Results 21  30
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101
Improved approximation of linear threshold functions
 In Proc. 24nd Annual IEEE Conference on Computational Complexity (CCC
, 2009
"... We prove two main results on how arbitrary linear threshold functions f(x) = sign(w · x − θ) over the ndimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every nvariable threshold function f is ɛclose to a threshold function depending only ..."
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Cited by 19 (12 self)
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We prove two main results on how arbitrary linear threshold functions f(x) = sign(w · x − θ) over the ndimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every nvariable threshold function f is ɛclose to a threshold function depending only on Inf(f) 2 · poly(1/ɛ) many variables, where Inf(f) denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut’s wellknown theorem [Fri98], which states that every Boolean function f is ɛclose to a function depending only on 2 O(Inf(f)/ɛ) many variables, for the case of threshold functions. We complement this upper bound by showing that Ω(Inf(f) 2 + 1/ɛ 2) many variables are required for ɛapproximating threshold functions. Our second result is a proof that every nvariable threshold function is ɛclose to a threshold function with integer weights at most poly(n) · 2 Õ(1/ɛ2/3). This is an improvement, in the dependence on the error parameter ɛ, on an earlier result of [Ser07] which gave a poly(n) · 2 Õ(1/ɛ2) bound. Our improvement is obtained via a new proof technique that uses strong anticoncentration bounds from probability theory. The new technique also gives a simple and modular proof of the original [Ser07] result, and extends to give lowweight approximators for threshold functions under a range of probability distributions other than the uniform distribution.
Partitioning Graphs into Balanced Components
, 2009
"... We consider the kbalanced partitioning problem, where the goal is to partition the vertices of an input graph G into k equally sized components, while minimizing the total weight of the edges connecting different components. We allow k to be part of the input and denote the cardinality of the verte ..."
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Cited by 18 (2 self)
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We consider the kbalanced partitioning problem, where the goal is to partition the vertices of an input graph G into k equally sized components, while minimizing the total weight of the edges connecting different components. We allow k to be part of the input and denote the cardinality of the vertex set by n. This problem is a natural and important generalization of wellknown graph partitioning problems, including minimum bisection and minimum balanced cut. We present a (bicriteria) approximation algorithm achieving an approximation of O ( √ log n log k), which matches or improves over previous algorithms for all relevant values of k. Our algorithm uses a semidefinite relaxation which combines ℓ 2 2 metrics with spreading metrics. Surprisingly, we show that the integrality gap of the semidefinite relaxation is Ω(log k) even for large values of k (e.g., k = n Ω(1)), implying that the dependence on k of the approximation factor is necessary. This is in contrast to previous approximation algorithms for kbalanced partitioning, which are based on linear programming relaxations and their approximation factor is independent of k.
Constant ratio fixedparameter approximation of the edge multicut problem
 In ESA 2009
, 2009
"... Abstract. The input of the Edge Multicut problem consists of an undirected graph G and pairs of terminals {s1, t1},..., {sm, tm}; the task is to remove a minimum set of edges such that si and ti are disconnected for every 1 ≤ i ≤ m. The parameterized complexity of the problem, parameterized by the ..."
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Cited by 18 (3 self)
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Abstract. The input of the Edge Multicut problem consists of an undirected graph G and pairs of terminals {s1, t1},..., {sm, tm}; the task is to remove a minimum set of edges such that si and ti are disconnected for every 1 ≤ i ≤ m. The parameterized complexity of the problem, parameterized by the maximum number k of edges that are allowed to be removed, is currently open. The main result of the paper is a parameterized 2approximation algorithm: in time f(k) · nO(1), we can either find a solution of size 2k or correctly conclude that there is no solution of size k. The proposed algorithm is based on a transformation of the Edge Multicut problem into a variant of parameterized Max2SAT problem, where the parameter is related to the number of clauses that are not satisfied. It follows from previous results that the latter problem can be 2approximated in a fixedparameter time; on the other hand, we show here that it is W[1]hard. Thus the additional contribution of the present paper is introducing the first natural W[1]hard problem that is constantratio fixedparameter approximable. 1
SPECTRAL ALGORITHMS FOR UNIQUE Games
"... We give a new algorithm for Unique Games which is based on purely spectral techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our algorithm is able to recover a good assignment. The appro ..."
