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87
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
Optimal bounds on approximation of submodular and xos functions by juntas
 CoRR
"... Abstract—We investigate the approximability of several classes of realvalued functions by functions of a small number of variables (juntas). Our main results are tight bounds on the number of variables required to approximate a function f: {0, 1}n → [0, 1] within `2error over the uniform distribu ..."
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Cited by 14 (5 self)
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Abstract—We investigate the approximability of several classes of realvalued functions by functions of a small number of variables (juntas). Our main results are tight bounds on the number of variables required to approximate a function f: {0, 1}n → [0, 1] within `2error over the uniform distribution: • If f is submodular, then it is close to a function of
CYCLE KILLER... QU’ESTCE QUE C’EST? ON THE COMPARATIVE APPROXIMABILITY OF HYBRIDIZATION NUMBER AND DIRECTED FEEDBACK VERTEX SET
"... We show that the problem of computing the hybridization number of two rooted binary phylogenetic trees on the same set of taxa X has a constant factor polynomialtime approximation if and only if the problem of computing a minimumsize feedback vertex set in a directed graph (DFVS) has a constant f ..."
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Cited by 11 (7 self)
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We show that the problem of computing the hybridization number of two rooted binary phylogenetic trees on the same set of taxa X has a constant factor polynomialtime approximation if and only if the problem of computing a minimumsize feedback vertex set in a directed graph (DFVS) has a constant factor polynomialtime approximation. The latter problem, which asks for a minimum number of vertices to be removed from a directed graph to transform it into a directed acyclic graph, is one of the problems in Karp’s seminal 1972 list of 21 NPcomplete problems. Despite considerable attention from the combinatorial optimization community, it remains to this day unknown whether a constant factor polynomialtime approximation exists for DFVS. Our result thus places the (in)approximability of hybridization number in a much broader complexity context, and as a consequence we obtain that it inherits inapproximability results from the problem Vertex Cover. On the positive side, we use results from the DFVS literature to give an O(log r log log r) approximation for the hybridization number where r is the correct value.
Connections in Networks: Hardness of Feasibility versus Optimality
, 2007
"... We study the complexity of combinatorial problems that consist of competing infeasibility and optimization components. In particular, we investigate the complexity of the connection subgraph problem, which occurs, e.g., in resource environment economics and social networks. We present results on i ..."
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Cited by 10 (7 self)
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We study the complexity of combinatorial problems that consist of competing infeasibility and optimization components. In particular, we investigate the complexity of the connection subgraph problem, which occurs, e.g., in resource environment economics and social networks. We present results on its worstcase hardness and approximability. We then provide a typicalcase analysis by means of a detailed computational study. First, we identify an easyhardeasy pattern, coinciding with the feasibility phase transition of the problem. Second, our experimental results reveal an interesting interplay between feasibility and optimization. They surprisingly show that proving optimality of the solution of the feasible instances can be substantially easier than proving infeasibility of the infeasible instances in a computationally hard region of the problem space. We also observe an intriguing easyhardeasy profile for the optimization component itself.
Vertex and Edge Covers with Clustering Properties: Complexity and Algorithms
, 2006
"... We consider the concepts of a ttotal vertex cover and a ttotal edge cover (t 1), which generalize the notions of a vertex cover and an edge cover, respectively. A ttotal vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of ..."
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Cited by 9 (2 self)
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We consider the concepts of a ttotal vertex cover and a ttotal edge cover (t 1), which generalize the notions of a vertex cover and an edge cover, respectively. A ttotal vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has least t vertices (edges). These denitions are motivated by combining the concepts of clustering and covering in graphs. Moreover they yield a spectrum of parameters that essentially range from a vertex cover to a connected vertex cover (in the vertex case) and from an edge cover to a spanning tree (in the edge case). For various values of t, we present NPcompleteness and approximability results (both upper and lower bounds) and FPT algorithms for problems concerned with nding the minimum size of a ttotal vertex cover, ttotal edge cover and connected vertex cover, in particular improving on a previous FPT algorithm for the latter problem.
Approximability and parameterized complexity of consecutive ones submatrix problems
 IN PROC. 4TH TAMC, VOLUME 4484 OF LNCS
, 2007
"... We develop a refinement of a forbidden submatrix characterization of 0/1matrices fulfilling the Consecutive Ones Property (C1P). This novel characterization finds applications in new polynomialtime approximation algorithms and fixedparameter tractability results for the problem to find a maximum ..."
