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Tighter Relaxations for HigherOrder Models based on Generalized Roof Duality
"... Many problems in computer vision can be turned into a largescale boolean optimization problem, which is in general NPhard. In this paper, we further develop one of the most successful approaches, namely roof duality, for approximately solving such problems for higherorder models. Two new methods ..."
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Many problems in computer vision can be turned into a largescale boolean optimization problem, which is in general NPhard. In this paper, we further develop one of the most successful approaches, namely roof duality, for approximately solving such problems for higherorder models. Two new methods that can be applied independently or in combination are investigated. The first one is based on constructing relaxations using generators of the submodular function cone. In the second method, it is shown that the roof dual bound can be applied in an iterated way in order to obtain a tighter relaxation. We also provide experimental results that demonstrate better performance with respect to the stateoftheart, both in terms of improved bounds and the number of optimally assigned variables.
The Complexity of Valued Constraint Satisfaction
"... We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems. We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify general features of the complexity landscape. ..."
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We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems. We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify general features of the complexity landscape.
Halfintegrality, LPbranching and FPT Algorithms
, 2014
"... A recent trend in parameterized algorithms is the application of polytope tools (specifically, LPbranching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). Though the list of work in this direction is short, the results are already interesting, yielding significant speedup ..."
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A recent trend in parameterized algorithms is the application of polytope tools (specifically, LPbranching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). Though the list of work in this direction is short, the results are already interesting, yielding significant speedups for a range of important problems. However, the existing approaches require the underlying polytope to have very restrictive properties, including halfintegrality and NemhauserTrotterstyle persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994). Taking a slightly different approach, we view halfintegrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from {0, 1}V to {0, 1/2, 1}V such that the new problem admits a polynomialtime exact solution. Using tools from CSP (in particular Thapper and Živný, 2012) to study the existence of such relaxations, we are able to provide a much broader class of halfintegral polytopes with the required properties. Our results unify and significantly extend the previously known cases. In addition to the new insight into problems with halfintegral relaxations, our results yield a range of new and improved FPT algo