Topic
Clique problem
About: Clique problem is a research topic. Over the lifetime, 1252 publications have been published within this topic receiving 31280 citations.
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Papers
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TL;DR: The approximation schemes for hierarchically specified unit disk graphs presented in this paper are among the first approximation schemes in the literature for natural PSPACE-hard optimization problems.
345 citations
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07 Jun 2015TL;DR: This paper formulate data association as a Generalized Maximum Multi Clique problem (GMMCP) and shows that this is the ideal case of modeling tracking in real world scenario where all the pairwise relationships between targets in a batch of frames are taken into account.
Abstract: Data association is the backbone to many multiple object tracking (MOT) methods. In this paper we formulate data association as a Generalized Maximum Multi Clique problem (GMMCP). We show that this is the ideal case of modeling tracking in real world scenario where all the pairwise relationships between targets in a batch of frames are taken into account. Previous works assume simplified version of our tracker either in problem formulation or problem optimization. However, we propose a solution using GMMCP where no simplification is assumed in either steps. We show that the NP hard problem of GMMCP can be formulated through Binary-Integer Program where for small and medium size MOT problems the solution can be found efficiently. We further propose a speed-up method, employing Aggregated Dummy Nodes for modeling occlusion and miss-detection, which reduces the size of the input graph without using any heuristics. We show that, using the speedup method, our tracker lends itself to real-time implementation which is plausible in many applications. We evaluated our tracker on six challenging sequences of Town Center, TUD-Crossing, TUD-Stadtmitte, Parking-lot 1, Parking-lot 2 and Parking-lot pizza and show favorable improvement against state of art.
328 citations
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TL;DR: Many important combinatorial optimization problems, such as the traveling salesman problem (TSP), the clique problem and many others, call for the optimization of a linear functional over some discrete set of vectors.
319 citations
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TL;DR: An algorithmic approach is developed exploiting the graph-theoretic properties of a k-plex that is effective in solving the problem to optimality on very large, sparse graphs such as the power law graphs frequently encountered in the applications of interest.
Abstract: This paper introduces and studies the maximum k-plex problem, which arises in social network analysis and has wider applicability in several important areas employing graph-based data mining. After establishing NP-completeness of the decision version of the problem on arbitrary graphs, an integer programming formulation is presented, followed by a polyhedral study to identify combinatorial valid inequalities and facets. A branch-and-cut algorithm is implemented and tested on proposed benchmark instances. An algorithmic approach is developed exploiting the graph-theoretic properties of a k-plex that is effective in solving the problem to optimality on very large, sparse graphs such as the power law graphs frequently encountered in the applications of interest.
300 citations
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13 Jun 2013TL;DR: The performance of a test is measured by the smallest signal strength that it can detect and a computationally efficient method based on semidefinite programming is proposed and it is proved that the statistical performance of this test cannot be strictly improved by any computationallyefficient method.
Abstract: In the context of sparse principal component detection, we bring evidence towards the existence of a statistical price to pay for computational efficiency. We measure the performance of a test by the smallest signal strength that it can detect and we propose a computationally efficient method based on semidefinite programming.We also prove that the statistical performance of this test cannot be strictly improved by any computationally efficient method. Our results can be viewed as complexity theoretic lower bounds conditionally on the assumptions that some instances of the planted clique problem cannot be solved in randomized polynomial time.
275 citations