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64
Expander Flows, Geometric Embeddings and Graph Partitioning
 IN 36TH ANNUAL SYMPOSIUM ON THE THEORY OF COMPUTING
, 2004
"... We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a ..."
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Cited by 319 (18 self)
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We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.
The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into `1
 In Proc. 46th IEEE Symp. on Foundations of Comp. Sci
, 2005
"... In this paper we disprove the following conjecture due to Goemans [17] and Linial [25] (also see [5, 27]): “Every negative type metric embeds into `1 with constant distortion. ” We show that for every δ> 0, and for large enough n, there is an npoint negative type metric which requires distortion ..."
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Cited by 180 (13 self)
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In this paper we disprove the following conjecture due to Goemans [17] and Linial [25] (also see [5, 27]): “Every negative type metric embeds into `1 with constant distortion. ” We show that for every δ> 0, and for large enough n, there is an npoint negative type metric which requires distortion atleast (log log n)1/6−δ to embed into `1. Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot [20], establishing a previously unsuspected connection between PCPs and the theory of metric embeddings. We first prove that the UGC implies superconstant hardness results for (nonuniform) Sparsest Cut and Minimum Uncut problems. It is already known that the UGC also implies an optimal hardness result for Maximum Cut [21]. Though these hardness results rely on the UGC, we demonstrate, nevertheless, that the corresponding PCP reductions can be used to construct “integrality gap instances ” for the respective problems. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of Unique Games. Then, we “simulate ” the PCP reduction, and “translate ” the integrality gap instance of Unique Games to integrality gap instances for the respective cut problems! This enables us to prove
SDP gaps and UGChardness for MaxCutGain
, 2008
"... Given a graph with maximum cut of (fractional) size c, the Goemans–Williamson semidefinite programming (SDP)based algorithm is guaranteed to find a cut of size at least.878 · c. However this guarantee becomes trivial when c is near 1/2, since making random cuts guarantees a cut of size 1/2 (i.e., ..."
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Cited by 26 (3 self)
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Given a graph with maximum cut of (fractional) size c, the Goemans–Williamson semidefinite programming (SDP)based algorithm is guaranteed to find a cut of size at least.878 · c. However this guarantee becomes trivial when c is near 1/2, since making random cuts guarantees a cut of size 1/2 (i.e., half of all edges). A few years ago, Charikar and Wirth (analyzing an algorithm of Feige and Langberg) showed that given a graph with maximum cut 1/2 + ε, one can find a cut of size 1/2 + Ω(ε/log(1/ε)). The main contribution of our paper is twofold: 1. We give a natural and explicit 1/2 + ε vs. 1/2 + O(ε/log(1/ε)) integrality gap for the MaxCut SDP based on Euclidean space with the Gaussian probability distribution. This shows that the SDProunding algorithm of CharikarWirth is essentially best possible. 2. We show how this SDP gap can be translated into a Long Code test with the same parameters. This implies that beating the CharikarWirth guarantee with any efficient algorithm is NPhard, assuming the Unique Games Conjecture (UGC). This result essentially settles the asymptotic approximability of MaxCut, assuming UGC. Building on the first contribution, we show how “randomness reduction ” on related SDP gaps for the QuadraticProgramming problem lets us make the Ω(log(1/ε)) gap as large as Ω(logn) for nvertex graphs. In addition to optimally answering an open question
Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs
"... We study the inapproximability of Vertex Cover and Independent Set on degree d graphs. We prove that: • Vertex Cover is Unique Gameshard to approximate log log d to within a factor 2−(2+od(1)). This exactly log d matches the algorithmic result of Halperin [1] up to the od(1) term. • Independent Set ..."
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Cited by 21 (0 self)
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We study the inapproximability of Vertex Cover and Independent Set on degree d graphs. We prove that: • Vertex Cover is Unique Gameshard to approximate log log d to within a factor 2−(2+od(1)). This exactly log d matches the algorithmic result of Halperin [1] up to the od(1) term. • Independent Set is Unique Gameshard to approxid mate to within a factor O( log2). This improves the d d logO(1) Unique Games hardness result of Samorod
Simpath: An efficient algorithm for influence maximization under the linear threshold model
 In Data Mining (ICDM), 2011 IEEE 11th International Conference on
, 2011
"... Abstract—There is significant current interest in the problem of influence maximization: given a directed social network with influence weights on edges and a number k, find k seed nodes such that activating them leads to the maximum expected number of activated nodes, according to a propagation mod ..."
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Cited by 19 (3 self)
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Abstract—There is significant current interest in the problem of influence maximization: given a directed social network with influence weights on edges and a number k, find k seed nodes such that activating them leads to the maximum expected number of activated nodes, according to a propagation model. Kempe et al. [1] showed, among other things, that under the Linear Threshold model, the problem is NPhard, and that a simple greedy algorithm guarantees the best possible approximation factor in PTIME. However, this algorithm suffers from various major performance drawbacks. In this paper, we propose SIMPATH, an efficient and effective algorithm for influence maximization under the linear threshold model that addresses these drawbacks by incorporating several clever optimizations. Through a comprehensive performance study on four real data sets, we show that SIMPATH consistently outperforms the state of the art w.r.t. running time, memory consumption and the quality of the seed set chosen, measured in terms of expected influence spread achieved.
