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The connectivity of boolean satisfiability: Computational and structural dichotomies
 in Proc. 33 rd Intl. Colloquium on Automata, Languages and Programming
"... Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we ..."
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Cited by 15 (4 self)
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Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivityrelated properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaeferâ€™s framework. On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and stconnectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACEcomplete, while the tractable side which includes but is not limited to all problems with polynomial time algorithms for satisfiability is in P for the stconnectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACEcomplete cases, whereas in all other cases it is linear; thus, small diameter and tractability of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space possess certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary. 1
Approximability of the Subset Sum Reconfiguration Problem
"... Abstract. The subset sum problem is a wellknown NPcomplete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this pa ..."
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Cited by 1 (1 self)
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Abstract. The subset sum problem is a wellknown NPcomplete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this paper, we study the problem of reconfiguring one packing into another packing by moving only one item at a time, while at all times maintaining the feasibility of packings. First we show that this decision problem is strongly NPhard, and is PSPACEcomplete if we are given a conflict graph for the set of items in which each vertex corresponds to an item and each edge represents a pair of items that are not allowed to be packed together into the knapsack. We then study an optimization version of the problem: we wish to maximize the minimum sum among all packings in the reconfiguration. We show that this maximization problem admits a polynomialtime approximation scheme (PTAS), while the problem is APXhard if we are given a conflict graph. 1