Fast algorithms for the Tron game on trees

Abstract TR.ON is the following game which can be played on any graph: Two players choose alternately a node of the graph subject to the requirement that each player must choose a node which is adjacent to his previously chosen node and such that every node is chosen only once. In this paper O(n) and O(ny'n) algorithms are given for deciding whether there is a winning strategy for the first player when TR.oN is played on a given tree, for the variants with and without specified starting nodes, respectively. The problem is shown to be both NP-hard and co-NP-hard for connected undirected graphs in general

Exact Weight Subgraphs and the k-Sum Conjecture

We consider the Exact-Weight-H problem of finding a (not necessarily induced) subgraph H of weight 0 in an edge-weighted graph G. We show that for every H, the complexity of this problem is strongly related to that of the infamous k-Sum problem. In particular, we show that under the k-Sum Conjecture, we can achieve tight upper and lower bounds for the Exact-Weight-H problem for various subgraphs H such as matching, star, path, and cycle. One interesting consequence is that improving on the O(n^3) upper bound for Exact-Weight-4-Path or Exact-Weight-5-Path will imply improved algorithms for 3-Sum, 5-Sum, All-Pairs Shortest Paths and other fundamental problems. This is in sharp contrast to the minimum-weight and (unweighted) detection versions, which can be solved easily in time O(n^2). We also show that a faster algorithm for any of the following three problems would yield faster algorithms for the others: 3-Sum, Exact-Weight-3-Matching, and Exact-Weight-3-Star

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

We consider the multivariate interlace polynomial introduced by Courcelle (2008), which generalizes several interlace polynomials defined by Arratia, Bollobas, and Sorkin (2004) and by Aigner and van der Holst (2004). We present an algorithm to evaluate the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. The best previously known result (Courcelle 2008) employs a general logical framework and leads to an algorithm with running time f(k)*n, where f(k) is doubly exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context of tree decompositions, we give a faster and more direct algorithm. Our algorithm uses 2^{3k^2+O(k)}*n arithmetic operations and can be efficiently implemented in parallel.Comment: v4: Minor error in Lemma 5.5 fixed, Section 6.6 added, minor improvements. 44 pages, 14 figure

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

On Structural Parameterizations of the Edge Disjoint Paths Problem

In this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain NP-hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the W[1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph

Deleting edges to restrict the size of an epidemic: a new application for treewidth

Motivated by applications in network epidemiology, we consider the problem of determining whether it is possible to delete at most k edges from a given input graph (of small treewidth) so that the resulting graph avoids a set FF of forbidden subgraphs; of particular interest is the problem of determining whether it is possible to delete at most k edges so that the resulting graph has no connected component of more than h vertices, as this bounds the worst-case size of an epidemic. While even this special case of the problem is NP-complete in general (even when h=3h=3 ), we provide evidence that many of the real-world networks of interest are likely to have small treewidth, and we describe an algorithm which solves the general problem in time 2O(|F|wr)n2O(|F|wr)n  on an input graph having n vertices and whose treewidth is bounded by a fixed constant w, if each of the subgraphs we wish to avoid has at most r vertices. For the special case in which we wish only to ensure that no component has more than h vertices, we improve on this to give an algorithm running in time O((wh)2wn)O((wh)2wn) , which we have implemented and tested on real datasets based on cattle movements

A Novel Cold-Regulated Cold Shock Domain Containing Protein from Scallop Chlamys farreri with Nucleic Acid-Binding Activity

Background: The cold shock domain (CSD) containing proteins (CSDPs) are one group of the evolutionarily conserved nucleic acid-binding proteins widely distributed in bacteria, plants, animals, and involved in various cellular processes, including adaptation to low temperature, cellular growth, nutrient stress and stationary phase. Methodology: The cDNA of a novel CSDP was cloned from Zhikong scallop Chlamys farreri (designated as CfCSP) by expressed sequence tag (EST) analysis and rapid amplification of cDNA ends (RACE) approach. The full length cDNA of CfCSP was of 1735 bp containing a 927 bp open reading frame which encoded an N-terminal CSD with conserved nucleic acids binding motif and a C-terminal domain with four Arg-Gly-Gly (RGG) repeats. The CSD of CfCSP shared high homology with the CSDs from other CSDPs in vertebrate, invertebrate and bacteria. The mRNA transcripts of CfCSP were mainly detected in the tissue of adductor and also marginally detectable in gill, hepatopancreas, hemocytes, kidney, mantle and gonad of healthy scallop. The relative expression level of CfCSP was up-regulated significantly in adductor and hemocytes at 1 h and 24 h respectively after low temperature treatment (P,0.05). The recombinant CfCSP protein (rCfCSP) could bind ssDNA and in vitro transcribed mRNA, but it could not bind dsDNA. BX04, a cold sensitive Escherichia coli CSP quadruple-deletion mutant, was used to examine the cold adaptation ability of CfCSP. After incubation at 17uC for 120 h, the strain of BX04 containing the vector pINIII showed growth defect and failed to form colonies, while strain containing pINIII-CSPA or pINIII

Formation of Amyloid-Like Fibrils by Y-Box Binding Protein 1 (YB-1) Is Mediated by Its Cold Shock Domain and Modulated by Disordered Terminal Domains

YB-1, a multifunctional DNA- and RNA-binding nucleocytoplasmic protein, is involved in the majority of DNA- and mRNA-dependent events in the cell. It consists of three structurally different domains: its central cold shock domain has the structure of a β-barrel, while the flanking domains are predicted to be intrinsically disordered. Recently, we showed that YB-1 is capable of forming elongated fibrils under high ionic strength conditions. Here we report that it is the cold shock domain that is responsible for formation of YB-1 fibrils, while the terminal domains differentially modulate this process depending on salt conditions. We demonstrate that YB-1 fibrils have amyloid-like features, including affinity for specific dyes and a typical X-ray diffraction pattern, and that in contrast to most of amyloids, they disassemble under nearly physiological conditions