On short paths interdiction problems : total and nodewise limited interdiction
 Theory of Computing Systems
"... nodewise limited interdiction. 1 by ..."
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Integrality gaps of 2 − o(1) for vertex cover sdps in the lovászschrijver hierarchy
 IN: ECCCTR: ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, TECHNICAL REPORTS
, 2006
"... Linear and semidefinite programming are highly successful approaches for obtaining good approximations for NPhard optimization problems. For example, breakthrough approximation algorithms for MAX CUT and SPARSEST CUT use semidefinite programming. Perhaps the most prominent NPhard problem whose e ..."
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Cited by 15 (6 self)
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Linear and semidefinite programming are highly successful approaches for obtaining good approximations for NPhard optimization problems. For example, breakthrough approximation algorithms for MAX CUT and SPARSEST CUT use semidefinite programming. Perhaps the most prominent NPhard problem whose exact approximation factor is still unresolved is VERTEX COVER. PCPbased techniques of Dinur and Safra [7] show that it is not possible to achieve a factor better than 1.36; on the other hand no known algorithm does better than the factor of 2 achieved by the simple greedy algorithm. Furthermore, there is a widespread belief that SDP techniques are the most promising methods available for improving upon this factor of 2. Following a line of study initiated by Arora et al. [3], our aim is to show that a large family of LP and SDP based algorithms fail to produce an approximation for VERTEX COVER better than 2. Lovász and Schrijver [21] introduced the systems LS and LS+ for systematically tightening LP and SDP relaxations, respectively, over many rounds. These systems naturally capture large classes of LP and SDP relaxations; indeed, LS+ captures the celebrated SDPbased algorithms for MAX CUT and SPARSEST CUT mentioned above. We rule out polynomialtime 2 − Ω(1) approximations for VERTEX COVER using LS+. In particular, we prove an integrality gap of 2−o(1) for VERTEX COVER SDPs obtained by tightening the standard LP relaxation with Ω( logn / log logn) rounds of LS+. While tight integrality gaps were known for VERTEX COVER in the weaker LS system [23], previous results did not rule out Funded in part by NSERC a 2−Ω(1) approximation after even two rounds of LS+.
Inapproximability of hypergraph vertex cover and applications to scheduling problems
 In Proc. ICALP (2010
"... Assuming the Unique Games Conjecture (UGC), we show optimal inapproximability results for two classic scheduling problems. We obtain a hardness of 2 − ε for the problem of minimizing the total weighted completion time in concurrent open shops. We also obtain a hardness of 2 − ε for minimizing the ma ..."
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Cited by 15 (3 self)
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Assuming the Unique Games Conjecture (UGC), we show optimal inapproximability results for two classic scheduling problems. We obtain a hardness of 2 − ε for the problem of minimizing the total weighted completion time in concurrent open shops. We also obtain a hardness of 2 − ε for minimizing the makespan in the assembly line problem. These results follow from a new inapproximability result for the Vertex Cover problem on kuniform hypergraphs that is stronger and simpler than previous results. We show that assuming the UGC, for every k ≥ 2, the problem is inapproximable within k − ε even when the hypergraph is almost kpartite. 1
A 1.875–Approximation Algorithm for the Stable Marriage Problem
, 2007
"... We consider the problem of finding a stable matching of maximum size when both ties and unacceptable partners are allowed in preference lists. This problem is known to be APXhard, and the current best known approximation algorithm achieves the approximation ratio 2 − c 1 √ N,where c is some positiv ..."
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Cited by 14 (2 self)
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We consider the problem of finding a stable matching of maximum size when both ties and unacceptable partners are allowed in preference lists. This problem is known to be APXhard, and the current best known approximation algorithm achieves the approximation ratio 2 − c 1 √ N,where c is some positive constant. In this paper, we give a 1.875– approximation algorithm, which is the first result on the approximation ratio better than two.
A primaldual approximation algorithm for partial vertex cover: Making educated guesses
 In 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, volume 3624 of LNCS
, 2005
"... We study the partial vertex cover problem. Given a graph G = (V,E), a weight function w: V → R +, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NPhard as it generalizes the wellknown vertex cover problem. We provide ..."
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Cited by 13 (2 self)
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We study the partial vertex cover problem. Given a graph G = (V,E), a weight function w: V → R +, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NPhard as it generalizes the wellknown vertex cover problem. We provide a primaldual 2approximation algorithm which runs in O(nlog n+m) time. This represents an improvement in running time from the previously known fastest algorithm. Our technique can also be used to get a 2approximation for a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity ku. A solution consists of a function x: V → N0 and an orientation of all but s edges, such that the number of edges oriented toward vertex u is at most xuku. Our objective is to find a cover that minimizes � v∈V xvwv. This is the first 2approximation for the problem and also runs in O(nlog n + m) time.