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48
Incentive compatible multi unit combinatorial auctions
 In TARK 03
, 2003
"... This paper deals with multiunit combinatorial auctions where there are n types of goods for sale, and for each good there is some fixed number of units. We focus on the case where each bidder desires a relatively small number of units of each good. In particular, this includes the case where each g ..."
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Cited by 112 (13 self)
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This paper deals with multiunit combinatorial auctions where there are n types of goods for sale, and for each good there is some fixed number of units. We focus on the case where each bidder desires a relatively small number of units of each good. In particular, this includes the case where each good has exactly k units, and each bidder desires no more than a single unit of each good. We provide incentive compatible mechanisms for combinatorial auctions for the general case where bidders are not limited to single minded valuations. The mechanisms we give have approximation ratios close to the best possible for both online and offline scenarios. This is the first result where nonVCG mechanisms are derived for nonsingle minded bidders for a natural model of combinatorial auctions.
Approximation Techniques for Utilitarian Mechanism Design
, 2005
"... This paper deals with the design of efficiently computable incentive compatible, or truthful, mechanisms for combinatorial optimization problems with multiparameter agents. We focus on approximation algorithms for NPhard mechanism design problems. These algorithms need to satisfy certain monotonic ..."
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Cited by 92 (5 self)
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This paper deals with the design of efficiently computable incentive compatible, or truthful, mechanisms for combinatorial optimization problems with multiparameter agents. We focus on approximation algorithms for NPhard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques. Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone FPTAS. This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for singleminded multiunit auctions. The best previous result for such auctions was a 2approximation. In addition, we present a monotone PTAS for the generalized assignment problem with any bounded number of parameters per agent. The most efficient way to solve packing integer programs (PIPs) is LPbased randomized rounding, which also is in general not monotone. We show that primaldual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like singleminded combinatorial auctions (CAs), unsplittable flow routing and multicast routing. Our approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle.
Improved Bounds for the Unsplittable Flow Problem
 In Proceedings of the 13th ACMSIAM Symposium on Discrete Algorithms
, 2002
"... In this paper we consider the unsplittable ow problem (UFP): given a directed or undirected network G = (V, E) with edge capacities and a set of terminal pairs (or requests) with associated demands, find a subset of the pairs of maximum total demand for which a single flow path can be chosen for eac ..."
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Cited by 56 (6 self)
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In this paper we consider the unsplittable ow problem (UFP): given a directed or undirected network G = (V, E) with edge capacities and a set of terminal pairs (or requests) with associated demands, find a subset of the pairs of maximum total demand for which a single flow path can be chosen for each pair so that for every edge, the sum of the demands of the paths crossing the edge does not exceed its capacity.
Hardness of the undirected edgedisjoint paths problem
 Proc. of STOC
, 2005
"... In the EdgeDisjoint Paths problem with Congestion (EDPwC), we are given a graph with n nodes, a set of terminal pairs and an integer c. The objective is to route as many terminal pairs as possible, subject to the constraint that at most c demands can be routed through any edge in the graph. When c ..."
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Cited by 56 (8 self)
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In the EdgeDisjoint Paths problem with Congestion (EDPwC), we are given a graph with n nodes, a set of terminal pairs and an integer c. The objective is to route as many terminal pairs as possible, subject to the constraint that at most c demands can be routed through any edge in the graph. When c = 1, the problem is simply referred to as the EdgeDisjoint Paths (EDP) problem. In this paper, we study the hardness of EDPwC in undirected graphs. We obtain an improved hardness result for EDP, and also show the first polylogarithmic integrality gaps and hardness of approximation results for EDPwC. Specifically, we prove that EDP is (log 1 2 −ε n)hard to approximate for any constant ε> 0, unless NP ⊆ ZP T IME(n polylog n). We also show that for any congestion c = o(log log n / log log log n), there is no (log 1−ε c+1 n)approximation algorithm for EDPwC, unless NP ⊆ ZP T IME(npolylog n). For larger congestion, where c ≤ η log log n / log log log n for some constant η, we obtain superconstant inapproximability ratios. All of our hardness results can be converted into integrality gaps for the multicommodity flow relaxation. We also present a separate elementary direct proof of this integrality gap result. Finally, we note that similar results can be obtained for the AllorNothing Flow (ANF) problem, a relaxation of EDP, in which the flow unit routed between the sourcesink pairs does not have follow a single path, so the resulting flow is not necessarily integral. Using standard transformations, our results also extend to the nodedisjoint versions of these problems as well as to the directed setting. 1
Approximation Algorithms for the Unsplittable Flow Problem
"... We present approximation algorithms for the unsplittable flow problem (UFP) on undirected graphs. As is standard in this line of research, we assume that the maximum demand is at most the minimum capacity. We focus on the nonuniform capacity case in which the edge capacities can vary arbitrarily ..."
