On Sparsification for Computing Treewidth

We investigate whether an n-vertex instance (G,k) of Treewidth, asking whether the graph G has treewidth at most k, can efficiently be made sparse without changing its answer. By giving a special form of OR-cross-composition, we prove that this is unlikely: if there is an e > 0 and a polynomial-time algorithm that reduces n-vertex Treewidth instances to equivalent instances, of an arbitrary problem, with O(n^{2-e}) bits, then NP is in coNP/poly and the polynomial hierarchy collapses to its third level. Our sparsification lower bound has implications for structural parameterizations of Treewidth: parameterizations by measures that do not exceed the vertex count, cannot have kernels with O(k^{2-e}) bits for any e > 0, unless NP is in coNP/poly. Motivated by the question of determining the optimal kernel size for Treewidth parameterized by vertex cover, we improve the O(k^3)-vertex kernel from Bodlaender et al. (STACS 2011) to a kernel with O(k^2) vertices. Our improved kernel is based on a novel form of treewidth-invariant set. We use the q-expansion lemma of Fomin et al. (STACS 2011) to find such sets efficiently in graphs whose vertex count is superquadratic in their vertex cover number.Comment: 21 pages. Full version of the extended abstract presented at IPEC 201

On the dynamics of the glass transition on Bethe lattices

The Glauber dynamics of disordered spin models with multi-spin interactions on sparse random graphs (Bethe lattices) is investigated. Such models undergo a dynamical glass transition upon decreasing the temperature or increasing the degree of constrainedness. Our analysis is based upon a detailed study of large scale rearrangements which control the slow dynamics of the system close to the dynamical transition. Particular attention is devoted to the neighborhood of a zero temperature tricritical point. Both the approach and several key results are conjectured to be valid in a considerably more general context.Comment: 56 pages, 38 eps figure

XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure

In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth

XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure

In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth