Fast Algorithms for Join Operations on Tree Decompositions

Treewidth is a measure of how tree-like a graph is. It has many important algorithmic applications because many NP-hard problems on general graphs become tractable when restricted to graphs of bounded treewidth. Algorithms for problems on graphs of bounded treewidth mostly are dynamic programming algorithms using the structure of a tree decomposition of the graph. The bottleneck in the worst-case run time of these algorithms often is the computations for the so called join nodes in the associated nice tree decomposition. In this paper, we review two different approaches that have appeared in the literature about computations for the join nodes: one using fast zeta and M\"obius transforms and one using fast Fourier transforms. We combine these approaches to obtain new, faster algorithms for a broad class of vertex subset problems known as the [\sigma,\rho]-domination problems. Our main result is that we show how to solve [\sigma,\rho]-domination problems in $O(s^{t+2} t n^2 (t\log(s)+\log(n)))$ arithmetic operations. Here, t is the treewidth, s is the (fixed) number of states required to represent partial solutions of the specific [\sigma,\rho]-domination problem, and n is the number of vertices in the graph. This reduces the polynomial factors involved compared to the previously best time bound (van Rooij, Bodlaender, Rossmanith, ESA 2009) of $O( s^{t+2} (st)^{2(s-2)} n^3 )$ arithmetic operations. In particular, this removes the dependence of the degree of the polynomial on the fixed number of states~$s$.Comment: An earlier version appeared in "Treewidth, Kernels, and Algorithms. Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th Birthday" LNCS 1216

5-Approximation for ?-Treewidth Essentially as Fast as ?-Deletion Parameterized by Solution Size

The notion of ?-treewidth, where ? is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of ?-treewidth at most k can be decomposed into (arbitrarily large) ?-subgraphs which interact only through vertex sets of size ?(k) which can be organized in a tree-like fashion. ?-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for ?-deletion problems, which ask to find a minimum vertex set whose removal from a given graph G turns it into a member of ?. The bottleneck in the current parameterized algorithms lies in the computation of suitable tree ?-decompositions. We present FPT-approximation algorithms to compute tree ?-decompositions for hereditary and union-closed graph classes ?. Given a graph of ?-treewidth k, we can compute a 5-approximate tree ?-decomposition in time f(?(k)) ? n^?(1) whenever ?-deletion parameterized by solution size can be solved in time f(k) ? n^?(1) for some function f(k) ? 2^k. The current-best algorithms either achieve an approximation factor of k^?(1) or construct optimal decompositions while suffering from non-uniformity with unknown parameter dependence. Using these decompositions, we obtain algorithms solving Odd Cycle Transversal in time 2^?(k) ? n^?(1) parameterized by bipartite-treewidth and Vertex Planarization in time 2^?(k log k) ? n^?(1) parameterized by planar-treewidth, showing that these can be as fast as the solution-size parameterizations and giving the first ETH-tight algorithms for parameterizations by hybrid width measures

Treewidth, Kernels, and Algorithms : Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th Birthday

This Festschrift was published in honor of Hans L. Bodlaender on the occasion of his 60th birthday. The 14 full and 5 short contributions included in this volume show the many transformative discoveries made by H.L. Bodlaender in the areas of graph algorithms, parameterized complexity, kernelization and combinatorial games. The papers are written by his former Ph.D. students and colleagues as well as by his former Ph.D. advisor, Jan van Leeuwen

5-Approximation for H\mathcal{H}-Treewidth Essentially as Fast as H\mathcal{H}-Deletion Parameterized by Solution Size

The notion of $\mathcal{H}$-treewidth, where $\mathcal{H}$ is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of $\mathcal{H}$-treewidth at most $k$ can be decomposed into (arbitrarily large) $\mathcal{H}$-subgraphs which interact only through vertex sets of size $O(k)$ which can be organized in a tree-like fashion. $\mathcal{H}$-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for $\mathcal{H}$-deletion problems, which ask to find a minimum vertex set whose removal from a given graph $G$ turns it into a member of $\mathcal{H}$. The bottleneck in the current parameterized algorithms lies in the computation of suitable tree $\mathcal{H}$-decompositions. We present FPT approximation algorithms to compute tree $\mathcal{H}$-decompositions for hereditary and union-closed graph classes $\mathcal{H}$. Given a graph of $\mathcal{H}$-treewidth $k$, we can compute a 5-approximate tree $\mathcal{H}$-decomposition in time $f(O(k)) \cdot n^{O(1)}$ whenever $\mathcal{H}$-deletion parameterized by solution size can be solved in time $f(k) \cdot n^{O(1)}$ for some function $f(k) \geq 2^k$. The current-best algorithms either achieve an approximation factor of $k^{O(1)}$ or construct optimal decompositions while suffering from non-uniformity with unknown parameter dependence. Using these decompositions, we obtain algorithms solving Odd Cycle Transversal in time $2^{O(k)} \cdot n^{O(1)}$ parameterized by $\mathsf{bipartite}$-treewidth and Vertex Planarization in time $2^{O(k \log k)} \cdot n^{O(1)}$ parameterized by $\mathsf{planar}$-treewidth, showing that these can be as fast as the solution-size parameterizations and giving the first ETH-tight algorithms for parameterizations by hybrid width measures.Comment: Conference version to appear at the European Symposium on Algorithms (ESA 2023

