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On nonapproximability for quadratic programs
 IN 46TH ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2005
"... This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x ∈ {−1, 1} n that maximizes x T Mx. This problem recently attracted attention due to its application in various clustering settings, as well ..."
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This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x ∈ {−1, 1} n that maximizes x T Mx. This problem recently attracted attention due to its application in various clustering settings, as well as an intriguing connection to the famous Grothendieck inequality. It is approximable to within a factor of O(log n), and known to be NPhard to approximate within any factor better than 13/11 − ɛ for all ɛ> 0. We show that it is quasiNPhard to approximate to a factor better than O(log γ n) for some γ> 0. The integrality gap of the natural semidefinite relaxation for this problem is known as the Grothendieck constant of the complete graph, and known to be Θ(log n). The proof of this fact was nonconstructive, and did not yield an explicit problem instance where this integrality gap is achieved. Our techniques yield an explicit instance for which the integrality gap is log n Ω ( log log n), essentially answering one of the open problems of Alon et al. [AMMN].
On Approximating Complex Quadratic Optimization Problems Via Semidefinite Programming Relaxations
 Mathematical Programming, Series B
, 2007
"... Abstract. In this paper we study semidefinite programming (SDP) models for a class of discrete and continuous quadratic optimization problems in the complex Hermitian form. These problems capture a class of well–known combinatorial optimization problems, as well as problems in control theory. For i ..."
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Cited by 28 (4 self)
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Abstract. In this paper we study semidefinite programming (SDP) models for a class of discrete and continuous quadratic optimization problems in the complex Hermitian form. These problems capture a class of well–known combinatorial optimization problems, as well as problems in control theory. For instance, they include Max–3–Cut with arbitrary edge weights (i.e. some of the edge weights might be negative). We present a generic algorithm and a unified analysis of the SDP relaxations which allow us to obtain good approximation guarantees for our models. Specifically, we give an (k sin(pi/k))2/(4pi)–approximation algorithm for the discrete problem where the decision variables are k– ary and the objective matrix is positive semidefinite. To the best of our knowledge, this is the first known approximation result for this family of problems. For the continuous problem where the objective matrix is positive semidefinite, we obtain the well–known pi/4 result due to [2], and independently, [12]. However, our techniques simplify their analyses and provide a unified framework for treating these problems. In addition, we show for the first time that the integrality gap of the SDP relaxation is precisely pi/4. We also show that the unified analysis can be used to obtain an O(1 / log n)–approximation algorithm for the continuous problem in the case where the objective matrix is not positive semidefinite. 1
Tight bounds for Parameterized Complexity of Cluster Editing
, 2013
"... In the Correlation Clustering problem, also known as Cluster Editing, we are given an undirected graph G and a positive integer k; the task is to decide whether G can be transformed into a cluster graph, i.e., a disjoint union of cliques, by changing at most k adjacencies, that is, by adding or dele ..."
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Cited by 6 (3 self)
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In the Correlation Clustering problem, also known as Cluster Editing, we are given an undirected graph G and a positive integer k; the task is to decide whether G can be transformed into a cluster graph, i.e., a disjoint union of cliques, by changing at most k adjacencies, that is, by adding or deleting at most k edges. The motivation of the problem stems from various tasks in computational biology (BenDor et al., Journal of Computational Biology 1999) and machine learning (Bansal et al., Machine Learning 2004). Although in general Correlation Clustering is APXhard (Charikar et al., FOCS 2003), the version of the problem where the number of cliques may not exceed a prescribed constant p admits a PTAS (Giotis and Guruswami, SODA 2006). We study the parameterized complexity of Correlation Clustering with this restriction on the number of cliques to be created. We give an algorithm that in time O(2 O( √ pk) + n + m) decides whether a graph G on n vertices and m edges can be transformed into a cluster graph with exactly p cliques by changing at most k adjacencies.
THE AIZENMANSIMSSTARR SCHEME FOR THE SK MODEL WITH MULTIDIMENSIONAL SPINS
, 2007
"... The nonhierarchical correlation structure of the SherringtonKirkpatrick (SK) model with multidimensional (e.g. Heisenberg) spins is studied at the level of the logarithmic asymptotic of the corresponding sum of the correlated exponentials – the thermodynamic pressure. For this purpose an abstrac ..."
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The nonhierarchical correlation structure of the SherringtonKirkpatrick (SK) model with multidimensional (e.g. Heisenberg) spins is studied at the level of the logarithmic asymptotic of the corresponding sum of the correlated exponentials – the thermodynamic pressure. For this purpose an abstract quenched large deviations principle (LDP) of GärtnerEllis type is obtained under an assumption of measure concentration. With the aid of this principle the framework of the AizenmanSimsStarr comparison scheme (AS 2 scheme) is extended to the case of the SK model with multidimensional spins. This extension, based the quenched LDP, shows how the Hadamard matrix products arise rigorously in the context of the Parisi formula. This allows one to relate the pressure of the nonhierarchical SK model with the pressure of the hierarchical GREM by a saddlepoint variational formula of the Parisi type including a negative remainder term. Let Σ ⊂
Sorting Noisy Data with Partial Information
"... In this paper, we propose two semirandom models for the Minimum Feedback Arc Set Problem and present approximation algorithms for them. In the first model, which we call the Random Edge Flipping model, an instance is generated as follows. We start with an arbitrary acyclic directed graph and then ..."
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In this paper, we propose two semirandom models for the Minimum Feedback Arc Set Problem and present approximation algorithms for them. In the first model, which we call the Random Edge Flipping model, an instance is generated as follows. We start with an arbitrary acyclic directed graph and then randomly flip its edges (the adversary may later unflip some of them). In the second model, which we call the Random Backward Edge model, again we start with an arbitrary acyclic graph but now add new random backward edges (the adversary may delete some of them). For the first model, we give an approximation algorithm that finds a solution of cost (1 + δ) optcost+npolylog n, where optcost is the cost of the optimal solution. For the second model, we give an approximation algorithm that finds a solution of cost O(plantedcost)+npolylog n, where plantedcost is the cost of the planted solution. Additionally, we present an approximation algorithm for semirandom instances of Minimum Directed Balanced Cut.
The Approximability of Learning and Constraint Satisfaction Problems
, 2010
"... International Business Machine. The views and conclusions contained in this document are those of the ..."
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International Business Machine. The views and conclusions contained in this document are those of the