Results 1  10
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22
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
Approximation Algorithms for MAX3CUT and Other Problems via Complex Semidefinite Programming
, 2002
"... ..."
An analysis of convex relaxations for MAP estimation of discrete MRFs
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2008
"... The problem of obtaining the maximum a posteriori estimate of a general discrete Markov random field (i.e., a Markov random field defined using a discrete set of labels) is known to be NPhard. However, due to its central importance in many applications, several approximation algorithms have been pr ..."
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Cited by 27 (1 self)
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The problem of obtaining the maximum a posteriori estimate of a general discrete Markov random field (i.e., a Markov random field defined using a discrete set of labels) is known to be NPhard. However, due to its central importance in many applications, several approximation algorithms have been proposed in the literature. In this paper, we present an analysis of three such algorithms based on convex relaxations: (i) LPS: the linear programming (LP) relaxation proposed by Schlesinger (1976) for a special case and independently in Chekuri et al. (2001), Koster et al. (1998), and Wainwright et al. (2005) for the general case; (ii) QPRL: the quadratic programming (QP) relaxation of Ravikumar and Lafferty (2006); and (iii) SOCPMS: the second order cone programming (SOCP) relaxation first proposed by Muramatsu and Suzuki (2003) for two label problems and later extended by Kumar et al. (2006) for a general label set. We show that the SOCPMS and the QPRL relaxations are equivalent. Furthermore, we prove that despite the flexibility in the form of the constraints/objective function offered by QP and SOCP, the LPS relaxation strictly dominates (i.e., provides a better approximation than) QPRL and SOCPMS. We generalize these results by defining a large class of SOCP (and equivalent QP) relaxations
THE OPERATOR Ψ FOR THE CHROMATIC NUMBER OF A GRAPH
, 2008
"... We investigate hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. We introduce an operator Ψ mapping any graph parameter β(G), nested between the stability number α(G) and χ(G), to a new graph parameter Ψβ(G), nested between α(G) and χ(G); Ψβ(G) is polynomial ti ..."
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Cited by 22 (6 self)
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We investigate hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. We introduce an operator Ψ mapping any graph parameter β(G), nested between the stability number α(G) and χ(G), to a new graph parameter Ψβ(G), nested between α(G) and χ(G); Ψβ(G) is polynomial time computable if β(G) is. As an application, there is no polynomial time computable graph parameter nested between the fractional chromatic number χ ∗ (·) and χ(·) unless P = NP. Moreover, based on the Motzkin–Straus formulation for α(G), we give (quadratically constrained) quadratic and copositive programming formulations for χ(G). Under some mild assumptions, n/β(G) ≤ Ψβ(G), but, while n/β(G) remains below χ ∗ (G), Ψβ(G) can reach χ(G) (e.g., for β(·) =α(·)). We also define new polynomial time computable lower bounds for χ(G), improving the classic Lovász theta number (and its strengthenings obtained by adding nonnegativity and triangle inequalities); experimental results on Hamming graphs, Kneser graphs, and DIMACS benchmark graphs will be given in the followup paper [N. Gvozdenović and M. Laurent, SIAM J. Optim., 19 (2008), pp. 592–615].
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
Semidefinite programming heuristics for surface reconstruction ambiguities
 In ECCV
, 2008
"... Abstract. We consider the problem of reconstructing a smooth surface under constraints that have discrete ambiguities. These problems arise in areas such as shape from texture, shape from shading, photometric stereo and shape from defocus. While the problem is computationally hard, heuristics based ..."
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Cited by 18 (1 self)
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Abstract. We consider the problem of reconstructing a smooth surface under constraints that have discrete ambiguities. These problems arise in areas such as shape from texture, shape from shading, photometric stereo and shape from defocus. While the problem is computationally hard, heuristics based on semidefinite programming may reveal the shape of the surface. 1
Maximally stable Gaussian partitions with discrete applications
 Israel J. Math
"... Gaussian noise stability results have recently played an important role in proving results in hardness of approximation in computer science and in the study of voting schemes in social choice. We prove a new Gaussian noise stability result generalizing an isoperimetric result by Borell on the heat k ..."
