y i 1 y i p i where p i = i. Every point y i i can be described as a inear combination of y i R m 1 p i, thus y i = p i =1 i y i. Each fractiona assignment of requests in CapFBDC to cache ocation i U corresponds to some some y i i, and therefore it can be represented as a inear combination y i = p i =1 i y i. Actuay, a y i with i > 0 must beong to the same poytop Z i for some Z i i. We reax that condition. In the inear reaxation we define beow, we aow to mix extreme points from different poytopes Z i in the same inear combination. Formay, we consider the inear programming reaxation of CapFBDC: maximize i U subject to =1 p i j H =1 f ij y i j i (6) p i i = 1 i U p i i U =1 y i j i 1 j H i 0 i U 1 p i Note that this way of deriving LP reaxations is a bit unconventiona. Usuay, we first formuate an integer programming formuation of the probem and reax the integraity condition afterwards. Here, the integraity condition corresponds to the constraint that we must use extreme points from the same poytop Z i in the inear combination. The dua of the inear program (6) is as foows: minimize q i + w j (7) i U j H subject to q i + y i j w j y i j f i j j H j H i U 1 p i w j 0 j H To find an approximate separation orace, given a set of variabes w j q i i U j H, we need to check if there is a vioated constraint in the inear program (7). Fixing a cache ocation i, we need to check if there exists y i for which j U y i j f ij w j is greater than q i. By setting v j = f i j w j as the vaue of request j, we need to check if there exists a point y i Z i i S i Z i such that j H v j y i j is greater than q i or

9 44 Mathematics of Operations Research 36(3), pp , 011 INFORMS not. Thus, the separation orace is to maximize i U y i j v j for y i Z i i S Z i. We coud reax this condition to y i Z i i Z i. This corresponds to the foowing two-dimensiona knapsack probem. We are given a bin with A i avaiabe space and B i avaiabe bandwidth. We are aso given a set S of items with a vaue v j = f i j w j, a type t j, and a bandwidth consumption b j. For each type t, we are given a size a t. We want to pack a set of items with the most vaue. We may pack items fractionay. The bandwidth consumption of these items can be at most B i and the sum of sizes of request types in this set is at most A i. We can easiy extend the PTAS for the separation orace of CapIBDC to give a PTAS for the separation orace for CapFBDC. A description of this PTAS is given in the appendix. Rounding the fractiona soution. Given a soution to the inear program (6), independenty for each bin i, choose one of the extreme points from the set i according to the probabiity distribution defined by variabes i. After that, we have an assignment of request types to the bins and fractiona packing of requests in each bin. In the resuting soution, for a request j et G j be the tota sum of a fractions appearing in different bins. Choose min 1 G j fractions for item j with highest f ij -vaue. Note that in this procedure, some fraction might be partitioned into two pieces and ony one of the pieces be taken to the fina soution. Theorem.5. > 0. There exists a poynomia-time 1 1/e -approximation agorithm for CapFBDC for any Proof. First we define the quantity E k n j for k = 1 n and 0 1. Given a soution to the inear program (6), independenty for each bin i, choose one of the extreme points from the set i according to the probabiity distribution defined by variabes i. After that we have an assignment of request types to the bins and fractiona packing of requests in each bin. In the resuting soution, for a request j et G j be the tota sum of a fractions appearing in bins k n. Choose min G j fractions for item j with highest f ij -vaue, i.e., we basicay consider our randomized rounding agorithm for the bins k n and we serve ony fraction of the request j. For simpicity assume that bins are ordered such that f 1j f j f nj. For item j and the bins i, et R ij = p i =1 y i j i be the tota fraction of service obtained from the bin i for item j in our fractiona soution. Obviousy, i U R ij 1. Using the definitions above we need to estimate E 1 n 1 j from beow. We caim that n 1 E 1 n 1 j f 1j R 1j + f j 1 R 1j R j + + f nj 1 R ij R nj (8) The inequaity (8) in conjunction with Lemma.1 impy the statement of the theorem. We now prove the inequaity (8). We caim that E k n j E k n 1 j for a 0 1. This foows from the way we defined E k n j and E k n 1 j, in both cases we have the same random process but in the first case we choose fraction of the most expensive service fractions and in the second we choose the most expensive fractions that sum up to one. Therefore, p k E k n 1 j = =1 p k =1 k k f kj y k j f kj y k j i=1 + E k + 1 n 1 y k j j + 1 y k j E k + 1 n 1 j = f kj R kj + 1 R kj E k + 1 n 1 j Noticing that E n n 1 j = f nj R nj we derive the inequaity (8). 3. Loca search agorithms. In this section, for any > 0, we give a simpe oca search / + 1 approximation agorithm for separabe assignment probems given a -approximation agorithm for the singe-bin subprobem. This, in turn, gives a combinatoria 1 -approximation agorithm for GAP and a variants of DCP. We aso show how to extend this agorithm to give a 1 -approximation agorithm for the maximum k-median probem with hard capacities and packing constraints. Fisher et a. [15] describe both a simpe greedy agorithm and a oca search agorithm that both yied 1 -approximation guarantees for the probem of maximizing a submoduar function subject to the constraint that the set is an independent set of a matroid, which we ca submoduar function maximization over matroids (SFMM). Chekuri [9] gives a reduction (beow) from SAP to SFMM, which impies that the agorithms in Fisher [15] yied 1 -approximation guarantees for SAP when = 1.