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Cited by 55 (9 self)
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We present approximation algorithms for the unsplittable flow problem (UFP) on undirected graphs. As is standard in this line of research, we assume that the maximum demand is at most the minimum capacity. We focus on the nonuniform capacity case in which the edge capacities can vary arbitrarily over the graph. Our results are: For undirected graphs we obtain a O(\Delta ff \Gamma 1 log2 n) approximation ratio, where n is the number of vertices, \Delta the maximum degree, and ff the expansion of the graph. Our ratio is capacity independent and improves upon the earlier O(\Delta ff \Gamma 1(c max=cmin) log n) bound [15] for large values of cmax=cmin. Furthermore, if we specialize to the case where all edges have the same capacity, our algorithm gives an O(\Delta ff \Gamma 1 log n) approximation, which matches the performance of the bestknown algorithm [15] for this special case. For certain strong constantdegree expanders considered by Frieze [10] we obtain an O(plog n) approximation for the uniform capacity case, improving upon the current O(log n) approximation. For UFP on the line and the ring, we give the first constantfactor approximation algorithms. Previous results addressed only the uniform capacity case. All of the above results improve if the maximum demand is bounded
The AllorNothing Multicommodity Flow Problem
 IN PROCEEDINGS OF THE 36TH ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC)
, 2004
"... ..., the same as that for edp [10]. Our algorithm extends to the case where each pair siti has a demand di associated with it and we need to completely route di to get credit for pair i. We also consider the online admission control version where pairs arrive online and the algorithm has to decide i ..."
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Cited by 42 (13 self)
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..., the same as that for edp [10]. Our algorithm extends to the case where each pair siti has a demand di associated with it and we need to completely route di to get credit for pair i. We also consider the online admission control version where pairs arrive online and the algorithm has to decide immediately on its arrival whether to accept it or not. We obtain a randomized algorithm with a competitive ratio that is similar to the approximation ratio for the offline algorithm.
Edge Disjoint Paths Revisited
 In Proceedings of the 14th ACMSIAM Symposium on Discrete Algorithms
, 2003
"... The approximability of the maximum edge disjoint paths problem (EDP) in directed graphs was seemingly settled by the )hardness result of Guruswami et al. [10] and the O( m) approximation achievable via both the natural LP relaxation [19] and the greedy algorithm [11, 12]. Here m is the numb ..."
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Cited by 40 (4 self)
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The approximability of the maximum edge disjoint paths problem (EDP) in directed graphs was seemingly settled by the )hardness result of Guruswami et al. [10] and the O( m) approximation achievable via both the natural LP relaxation [19] and the greedy algorithm [11, 12]. Here m is the number of edges in the graph. However, we observe that the hardness of approximation shown in [10] applies to sparse graphs and hence when expressed as a function of n, the number of vertices, only an \Omega\Gamma n )hardness follows. On the other hand, the O( m)approximation algorithms do not guarantee a sublinear (in terms of n) approximation algorithm for dense graphs. We note that a similar gap exists in the known results on the integrality gap of the natural LP relaxation: an \Omega\Gamma n) lower bound and an O( m) upper bound. Motivated by this discrepancy in the upper and lower bounds we study algorithms for the EDP in directed and undirected graphs obtaining improved approximation ratios. We show that the greedy algorithm has an approximation ratio of O(min(n m)) in undirected graphs and a ratio of O(min(n m)) in directed graphs. For ayclic graphs we give an O( n log n) approximation via LP rounding. These are the first sublinear approximation ratios for EDP. Our results also extend to EDP with weights and to the unsplittable flow problem with uniform edge capacities.
On the kSplittable Flow Problem
, 2002
"... In traditional multicommodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However, ..."
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Cited by 31 (3 self)
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In traditional multicommodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However,
New hardness results congestion minimization and machine scheduling
 PROC. 36TH. ANNUAL ACM SYMPOSIUM ON THEORY OF COMPTING
"... We study the approximability of two natural NPhard problems. The first problem is congestion minimization in directed networks. In this problem, we are given a directed graph and a set of sourcesink pairs. The goal is to route all the pairs with minimum congestion on the network edges. The second ..."
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Cited by 23 (3 self)
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We study the approximability of two natural NPhard problems. The first problem is congestion minimization in directed networks. In this problem, we are given a directed graph and a set of sourcesink pairs. The goal is to route all the pairs with minimum congestion on the network edges. The second problem is machine scheduling, where we are given a set of jobs, and for each job, there is a list of intervals on which it can be scheduled. The goal is to find the smallest number of machines on which all jobs can be scheduled such that no two jobs overlap in their execution on any machine. Both problems are known to be O(log n/loglog n)approximable via the randomized rounding technique of Raghavan and Thompson. However, until recently, only Max SNP hardness was known for each problem. We make progress in closing this gap by showing that both problems are Ω(log log n)hard to approximate unless NP ⊆ DTIME(n O(log log log n)).