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39
Optimal inapproximability results for MAXCUT and other 2variable CSPs?
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games ..."
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Cited by 178 (28 self)
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In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games
Optimal algorithms and inapproximability results for every CSP
 In Proc. 40 th ACM STOC
, 2008
"... Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the ..."
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Cited by 86 (12 self)
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Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the following result: If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains. Using this connection between integrality gaps and hardness results we obtain a generic polynomialtime algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP. Unconditionally, for all 2CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut
Inferring AS relationships: Dead end or lively beginning
 In Proceedings of 4th Workshop on Efficient and Experimental Algorithms (WEA’ 05
, 2005
"... Recent techniques for inferring business relationships between ASs [3, 8] have yielded maps that have extremely few invalid BGP paths in the terminology of Gao [9]. However, some relationships inferred by these newer algorithms are incorrect, leading to the deduction of unrealistic AS hierarchies. W ..."
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Cited by 26 (9 self)
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Recent techniques for inferring business relationships between ASs [3, 8] have yielded maps that have extremely few invalid BGP paths in the terminology of Gao [9]. However, some relationships inferred by these newer algorithms are incorrect, leading to the deduction of unrealistic AS hierarchies. We investigate this problem and discover what causes it. Having obtained such insight, we generalize the problem of AS relationship inference as a multiobjective optimization problem with nodedegreebased corrections to the original objective function of minimizing the number of invalid paths. We solve the generalized version of the problem using the semidefinite programming relaxation of the MAX2SAT problem. Keeping the number of invalid paths small, we obtain a more veracious solution than that yielded by recent heuristics.
Towards Sharp Inapproximability For Any 2CSP
, 2008
"... We continue the recent line of work on the connection between semidefinite programmingbased approximation algorithms and the Unique Games Conjecture. Given any boolean 2CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good ..."
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Cited by 22 (2 self)
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We continue the recent line of work on the connection between semidefinite programmingbased approximation algorithms and the Unique Games Conjecture. Given any boolean 2CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good inapproximability result to a certain numeric minimization problem. The key objects in our analysis are the vector triples arising when doing clausebyclause analysis of algorithms based on semidefinite programming. Given a weighted set of such triples of a certain restricted type, which are “hard” to round in a certain sense, we obtain a Unique Gamesbased inapproximability matching this “hardness ” of rounding the set of vector triples. Conversely, any instance together with an SDP solution can be viewed as a set of vector triples, and we show that we can always find an assignment to the instance which is at least as good as the “hardness ” of rounding the corresponding set of vector triples. We conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly. This conjecture is supported by all existing results for specific 2CSPs. As an application, we show that MAX 2AND is hard to approximate within 0.87435. Thisimproves upon the best previous hardness of αGW + ɛ ≈ 0.87856, and comes very close to matching the approximation ratio of the best algorithm known, 0.87401. It also establishes that balanced instances of MAX 2AND, i.e., instances in which each variable occurs positively and negatively equally often, are not the hardest to approximate, as these can be approximated within a factor αGW.
Classifying CustomerProvider Relationships in the Internet
, 2002
"... The problem of inferring customerprovider relationships in the autonomous system topology of the Internet leads to the following optimization problem: given an undirected graph G and a set P of paths in G, orient the edges of G such that as many paths as possible are valid, meaning that they do not ..."
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Cited by 21 (4 self)
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The problem of inferring customerprovider relationships in the autonomous system topology of the Internet leads to the following optimization problem: given an undirected graph G and a set P of paths in G, orient the edges of G such that as many paths as possible are valid, meaning that they do not contain an internal node with both incident edges on the path directed away from that node. The complexity of this problem was left open by Subramanian et al. ("Characterizing the Internet hierarchy from multiple vantage points," INFOCOM 2002). We show
Nearoptimal algorithms for maximum constraint satisfaction problems
 In SODA ’07: Proceedings of the eighteenth annual ACMSIAM symposium on Discrete algorithms
, 2007
"... In this paper we present approximation algorithms for the maximum constraint satisfaction problem with k variables in each constraint (MAX kCSP). Given a (1 − ε) satisfiable 2CSP our first algorithm finds an assignment of variables satisfying a 1 − O ( √ ε) fraction of all constraints. The best pr ..."
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Cited by 17 (3 self)
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In this paper we present approximation algorithms for the maximum constraint satisfaction problem with k variables in each constraint (MAX kCSP). Given a (1 − ε) satisfiable 2CSP our first algorithm finds an assignment of variables satisfying a 1 − O ( √ ε) fraction of all constraints. The best previously known result, due to Zwick, was 1 − O(ε 1/3). The second algorithm finds a ck/2 k approximation for the MAX kCSP problem (where c> 0.44 is an absolute constant). This result improves the previously best known algorithm by Hast, which had an approximation guarantee of Ω(k/(2 k log k)). Both results are optimal assuming the Unique Games Conjecture and are based on rounding natural semidefinite programming relaxations. We also believe that our algorithms and their analysis are simpler than those previously known. 1
An Algorithm for Orienting Graphs Based on CauseEffect Pairs and Its Applications to Orienting Protein Networks
"... We consider a graph orientation problem arising in the study of biological networks. Given an undirected graph and a list of ordered sourcetarget pairs, the goal is to orient the graph so that a maximum number of pairs will admit a directed path from the source to the target. We show that the prob ..."
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Cited by 14 (3 self)
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We consider a graph orientation problem arising in the study of biological networks. Given an undirected graph and a list of ordered sourcetarget pairs, the goal is to orient the graph so that a maximum number of pairs will admit a directed path from the source to the target. We show that the problem is NPhard and hard to approximate to within a constant ratio. We then study restrictions of the problem to various graph classes, and provide an O(log n) approximation algorithm for the general case. We show that this algorithm achieves very tight approximation ratios in practice and is able to infer edge directions with high accuracy on both simulated and real network data.
How to Round Any CSP
"... A large number of interesting combinatorial optimization ..."
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Cited by 14 (3 self)
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A large number of interesting combinatorial optimization
On the unique games conjecture
 In FOCS
, 2005
"... This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1 ..."
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Cited by 11 (0 self)
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This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1
Improved Orientations of Physical Networks
"... Abstract. The orientation of physical networks is a prime task in deciphering the signalingregulatory circuitry of the cell. One manifestation of this computational task is as a maximum graph orientation problem, where given an undirected graph on n vertices and a collection of vertex pairs, the go ..."
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Cited by 10 (4 self)
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Abstract. The orientation of physical networks is a prime task in deciphering the signalingregulatory circuitry of the cell. One manifestation of this computational task is as a maximum graph orientation problem, where given an undirected graph on n vertices and a collection of vertex pairs, the goal is to orient the edges of the graph so that a maximum number of pairs are connected by a directed path. We develop a novel approximation algorithm for this problem with a performance guarantee of O(logn/loglogn), improving on the current logarithmic approximation. In addition, motivated by interactions whose direction is preset, such as proteinDNA interactions, we extend our algorithm to handle mixed graphs, a major open problem posed by earlier work. In this setting, we show that a polylogarithmic approximation ratio is achievable under biologicallymotivated assumptions on the sought paths. 1