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Cited by 17 (1 self)
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We give a new algorithm for Unique Games which is based on purely spectral techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our algorithm is able to recover a good assignment. The approximation guarantee depends only on the completeness of the game, and not on the alphabet size, while the running time depends on spectral properties of the LabelExtended graph associated with the instance of Unique Games. We further show that on input the integrality gap instance of Khot and Vishnoi, our algorithm runs in quasipolynomial time and decides that the instance if highly unsatisfiable. Notably, when run on this instance, the standard SDP relaxation of Unique Games fails. As a special case, we also rederive a polynomial time algorithm for Unique Games on expander constraint graphs. The main ingredient of our algorithm is a technique to effectively use the full spectrum of the underlying graph instead of just the second eigenvalue, which is of independent interest. The question of how to take advantage of the full spectrum of a graph in the design of algorithms has been often studied, but no significant progress was made prior to this work.
LOCAL VERSUS GLOBAL PROPERTIES OF METRIC SPACES
, 2012
"... Motivated by applications in combinatorial optimization, we study the extent to which the global properties of a metric space, and especially its embeddability into ℓ1 with low distortion, are determined by the properties of its small subspaces. We establish both upper and lower bounds on the disto ..."
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Cited by 17 (0 self)
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Motivated by applications in combinatorial optimization, we study the extent to which the global properties of a metric space, and especially its embeddability into ℓ1 with low distortion, are determined by the properties of its small subspaces. We establish both upper and lower bounds on the distortion of embedding locally constrained metrics into various target spaces. Other aspects of locally constrained metrics are studied as well, in particular, how far are those metrics from general metrics.
Multicut is FPT
 In STOC
, 2011
"... Let G = (V,E) be a graph on n vertices and R be a set of pairs of vertices in V called requests. A multicut is a subset F of E such that every request xy of R is separator by F, i.e.every xypath of G intersects F. We show that there exists an O(f(k)nc) algorithm which decides if there exists a mult ..."
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Cited by 17 (0 self)
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Let G = (V,E) be a graph on n vertices and R be a set of pairs of vertices in V called requests. A multicut is a subset F of E such that every request xy of R is separator by F, i.e.every xypath of G intersects F. We show that there exists an O(f(k)nc) algorithm which decides if there exists a multicut of size at most k. In other words, the MULTICUT problem parameterized by the solution size k is FixedParameter Tractable. 1
Hardness of Robust Network Design
"... We settle the complexity status of the robust network design problem in undirected graphs. The fact that the flowcut gap in general graphs can be large, poses some difficulty in establishing a hardness result. Instead we introduce a singlesource version of the problem where the flowcut gap is k ..."
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Cited by 16 (3 self)
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We settle the complexity status of the robust network design problem in undirected graphs. The fact that the flowcut gap in general graphs can be large, poses some difficulty in establishing a hardness result. Instead we introduce a singlesource version of the problem where the flowcut gap is known to be one. We then show that this restricted problem is coNPHard. This version also captures, as special cases, the fractional relaxations of several problems including the spanning tree problem, the Steiner tree problem, and the shortest path problem.
The unique game conjecture with entangled provers is false
, 2007
"... We consider oneround games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are ‘unique ’ constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only a ..."
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Cited by 15 (4 self)
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We consider oneround games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are ‘unique ’ constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only algorithm known previously was for the special case of binary answers, as follows from the work of Tsirelson in 1980. Among other things, our result implies that the variant of the unique games conjecture where we allow the provers to share entanglement is false. Our proof is based on a novel ‘quantum rounding technique’, showing how to take a solution to an SDP and transform it to a strategy for entangled provers. 1
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 15 (1 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
A (log n)Ω(1) integrality gap for the Sparsest Cut SDP
 In Proceedings of 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009
, 2009
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