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38
Submodular function maximization via the multilinear relaxation and contention resolution schemes
 IN ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2011
"... We consider the problem of maximizing a nonnegative submodular set function f: 2 N → R+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that all ..."
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Cited by 40 (2 self)
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We consider the problem of maximizing a nonnegative submodular set function f: 2 N → R+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular when f may be a nonmonotone function. Our algorithms are based on (approximately) solving the multilinear extension F of f [5] over a polytope P that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully in some settings [6, 22, 24, 13, 3], it has been limited in some important ways. We overcome these limitations as follows. First, we give constant factor approximation algorithms to maximize
Approximating the kMulticut Problem
"... We study the kmulticut problem: Given an edgeweighted undirected graph, a set of l pairs of vertices, and a target k ≤ l, find the minimum cost set of edges whose removal disconnects at least k pairs. This generalizes the well known multicut problem, where k = l. We show that the kmulticut problem ..."
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Cited by 20 (1 self)
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We study the kmulticut problem: Given an edgeweighted undirected graph, a set of l pairs of vertices, and a target k ≤ l, find the minimum cost set of edges whose removal disconnects at least k pairs. This generalizes the well known multicut problem, where k = l. We show that the kmulticut problem on trees can be approximated within a factor of 8 3 + ɛ, for any fixed ɛ> 0, and within O(log 2 n log log n) on general graphs, where n is the number of vertices in the graph. For any fixed ɛ> 0, we also obtain a polynomial time algorithm for kmulticut on trees which returns a solution of cost at most (2 + ɛ) · OP T, that separates at least (1 − ɛ) · k pairs, where OP T is the cost of the optimal solution separating k pairs. Our techniques also give a simple 2approximation algorithm for the multicut problem on trees using total unimodularity, matching the best known algorithm [8].
Approximation algorithms for singleminded envyfree profitmaximization problems with limited supply
 FOCS
"... We present the first polynomialtime approximation algorithms for singleminded envyfree profitmaximization problems [13] with limited supply. Our algorithms return a pricing scheme and a subset of customers that are designated the winners, which satisfy the envyfreeness constraint, whereas in ou ..."
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We present the first polynomialtime approximation algorithms for singleminded envyfree profitmaximization problems [13] with limited supply. Our algorithms return a pricing scheme and a subset of customers that are designated the winners, which satisfy the envyfreeness constraint, whereas in our analyses, we compare the profit of our solution against the optimal value of the corresponding socialwelfaremaximization (SWM) problem of finding a winnerset with maximum total value. Our algorithms take any LPbased αapproximation algorithm for the corresponding SWM problem as input and return a solution that achieves profit at least OPT /O(α · log umax), where OPT is the optimal value of the SWM problem, and umax is the maximum supply of an item. This immediately yields approximation guarantees of O ( √ m log umax) for the general singleminded envyfree problem; and O(log umax) for the tollbooth and highway problems [13], and the graphvertex pricing problem [3] (α = O(1) for all the corresponding SWM problems). Since OPT is an upper bound on the maximum profit achievable by any solution (i.e., irrespective of whether the solution satisfies the envyfreeness constraint), our results directly carry over to the nonenvyfree versions of these problems too. Our result also thus (constructively) establishes an upper bound of O(α · log umax) on the ratio of (i) the optimum value of the profitmaximization problem and OPT; and (ii) the optimum profit achievable with and without the constraint of envyfreeness. 1.
On kColumn Sparse Packing Programs
, 2009
"... We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an ek+o(k)approximation algorithm for kcolumn sparse PIPs, improving on recent results of k2 · 2k [14] and O(k ..."
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We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an ek+o(k)approximation algorithm for kcolumn sparse PIPs, improving on recent results of k2 · 2k [14] and O(k2) [3, 5]. We also show that the integrality gap of our linear programming relaxation is at least 2k − 1; it is known that kcolumn sparse PIPs are Ω(k log k)hard to approximate [8]. We also extend our result (at the loss of a small constant factor) to the more general case of maximizing a submodular objective over kcolumn sparse packing constraints.
A constant factor approximation algorithm for unsplittable flow on paths
 In Proceedings of the 52th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011
, 2011
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Edgedisjoint paths in planar graphs with constant congestion
 IN PROCEEDINGS OF THE 38TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING, 2006
, 2009
"... We study the maximum edgedisjoint paths problem in undirected planar graphs: given a graph G and node pairs (demands) s1t1, s2t2,..., sktk, the goal is to maximize the number of demands that can be connected (routed) by edgedisjoint paths. The natural multicommodity flow relaxation has an Ω ( √ ..."