10 Mathematics of Operations Research 36(3), pp , 011 INFORMS 45 Reduction to SFMM. Let I i be the set of a feasibe packings of bin i and et I = i I i (with repeated copies for sets feasibe for different bins), so that each eement S I corresponds to a set of items that can be packed in some bin. A feasibe soution for SAP is a set of sets A I such that A I i 1 for a i. The goa is to maximize f A where f A is the profit obtained by the sets in A. Define this as j H max S A f S j, where f S j = f ij if j S and S I i and f S j = 0 otherwise. It can be verified that f A is a submoduar set function over I. The constraint that one picks at most one eement of I from each I i is a partition matroid constraint Our oca search agorithm. We first give a naive oca search / + 1 -approximation agorithm whose running time might be exponentia. Then, we refine the agorithm and change it to a poynomia-time agorithm. Let = S 1 S n be an assignment of items to bins, where S i is the set of items in bin i. For an assignment = S 1 S n of items to bins, we denote the vaue of this assignment by v. Aso, et i be the tota vaue of items satisfied by bin i in. For an item j, et v j the vaue of item j in. The naive agorithm repeatedy iterates over the bins. For bin i, it runs procedure Loca i. Loca i, given current soution, finds a repacking S i of bin i. If repacing S i with S i improves the soution then this repacement is made. When no further improvements can be made on any bin, the agorithm hats. We ca the resut an -approximate oca optima soution. Specificay, Loca i does the foowing: (i) For each item j, et vaue j be equa to f i j if j is assigned to a bin i i in, and be equa to zero if j is unassigned or is assigned to bin i in. (ii) For each item j, et the margina vaue of j be w j = f ij vaue j. (iii) Use the -approximation agorithm for the singe-bin subprobem for bin i to pack a subset of items in bin i with the maximum margina vaue. Lemma 3.1. Let = S 1 S n be a -approximate oca optima soution and OPT the vaue of the optimum assignment. Then v OPT/ + 1. Proof. Let R be the set of items that are better served in the optima soution than in and L be the rest of the items. Let R i be the set of items in R served by bin i in the optimum. For any set T of items, et o T be the vaue of the items in T in and T be the vaue of items of T in. For a i and a items j R i, the margina vaue for bin i is positive (w j > 0), because the vaue of j for bin i is greater than the current vaue of j in. For each i and each item j R i, the margina vaue of j for soution is f ij vaue j f ij v j. Thus, the tota margina vaue of items in set R i for bin i is at east j R i f ij v j = o R i R i. Because is a -approximate oca soution, the operation Loca i cannot find a soution with margina vaue greater than i ; otherwise, this operation coud increase the tota vaue. Because we use a -approximation agorithm for the singe-bin subprobem and there exists a soution with margina vaue o R i R i, it foows that i o R i R i. Therefore, o R i R i + 1/s i. Furthermore, for items in set L, o L L by definition. Therefore, OPT = v = o L + o R = o L + i U o R i L + ( R i + 1 ) i U i ( ) v Thus v / + 1 v. Lemma 3.1 shows that if we can find an -approximate oca soution, then we have a / +1 -approximation agorithm. Mirrokni [3] showed PLS-competeness of a distributed caching game. As far as we can see a simiar PLS-competeness proof works in our setting, too. An impication of the PLS-hardness of this probem is that there exists a set of instances for which the above oca search agorithm coud take exponentia time to converge to a oca optima soution. Beow, we modify the naive agorithm to get a poynomia-time / + 1 approximation agorithm. The anaysis of this agorithm uses the foowing fact (which we prove in the proof of Theorem 3.1). Using this fact, we show that after a poynomia number of oca improvements the vaue of the soution is a good approximate soution. Proposition 3.1. If v / + 1 OPT then there is a bin i for which Loca i finds a packing with increase in vaue at east /n OPT 1 + /n v.