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Cited by 8 (4 self)
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We develop a refinement of a forbidden submatrix characterization of 0/1matrices fulfilling the Consecutive Ones Property (C1P). This novel characterization finds applications in new polynomialtime approximation algorithms and fixedparameter tractability results for the problem to find a maximumsize submatrix of a 0/1matrix such that the submatrix has the C1P. Moreover, we achieve a problem kernelization based on simple data reduction rules and provide several search tree algorithms. Finally, we derive inapproximability results.
On New Approaches of Assessing Network Vulnerability: Hardness and Approximation
"... Society relies heavily on its networked physical infrastructure and information systems. Accurately assessing the vulnerability of these systems against disruptive events is vital for planning and risk management. Existing approaches to vulnerability assessments of largescale systems mainly focus o ..."
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Cited by 8 (4 self)
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Society relies heavily on its networked physical infrastructure and information systems. Accurately assessing the vulnerability of these systems against disruptive events is vital for planning and risk management. Existing approaches to vulnerability assessments of largescale systems mainly focus on investigating inhomogeneous properties of the underlying graph elements. These measures and the associated heuristic solutions are limited in evaluating the vulnerability of largescale network topologies. Furthermore, these approaches often fail to provide performance guarantees of the proposed solutions. In this paper, we propose a vulnerability measure, pairwise connectivity, and use it to formulate network vulnerability assessment as a graphtheoretical optimization problem, referred to as βdisruptor. The objective is to identify the minimum set of critical network elements, namely nodes and edges, whose removal results in a specific degradation of the network global pairwise connectivity. We prove the NPCompleteness and inapproximability of this problem, and propose an O(log n log log n) pseudoapproximation algorithm to computing the set of critical nodes and an O(log 1.5 n) pseudoapproximation algorithm for computing the set of critical edges. The results of an extensive simulationbased experiment show the feasibility of our proposed vulnerability assessment framework and the efficiency of the proposed approximation algorithms in comparison with other approaches.
EWLS: A New Local Search for Minimum Vertex Cover
 PROCEEDINGS OF THE TWENTYFOURTH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE (AAAI10)
, 2010
"... A number of algorithms have been proposed for the Minimum Vertex Cover problem. However, they are far from satisfactory, especially on hard instances. In this paper, we introduce Edge Weighting Local Search (EWLS), a new local search algorithm for the Minimum Vertex Cover problem. EWLS is based on t ..."
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Cited by 7 (3 self)
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A number of algorithms have been proposed for the Minimum Vertex Cover problem. However, they are far from satisfactory, especially on hard instances. In this paper, we introduce Edge Weighting Local Search (EWLS), a new local search algorithm for the Minimum Vertex Cover problem. EWLS is based on the idea of extending a partial vertex cover into a vertex cover. A key point of EWLS is to find a vertex set that provides a tight upper bound on the size of the minimum vertex cover. To this purpose, EWLS employs an iterated local search procedure, using an edge weighting scheme which updates edge weights when stuck in local optima. Moreover, some sophisticated search strategies have been taken to improve the quality of local optima. Experimental results on the broadly used DIMACS benchmark show that EWLS is competitive with the current best heuristic algorithms, and outperforms them on hard instances. Furthermore, on a suite of difficult benchmarks, EWLS delivers the best results and sets a new record on the largest instance.
The Complexity of Finding Subgraphs Whose Matching Number Equals the Vertex Cover Number
 IN PROCEEDINGS OF THE 18TH INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION (ISAAC 2007), SPRINGER LNCS
, 2007
"... The class of graphs where the size of a minimum vertex cover equals that of a maximum matching is known as KönigEgerváry graphs. KönigEgerváry graphs have been studied extensively from a graph theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding ..."
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Cited by 7 (3 self)
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The class of graphs where the size of a minimum vertex cover equals that of a maximum matching is known as KönigEgerváry graphs. KönigEgerváry graphs have been studied extensively from a graph theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding maximumKönigEgerváry subgraphs of a given graph. More specifically, we look at the problem of finding a minimum number of vertices or edges to delete to make the resulting graph KönigEgerváry. We show that both these versions are NPcomplete and study their complexity from the points of view of approximation and parameterized complexity. En route, we point out an interesting connection between the vertex deletion version and the A G V C problem where one is interested in the parameterized complexity of the V C problem when parameterized by the ‘additional number of vertices’ needed beyond the matching size. This connection is of independent interest and could be useful in establishing the parameterized complexity of A G V C problem.