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Cited by 14 (2 self)
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We study the maximum edgedisjoint paths problem in undirected planar graphs: given a graph G and node pairs (demands) s1t1, s2t2,..., sktk, the goal is to maximize the number of demands that can be connected (routed) by edgedisjoint paths. The natural multicommodity flow relaxation has an Ω ( √ n) integrality gap, where n is the number of nodes in G. Motivated by this, we consider solutions with small constant congestion c>1, that is, solutions in which up to c paths are allowed to use an edge (alternatively, each edge has a capacity of c). In previous work we obtained an O(log n) approximation with congestion 2 via the flow relaxation. This was based on a method of decomposing into welllinked subproblems. In this paper we obtain an O(1) approximation with congestion 4. To obtain this improvement we develop an alternative decomposition that is specific to planar graphs. The decomposition produces instances that we call Okamura–Seymour (OS) instances. These have the property that all terminals lie on a single face. Another ingredient we develop is a constant factor approximation for the allornothing flow problem on OS instances via the flow relaxation.
Pricing on Paths: A PTAS for the Highway Problem
"... In the highway problem, we are given an nedge line graph (the highway), and a set of paths (the drivers), each one with its own budget. For a given assignment of edge weights (the tolls), the highway owner collects from each driver the weight of the associated path, when it does not exceed the budg ..."
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Cited by 13 (1 self)
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In the highway problem, we are given an nedge line graph (the highway), and a set of paths (the drivers), each one with its own budget. For a given assignment of edge weights (the tolls), the highway owner collects from each driver the weight of the associated path, when it does not exceed the budget of the driver, and zero otherwise. The goal is choosing weights so as to maximize the profit. A lot of research has been devoted to this apparently simple problem. The highway problem was shown to be strongly NPhard only recently [Elbassioni,Raman,Ray,Sitters’09]. The bestknown approximation is O(log n / log log n) [Gamzu,Segev’10], which improves on the previousbest O(log n) approximation [Balcan,Blum’06]. Better approximations are known for a number of special cases. Finding a constant (or better!) approximation algorithm for the general case is a challenging open problem. In this paper we present a PTAS for the highway problem, hence closing the complexity status of the problem. Our result is based on a novel randomized dissection approach, which has some points in common with Arora’s quadtree dissection for Euclidean network design [Arora’98]. The basic idea is enclosing the highway in a bounding path, such that both the size of the bounding path and the position of the highway in it are random variables. Then we consider a recursive O(1)ary dissection of the bounding path, in subpaths of uniform optimal weight. Since the optimal weights are unknown, we construct the dissection in a bottomup fashion via dynamic programming, while computing the approximate solution at the same time. Our algorithm can be easily derandomized. The same basic approach provides PTASs also for two generalizations of the problem: the tollbooth problem with a constant number of leaves and the maximumfeasibility subsystem problem on interval matrices. In both cases the previous best approximation factors are polylogarithmic [Gamzu,Segev’10,Elbassioni,Raman,Ray,Sitters’09].
On Columnrestricted and Priority Covering Integer Programs ⋆
"... Abstract. In a columnrestricted covering integer program (CCIP), all the nonzero entries of any column of the constraint matrix are equal. Such programs capture capacitated versions of covering problems. In this paper, we study the approximability of CCIPs, in particular, their relation to the int ..."
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Abstract. In a columnrestricted covering integer program (CCIP), all the nonzero entries of any column of the constraint matrix are equal. Such programs capture capacitated versions of covering problems. In this paper, we study the approximability of CCIPs, in particular, their relation to the integrality gaps of the underlying 0,1CIP. If the underlying 0,1CIP has an integrality gap O(γ), and assuming that the integrality gap of the priority version of the 0,1CIP is O(ω), we give a factor O(γ + ω) approximation algorithm for the CCIP. Priority versions of 0,1CIPs (PCIPs) naturally capture quality of service type constraints in a covering problem. We investigate priority versions of the line (PLC) and the (rooted) tree cover (PTC) problems. Apart from being natural objects to study, these problems fall in a class of fundamental geometric covering problems. We bound the integrality of certain classes of this PCIP by a constant. Algorithmically, we give a polytime exact algorithm for PLC, show that the PTC problem is APXhard, and give a factor 2approximation algorithm for it. 1