11 46 Mathematics of Operations Research 36(3), pp , 011 INFORMS Loca search agorithm (i) Start with the empty soution, i.e., = S 1 S n and S i = for a i U. (ii) For an appropriate > 0, run the foowing oop for 1/ n n 1/ times: (a) Let the current assignment be = S 1 S n. (b) For each bin i, run Loca i. Let the margina vaue of this soution for bin i be W i, and et S i be the set of items with margina vaue W i. (c) For each bin i, et i = W i i. (d) Let bin i be the bin with the maximum i, i.e., i i for any bin i. (e) If i > 0, change the set S i of items for bin i to S i. For any > 0, the above oca search agorithm is a poynomia-time / Theorem 3.1. approximation. Proof. Let be an optima assignment, and et be an intermediate assignment obtained in the oca search agorithm. Let R be the set of items that are better served in than in and L be the rest of the items. Let R i be the set of items in R satisfied by bin i in. For any set T of items, et o T be the vaue of the items in T in assignment and T be the vaue of items of T in assignment. Thus, we have OPT = o R + o L and v = R + L. For each item j R i, the margina vaue of j is f ij vaue j f ij v j. Thus, the tota margina vaue of items in set R i for bin i is at east j R i f ij v j = o R i R i and R i is a feasibe soution for bin i. Because we use a -approximation agorithm to find W i, W i o R i R i. Therefore, i U W i i U o R i R i = o R R. Because o L L, i U W i o R R + o L L = OPT v. Thus i = W i i S i U i U i U OPT v v = OPT 1 + v In particuar, i /n OPT 1 + /n v. Let be the assignment after changing the set of items of i to S i, i.e., = S 1 \S i S \S i S i 1\S i S i S i +1\S i S n\s i. As a resut, v = v + i v OPT n n v ( = v ) + n n OPT Let y k be the tota vaue of the assignment after the kth execution of Step. From the above discussion, y k /n y k 1 + /n OPT and y 0 = 0. Using induction, we get that for any 1 i k y k / /n k OPT. By setting k = n n 1/ / + 1, we get y k / /e n 1/ OPT = / OPT. Therefore, for = 1 + /, the vaue of the output of the above agorithm is at east / 1 + OPT as desired. 3.. Maximum k-median with hard capacities and packing constraints. We can extend the oca search agorithm for SAP to the maximum k-median probem with hard capacities and packing constraints (KMed). The KMed is as foows: Given a set U of n bins, a set H of m items with a vaue f ij for each item j and each bin i, and aso a singe-bin subprobem for each bin i, i.e., a ower-idea famiy I i of subsets for bin i, choose at most K bins and pack a set of items in each seected bin to maximize the tota vaue packed. The oca search agorithm for the KMed probem is very simiar to the oca search agorithm for SAP. At each step of the agorithm, we try to unpack a used bin, and pack a (possiby different) bin to increase the tota vaue. The forma description of the agorithm is as foows: Loca search agorithm for KMed (i) Start with the empty soution, i.e., = S 1 S n and S i = for a i U. (ii) Let P = 1 K. (iii) For an appropriate, run the foowing oop for 1/ K n 1/ times: (a) Let the current assignment be = S 1 S n where S i = for i P.

