Results 1 
3 of
3
Shortest reconfiguration paths in the solution space of Boolean formulas
"... Abstract. Given a Boolean formula and a satisfying assignment, a flip is an operation that changes the value of a variable in the assignment so that the resulting assignment remains satisfying. We study the problem of computing the shortest sequence of flips (if one exists) that transforms a given s ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Given a Boolean formula and a satisfying assignment, a flip is an operation that changes the value of a variable in the assignment so that the resulting assignment remains satisfying. We study the problem of computing the shortest sequence of flips (if one exists) that transforms a given satisfying assignment s to another satisfying assignment t of the Boolean formula. Earlier work characterized the complexity of deciding the existence of a sequence of flips between two given satisfying assignments using Schaefer’s framework for classification of Boolean formulas. We build on it to provide a trichotomy for the complexity of finding the shortest sequence of flips and show that it is either in P, NPcomplete, or PSPACEcomplete. Our result adds to the small set of complexity results known for shortest reconfiguration sequence problems by providing an example where the shortest sequence can be found in polynomial time even though the path flips variables that have the same value in both s and t. This is in contrast to all reconfiguration problems studied so far, where polynomial time algorithms for computing the shortest path were known only for cases where the path modified the symmetric difference only. Our proof uses Birkhoff’s representation theorem on a set system that we show to be a distributive lattice. The technique is insightful and can perhaps be used for other reconfiguration problems as well.
Reconfiguration on sparse graphs
"... Abstract. A vertexsubset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertexsubset problem asks, given two feasible solutions Ss and St of size k, whether it is possible to transform Ss into St by a sequence of verte ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. A vertexsubset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertexsubset problem asks, given two feasible solutions Ss and St of size k, whether it is possible to transform Ss into St by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We study reconfiguration variants of two classical vertexsubset problems, namely Independent Set and Dominating Set. We denote the former by ISR and the latter by DSR. Both ISR and DSR are PSPACEcomplete on graphs of bounded bandwidth and W[1]hard parameterized by k on general graphs. We show that ISR is fixedparameter tractable parameterized by k when the input graph is of bounded degeneracy or nowheredense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that ISR is fixedparameter tractable on planar graphs and graphs of bounded degree. For DSR, we show the problem fixedparameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes graphs of bounded degeneracy and nowheredense graphs. 1