Reconfiguration over tree decompositions

Amer E. Mouawad; Naomi Nishimura; Venkatesh Raman; Marcin Wrochna

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arXiv · 1405.2447

Reconfiguration over tree decompositions

Amer E. Mouawad·Naomi Nishimura·Venkatesh Raman·Marcin Wrochna

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Abstract

A vertex-subset graph problem $Q$ defines which subsets of the vertices of an input graph are feasible solutions. The reconfiguration version of a vertex-subset problem $Q$ asks whether it is possible to transform one feasible solution for $Q$ into another in at most $\ell$ steps, where each step is a vertex addition or deletion, and each intermediate set is also a feasible solution for $Q$ of size bounded by $k$. Motivated by recent results establishing W[1]-hardness of the reconfiguration versions of most vertex-subset problems parameterized by $\ell$, we investigate the complexity of such problems restricted to graphs of bounded treewidth. We show that the reconfiguration versions of most vertex-subset problems remain PSPACE-complete on graphs of treewidth at most $t$ but are fixed-parameter tractable parameterized by $\ell + t$ for all vertex-subset problems definable in monadic second-order logic (MSOL). To prove the latter result, we introduce a technique which allows us to circumvent cardinality constraints and define reconfiguration problems in MSOL.

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Amer E. Mouawad, Naomi Nishimura, Venkatesh Raman, Marcin Wrochna. 2014-05-10. Reconfiguration over tree decompositions. https://arxiv.org/abs/1405.2447

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