On the Boolean connectivity problem for Horn relations
Discrete Applied Mathematics
On the complexity of reconfiguration problems
Theoretical Computer Science
Shortest paths between shortest paths and independent sets
IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
Approximability of the subset sum reconfiguration problem
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
An improved sufficient condition for reconfiguration of list edge-colorings in a tree
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
An exact algorithm for the Boolean connectivity problem for k-CNF
Theoretical Computer Science
An exact algorithm for the boolean connectivity problem for k-CNF
SAT'10 Proceedings of the 13th international conference on Theory and Applications of Satisfiability Testing
Complexity of independent set reconfigurability problems
Theoretical Computer Science
Reconfiguration of list edge-colorings in a graph
Discrete Applied Mathematics
The complexity of rerouting shortest paths
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
The complexity of rerouting shortest paths
Theoretical Computer Science
Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs
Journal of Combinatorial Optimization
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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 connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer'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 $st$-connectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACE-complete, while the tractable side—which includes but is not limited to all problems with polynomial-time algorithms for satisfiability—is in P for the $st$-connectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACE-complete cases, whereas in all other cases it is linear; thus, diameter and complexity 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.