Edge exploration of temporal graphs

We introduce a natural temporal analogue of Eulerian circuits and prove that, in contrast with the static case, it is NP-hard to determine whether a given temporal graph is temporally Eulerian even if strong restrictions are placed on the structure of the underlying graph and each edge is active at only three times. However, we do obtain an FPT-algorithm with respect to a new parameter called interval-membership-width which restricts the times assigned to different edges; we believe that this parameter will be of independent interest for other temporal graph problems. Our techniques also allow us to resolve two open question of Akrida, Mertzios and Spirakis [CIAC 2019] concerning a related problem of exploring temporal stars. Furthermore, we introduce a vertex-variant of interval-membership-width (which can be arbitrarily larger than its edge-counterpart) and use it to obtain an FPT-time algorithm for a natural vertex-exploration problem that remains hard even when interval-membership-width is bounded.Comment: Extended abstract of this paper appeared in IWOCA 2021: Combinatorial Algorithms pp 107-121 (doi: https://doi.org/10.1007/978-3-030-79987-8_8

Treewidth, Kernels, and Algorithms: Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th Birthday

This Festschrift was published in honor of Hans L. Bodlaender on the occasion of his 60th birthday. The 14 full and 5 short contributions included in this volume show the many transformative discoveries made by H.L. Bodlaender in the areas of graph algorithms, parameterized complexity, kernelization and combinatorial games. The papers are written by his former Ph.D. students and colleagues as well as by his former Ph.D. advisor, Jan van Leeuwen

Parameterized complexity of Bandwidth of Caterpillars and Weighted Path Emulation

In this paper, we show that Bandwidth is hard for the complexity class $W[t]$ for all $t\in {\bf N}$, even for caterpillars with hair length at most three. As intermediate problem, we introduce the Weighted Path Emulation problem: given a vertex-weighted path $P_N$ and integer $M$, decide if there exists a mapping of the vertices of $P_N$ to a path $P_M$, such that adjacent vertices are mapped to adjacent or equal vertices, and such that the total weight of the image of a vertex from $P_M$ equals an integer $c$. We show that {\sc Weighted Path Emulation}, with $c$ as parameter, is hard for $W[t]$ for all $t\in {\bf N}$, and is strongly NP-complete. We also show that Directed Bandwidth is hard for $W[t]$ for all $t\in {\bf N}$, for directed acyclic graphs whose underlying undirected graph is a caterpillar.Comment: 31 pages; 9 figure

Planar Disjoint Paths, Treewidth, and Kernels

In the Planar Disjoint Paths problem, one is given an undirected planar graph with a set of $k$ vertex pairs $(s_i,t_i)$ and the task is to find $k$ pairwise vertex-disjoint paths such that the $i$-th path connects $s_i$ to $t_i$. We study the problem through the lens of kernelization, aiming at efficiently reducing the input size in terms of a parameter. We show that Planar Disjoint Paths does not admit a polynomial kernel when parameterized by $k$ unless coNP $\subseteq$ NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e}, Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel unless the WK-hierarchy collapses. Our reduction carries over to the setting of edge-disjoint paths, where the kernelization status remained open even in general graphs. On the positive side, we present a polynomial kernel for Planar Disjoint Paths parameterized by $k + tw$, where $tw$ denotes the treewidth of the input graph. As a consequence of both our results, we rule out the possibility of a polynomial-time (Turing) treewidth reduction to $tw= k^{O(1)}$ under the same assumptions. To the best of our knowledge, this is the first hardness result of this kind. Finally, combining our kernel with the known techniques [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver, SICOMP'94] yields an alternative (and arguably simpler) proof that Planar Disjoint Paths can be solved in time $2^{O(k^2)}\cdot n^{O(1)}$, matching the result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].Comment: To appear at FOCS'23, 82 pages, 30 figure