12 Mathematics of Operations Research 36(3), pp , 011 INFORMS 47 (b) For each bin i 1 P and bin i U P i 1 do i. For each item j, et vaue j be f i j if j is assigned to a bin i i 1 in, and be equa to zero if j is unassigned or is assigned to bin i 1 in. ii. For each item j, et the margina vaue of j (with respect to bins i 1 and i ) be w j = f i j vaue j. iii. Use the -approximation agorithm for the singe-bin subprobem for bin i to pack a subset of items in bin i with the maximum margina vaue. Let the margina vaue of this soution for bin i be W i1 i, and et S i 1 i be the set of items with margina vaue W i1 i. (c) For every two bins i 1 P and i U P i 1, et i1 i = W i1 i i1. (d) Let bins i1 and i be the bins with the maximum i 1 i, i.e., i 1 i i 1 i for any bin i 1 and i. (e) If i 1 i > 0, unpack bin i 1 and pack the set S i of items in bin i 1 i (i.e., set P = P i 1 i and set S i 1 = and S i = S i1 ). i Theorem 3.. For any > 0, the above oca search agorithm for KMed is a poynomia-time / + 1 -approximation agorithm. Proof. The output of the oca search agorithm is a feasibe soution of the KMed probem, because the number of nonempty bins in the output is at most K. Let be an optima assignment. Consider the current assignment in the process of the agorithm. Let P and P be the set of used bins in and, respectivey. The set of used bins in and are not necessariy the same. Let U U be a permutation function such that if i U is used in both and, then i = i and if i is used in but not in, then i = i where i is used in but not in. As the number of used bins in is K, such a permutation exists. Let R be the set of items that are better served in than in and L be the rest of the items. Let R i be the set of items in R satisfied by bin i in. For any set T of items, et o T be the vaue of the items in T in and T be the vaue of items of T in. Thus, we have OPT = o R + o L and v = R + L. For an item j, et v j the vaue of item j in. Consider two bins i and i P. For each item j R i, the margina vaue of j with respect to bins i and i is f ij vaue j f ij v j. Thus the margina vaue of set R i with respect to bins i and i is at east i R i f ij v j = o R i R i and R i is a feasibe soution for bin i. Because we use a -approximation agorithm to find W i i, W i i o R i R i. Therefore, i P W i i i P o R i R i = o R R. Since o L L, i P W i i o R R + o L L = OPT v. Thus i P i i = i P W i i i P i S OPT v v = OPT 1 + v. In particuar, i 1 i /K OPT 1 + /K v. Let be the assignment after setting S i 1 to and changing the set of items of i to S i As a resut, 1. i v = v + i 1 i v OPT K K v ( = v ) + K K OPT Now, et y k be the tota vaue of the assignment after the kth execution of Loop (iii)b. From the above discussion, y k /K y k 1 + /K OPT and y 0 = 0. Simiar to the proof of Theorem 3.1 and soving this recurrence reation, we get y k / /K k OPT. By setting k = K n 1/ / + 1, we get y k / /e n 1/ OPT = / OPT. Therefore, the vaue of the output of the above agorithm is at east / 1 + OPT as desired. Finay, we note that a modified variant of the reduction given from GAP to SFMM in 3 aso gives an aternative 1 -approximation agorithm for KMed (Cainescu et a. [6]). 4. A hardness resut. In this section, we show a hardness resut for the CapDC probem and specia cases of SAP. We prove that the CapDC probem is not approximabe better than a factor of 1 1/e uness NP DTIME n O og og n showing that the 1 1/e-approximation agorithm for CapDC is tight. This hardness resut uses a hardness resut by Feige et a. [1] for the domatic number of graphs. The domatic number of a graph is the maximum number of disjoint dominating sets in the graph. A subset S of vertices of a graph G V E is a dominating set if for any vertex v S, there exists a vertex u S which is connected to v, i.e., u v E G. We first define a set of probems that are used in the reduction and restate the resut of Feige et a. [1]. The Max 3-coorabiity probem is as foows: Given a graph G = V E coor the vertices of G with 3 coors to maximize the number of egay coored edges (edges whose endpoints are coored differenty). The Max 3-coorabiity-5 probem is the Max 3-coorabiity probem for 5-reguar graphs. Petrank [4] proved that the