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Cited by 13 (3 self)
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Gaussian noise stability results have recently played an important role in proving results in hardness of approximation in computer science and in the study of voting schemes in social choice. We prove a new Gaussian noise stability result generalizing an isoperimetric result by Borell on the heat kernel and derive as applications: • An optimality result for majority in the context of Condorcet voting. • A proof of a conjecture on “cosmic coin tossing ” for low influence functions. We also discuss a Gaussian noise stability conjecture which may be viewed as a generalization of the “Double Bubble ” theorem and show that it implies: • A proof of the “Plurality is Stablest Conjecture”. • That the FriezeJerrum SDP for MAXqCUT achieves the optimal approximation factor assuming the Unique Games Conjecture.
Approximability distance in the space of hcolourability problems
 Proceedings of the 4th International Computer Science Symposium in Russia (CSR2009), arXiv:0802.0423. 13 Chris Godsil and Gordon Royle, Algebraic graph theory, Graduate Texts in Mathematics
, 2001
"... Abstract. A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. We study the approximability properties of the Weighted Maximum HColourable Subgraph problem (MAX HCOL). The instances of this problem are edgeweighted graphs G and the objective is ..."
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Cited by 5 (3 self)
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Abstract. A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. We study the approximability properties of the Weighted Maximum HColourable Subgraph problem (MAX HCOL). The instances of this problem are edgeweighted graphs G and the objective is to find a subgraph of G that has maximal total edge weight, under the condition that the subgraph has a homomorphism to H; note that for H = Kk this problem is equivalent to MAX kCUT. To this end, we introduce a metric structure on the space of graphs which allows us to extend previously known approximability results to larger classes of graphs. Specifically, the approximation algorithms for MAX CUT by Goemans and Williamson and MAX kCUT by Frieze and Jerrum can be used to yield nontrivial approximation results for MAX HCOL. For a variety of graphs, we show nearoptimality results under the Unique Games Conjecture. We also use our method for comparing the performance of Frieze & Jerrum’s algorithm with Håstad’s approximation algorithm for general MAX 2CSP. This comparison is, in most cases, favourable to Frieze & Jerrum.
A BranchandCut Algorithm based on Semidefinite Programming for the Minimum kPartition Problem
, 2007
"... The minimum kpartition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution of this paper is the design and implementation of a branchandcut ..."
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Cited by 4 (1 self)
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The minimum kpartition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution of this paper is the design and implementation of a branchandcut algorithm based on semidefinite programming (SBC) for the MkP problem. The two key ingredients for this algorithm are: the combination of semidefinite programming (SDP) with polyhedral results; and the iterative clustering heuristic (ICH) that finds feasible solutions for the MkP problem. We compare ICH to the hyperplane rounding techniques of Goemans and Williamson and of Frieze and Jerrum, and the computational results support the conclusion that ICH consistently provides better feasible solutions for the MkP problem. ICH is used in our SBC algorithm to provide feasible solutions at each node of the branchandbound tree. The SBC algorithm computes globally optimal solutions for dense graphs with up to 60 vertices, for grid graphs with up to 100 vertices, and for different values of k, providing the best exact approach to date for k ≥ 3.
New Maximally Stable Gaussian Partitions with Discrete Applications
, 2009
"... Gaussian noise stability results have recently played an important role in proving fundamental results in hardness of approximation in computer science and in the study of voting schemes in social choice. We propose two Gaussian noise stability conjectures and derive consequences of the conjectures ..."
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Cited by 4 (1 self)
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Gaussian noise stability results have recently played an important role in proving fundamental results in hardness of approximation in computer science and in the study of voting schemes in social choice. We propose two Gaussian noise stability conjectures and derive consequences of the conjectures in hardness of approximation and social choice. Both conjectures generalize isoperimetric results by Borell on the heat kernel. One of the conjectures may be also be viewed as a generalization of the ”Double Bubble ” theorem. The applications of the conjectures include an optimality result for majority in the context of Condorcet voting and a proof that the FriezeJerrum SDP for MAXqCUT achieves the optimal approximation factor assuming the Unique Games Conjecture. We finally derive a short proof of the first conjecture based on the extended Riesz inequality. 1