
Finding Efficient Domination for S_1,3,3Free Bipartite Graphs in Polynomial Time
A vertex set D in a finite undirected graph G is an efficient dominating...
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Finding Efficient Domination for S_1,1,5Free Bipartite Graphs in Polynomial Time
A vertex set D in a finite undirected graph G is an efficient dominating...
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Approximability of the Eightvertex Model
We initiate a study of the classification of approximation complexity of...
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Approximability of the Sixvertex Model
In this paper we take the first step toward a classification of the appr...
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Torpid Mixing of Markov Chains for the Sixvertex Model on Z^2
In this paper, we study the mixing time of two widely used Markov chain ...
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Perfect Italian domination on planar and regular graphs
A perfect Italian dominating function of a graph G=(V,E) is a function f...
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Orientations and bijections for toroidal maps with prescribed facedegrees and essential girth
We present unified bijections for maps on the torus with control on the ...
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FPRAS via MCMC where it mixes torpidly (and very little effort)
Is Fully Polynomialtime Randomized Approximation Scheme (FPRAS) for a problem via an MCMC algorithm possible when it is known that rapid mixing provably fails? We introduce several weightpreserving maps for the eightvertex model on planar and on bipartite graphs, respectively. Some are onetoone, while others are holographic which map superpositions of exponentially many states from one setting to another, in a quantumlike manytomany fashion. In fact we introduce a set of such mappings that forms a group in each case. Using some holographic maps and their compositions we obtain FPRAS for the eightvertex model at parameter settings where it is known that rapid mixing provably fails due to an intrinsic barrier. This FPRAS is indeed the same MCMC algorithm, except its state space corresponds to superpositions of the given states, where rapid mixing holds. FPRAS is also given for torus graphs for parameter settings where natural Markov chains are known to mix torpidly. Our results show that the eightvertex model is the first problem with the provable property that while NPhard to approximate on general graphs (even #Phard for planar graphs in exact complexity), it possesses FPRAS on both bipartite graphs and planar graphs in substantial regions of its parameter space.
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