13 48 Mathematics of Operations Research 36(3), pp , 011 INFORMS Max 3-coorabiity probem is APX-hard. Using this proof, Feige et a. [1] proved that the Max 3-coorabiity-5 probem is APX-hard. Formay, they showed that for some number < 1, it is NP-hard to distinguish between 5-reguar graphs that have a ega 3-cooring, and 5-reguar graphs in which every 3-cooring egay coors at most fraction of the edges. The foowing caim is impicit in the hardness resut of Feige et a. [1]: Given an instance G = V E of the Max 3-coorabiity-5 probem, we can construct an instance of a set cover probem with m = O V O og og V E O og og V eements and n = O V O og og V E O og og V sets of size m/p where m/p = O V O og og V E O og og V such that: (i) If the vertices of graph G are (egay) 3-coorabe, then there exist n/p disjoint set covers, each with p sets 8 in the set cover instance; and (ii) If any 3-cooring of G has ess than E egay coored edges then any coection of p sets cover at most 1 1 1/p p m eements. 9 From an instance of the set cover probem with n sets and m eements, we construct an instance of CapDC with n/p types, m n/p requests, and n cache ocations as foows: For each eement j in the ground set of the set cover, we put n/p requests j 1 j j n/p of different types in the CapDC instance. For each set i in the set cover instance, we put a cache ocation i in the CapDC instance. The capacity A i for cache ocation i is A i = 1 and the size of each request type is equa to 1. Thus we can ocate at most one type in each cache ocation. The profit of assigning request j e to cache ocation i is 1 if the corresponding eement j is in the corresponding set in the set cover instance. If the set cover instance has n/p disjoint set covers then in the instance of the CapDC probem, we can satisfy a requests of a particuar type using one set cover, and thus we can find a soution to the instance of the CapDC probem with a tota profit of mn/p. Moreover, we caim that if any coection of p sets in the set cover probem cover at most 1 1 1/p p m of eements, then the profit of any assignment to the CapDC probem is at most 1 1 1/p p mn/p. Assume that in the set cover instance, any coection of p sets cover at most 1 1 1/p p m of the eements. Consider a soution with the maximum profit for the CapDC probem. For 1 t n/p, et t p be the number of cache ocations that keep the request type t in soution. We know that n/p t=1 t p = n. Aso from the inequaity 1 1 1/p xp /p tp 1 1 1/p yp /p zp where x y z t and x + t = y + z, it foows that the profit of maximizes when a t for 1 t n/p are the same, i.e., t = 1. By setting t = 1, we have that the profit of is at most n/p t= /p tp m n/p t= /p p m 1 1 1/p p mn/p. Therefore, if we appy the Feige et a. reduction from Max 3-coorabiity-5 to the set cover and the above reduction from the set cover to the CapDC probem, we have the foowing: Given an instance of Max 3-coorabiity-5 probem, we can construct an instance of the CapDC probem with n/p types, m n/p requests, and n cache ocations such that: (i) If vertices of graph G are (egay) 3-coorabe, then there exists a soution with profit mn/p for the CapDC instance; and (ii) If any 3-cooring of G has ess than E egay coored edges, the maximum possibe profit of the CapDC instance is at most 1 1 1/p p of the number of requests, i.e., 1 1 1/p p mn/p. Note that 1 1 1/p p tends to 1 1/e as p tends to. Therefore, for any > 0, there exists a sufficienty arge p such that 1 1 1/p p 1 1/e +. Hence, for any > 0, for any sufficienty arge instance of Max 3-coorabiity-5 probem, we can construct an instance of the CapDC probem with n/p types, m n/p requests, and n cache ocations such that: If vertices of graph G are (egay) 3-coorabe, then there exists a soution with profit mn/p for the CapDC instance. If any 3-cooring of G has ess than E egay coored edges, the maximum possibe profit of the CapDC instance is at most 1 1/e + of the number of requests, i.e., 1 1/e + mn/p. This shows that for any > 0, if we can approximate the CapDC probem within a factor better than 1 1/e +, then we can distinguish between the aforementioned cases of the sufficienty arge instances of the Max 3-coorabiity-5 probem in time O V O og og V E O og og V. Because distinguishing between these two cases of the Max 3-coorabiity-5 probem is NP-hard, if we can approximate the CapDC probem in poynomia time within a factor of 1 1/e + for <, then NP DTIME n O og og n. Note that in the above reduction, we used 8 See Lemma 17 of Feige et a. [1]. In fact, Feige et a. [1] present their resut in terms of the dominating set and the domatic number probem. We restate their resut for the set cover probem. 9 The proof of this caim comes from the proof of Lemma 18 of Feige et a. [1], in which authors refer to the hardness resut for the set cover probem by Feige (e.g., see Proposition 4.3 of Feige [11]). Essentiay, the proof of our caim comes from the fact that for the construction in Feige [11] appied to our probem, any coection of p sets cover at most 1 1 1/p p m eements Feige and Kortsarz [13]. The reason is that the number of eements that p sets cover is at most the expected number of eements that p random sets of size m/p each cover where a random set is a set in which each eement is picked uniformy at random and independent of other eements. We aso note that p is not a constant. In particuar, as V tends to, p aso tends to.