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177
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.
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 238 (32 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 144 (13 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
Euclidean distortion and the Sparsest Cut
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... BiLipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] ..."
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Cited by 120 (25 self)
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BiLipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] which shows that every
Improved approximation algorithms for minimum weight vertex separators
 In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, FOCS’89
, 1989
"... vertex separators ..."
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On the Hardness of Approximating Multicut and SparsestCut
 In Proceedings of the 20th Annual IEEE Conference on Computational Complexity
, 2005
"... We show that the MULTICUT, SPARSESTCUT, and MIN2CNF ≡ DELETION problems are NPhard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1. ..."
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Cited by 102 (5 self)
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We show that the MULTICUT, SPARSESTCUT, and MIN2CNF ≡ DELETION problems are NPhard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1.
Gowers uniformity, influence of variables, and PCPs
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... Gowers [Gow98, Gow01] introduced, for d ≥ 1, the notion of dimensiond uniformity U d (f) of a function f: G → C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it “looks random ” on certain structured subsets of the inputs. We pro ..."
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Cited by 65 (2 self)
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Gowers [Gow98, Gow01] introduced, for d ≥ 1, the notion of dimensiond uniformity U d (f) of a function f: G → C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it “looks random ” on certain structured subsets of the inputs. We prove the following inverse theorem. Write G = G1 × · · · × Gn as a product of groups. If a bounded balanced function f: G1 × · · · Gn → C is such that U d (f) ≥ ε, then one of the coordinates of f has influence at least ε/2 O(d). Other inverse theorems are known [Gow98, Gow01, GT05, Sam05], and U 3 is especially well understood, but the properties of functions f with large U d (f), d ≥ 4, are not yet well characterized. The dimensiond Gowers inner product 〈{fS} 〉 U d of a collection {fS} S⊆[d] of functions is a related measure of pseudorandomness. The definition is such that if all the functions fS are equal to the same fixed function f, then 〈{fS} 〉 U d = U d (f). We prove that if fS: G1 × · · · × Gn → C is a collection of bounded functions such that 〈{fS} 〉 U d  ≥ ε and at least one of the fS is balanced, then there is a variable that has influence at least ε 2 /2 O(d) for at least four functions in the collection. Finally, we relate the acceptance probability of the “hypergraph longcode test ” proposed by Samorodnitsky and Trevisan to the Gowers inner product of the functions being tested and we deduce the following result: if the Unique Games Conjecture is true, then for every q ≥ 3 there is a PCP characterization of NP where the verifier makes q queries, has almost perfect completeness, and soundness at most 2q/2 q. For infinitely many q, the soundness is (q + 1)/2 q, which might be a tight result. Two applications of this results are that, assuming that the unique games conjecture is true, it is hard to approximate Max kCSP within a factor 2k/2 k ((k + 1)/2 k for infinitely many k), and it is hard to approximate Independent Set in graphs of degree D within a factor (log D) O(1) /D. 1
Nonembeddability theorems via Fourier analysis
"... Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group ac ..."
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Cited by 53 (12 self)
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Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.
Nearoptimal algorithms for Unique Games
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... Unique games are constraint satisfaction problems that can be viewed as a generalization of MaxCut to a larger domain size. The Unique Games Conjecture states that it is hard to distinguish between instances of unique games where almost all constraints are satisfiable and those where almost none ar ..."
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Cited by 47 (8 self)
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Unique games are constraint satisfaction problems that can be viewed as a generalization of MaxCut to a larger domain size. The Unique Games Conjecture states that it is hard to distinguish between instances of unique games where almost all constraints are satisfiable and those where almost none are satisfiable. It has been shown to imply a number of inapproximability results for fundamental problems that seem difficult to obtain by more standard complexity assumptions. Thus, proving or refuting this conjecture is an important goal. We present significantly improved approximation algorithms for unique games. For instances with domain size k where the optimal solution satisfies 1 − ε fraction of all constraints, our algorithms satisfy roughly k −ε/(2−ε) and 1 − O ( √ ε log k) fraction of all constraints. Our algorithms are based on rounding a natural semidefinite programming relaxation for the problem and their performance almost matches the integrality gap of this relaxation. Our results are near optimal if the Unique Games Conjecture is true, i.e. any improvement (beyond low order terms) would refute the conjecture. 1