Detecting and counting small subgraphs, and evaluating a parameterized Tutte polynomial: lower bounds via toroidal grids and Cayley graph expanders

Given a graph property $\Phi$, we consider the problem $\mathtt{EdgeSub}(\Phi)$, where the input is a pair of a graph $G$ and a positive integer $k$, and the task is to decide whether $G$ contains a $k$-edge subgraph that satisfies $\Phi$. Specifically, we study the parameterized complexity of $\mathtt{EdgeSub}(\Phi)$ and of its counting problem $\#\mathtt{EdgeSub}(\Phi)$ with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties $\Phi$: the decision problem $\mathtt{EdgeSub}(\Phi)$ always admits an FPT algorithm and the counting problem $\#\mathtt{EdgeSub}(\Phi)$ always admits an FPTRAS. For exact counting, we present an exhaustive and explicit criterion on the property $\Phi$ which, if satisfied, yields fixed-parameter tractability and otherwise $\#\mathsf{W[1]}$-hardness. Additionally, most of our hardness results come with an almost tight conditional lower bound under the so-called Exponential Time Hypothesis, ruling out algorithms for $\#\mathtt{EdgeSub}(\Phi)$ that run in time $f(k)\cdot|G|^{o(k/\log k)}$ for any computable function $f$. As a main technical result, we gain a complete understanding of the coefficients of toroidal grids and selected Cayley graph expanders in the homomorphism basis of $\#\mathtt{EdgeSub}(\Phi)$. This allows us to establish hardness of exact counting using the Complexity Monotonicity framework due to Curticapean, Dell and Marx (STOC'17). Our methods can also be applied to a parameterized variant of the Tutte Polynomial $T^k_G$ of a graph $G$, to which many known combinatorial interpretations of values of the (classical) Tutte Polynomial can be extended. As an example, $T^k_G(2,1)$ corresponds to the number of $k$-forests in the graph $G$. Our techniques allow us to completely understand the parametrized complexity of computing the evaluation of $T^k_G$ at every pair of rational coordinates $(x,y)$

Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths

We systematically investigate the complexity of counting subgraph patterns modulo fixed integers. For example, it is known that the parity of the number of $k$-matchings can be determined in polynomial time by a simple reduction to the determinant. We generalize this to an $n^{f(t,s)}$-time algorithm to compute modulo $2^t$ the number of subgraph occurrences of patterns that are $s$ vertices away from being matchings. This shows that the known polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry over into the setting of counting modulo $2^t$. Complementing our algorithm, we also give a simple and self-contained proof that counting $k$-matchings modulo odd integers $q$ is Mod_q-W[1]-complete and prove that counting $k$-paths modulo $2$ is Parity-W[1]-complete, answering an open question by Bj\"orklund, Dell, and Husfeldt (ICALP 2015).Comment: 23 pages, to appear at ESA 202

Linear Orderings of Sparse Graphs

The Linear Ordering problem consists in finding a total ordering of the vertices of a directed graph such that the number of backward arcs, i.e., arcs whose heads precede their tails in the ordering, is minimized. A minimum set of backward arcs corresponds to an optimal solution to the equivalent Feedback Arc Set problem and forms a minimum Cycle Cover. Linear Ordering and Feedback Arc Set are classic NP-hard optimization problems and have a wide range of applications. Whereas both problems have been studied intensively on dense graphs and tournaments, not much is known about their structure and properties on sparser graphs. There are also only few approximative algorithms that give performance guarantees especially for graphs with bounded vertex degree. This thesis fills this gap in multiple respects: We establish necessary conditions for a linear ordering (and thereby also for a feedback arc set) to be optimal, which provide new and fine-grained insights into the combinatorial structure of the problem. From these, we derive a framework for polynomial-time algorithms that construct linear orderings which adhere to one or more of these conditions. The analysis of the linear orderings produced by these algorithms is especially tailored to graphs with bounded vertex degrees of three and four and improves on previously known upper bounds. Furthermore, the set of necessary conditions is used to implement exact and fast algorithms for the Linear Ordering problem on sparse graphs. In an experimental evaluation, we finally show that the property-enforcing algorithms produce linear orderings that are very close to the optimum and that the exact representative delivers solutions in a timely manner also in practice. As an additional benefit, our results can be applied to the Acyclic Subgraph problem, which is the complementary problem to Feedback Arc Set, and provide insights into the dual problem of Feedback Arc Set, the Arc-Disjoint Cycles problem

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets $\sigma,\rho$ of non-negative integers, a $(\sigma,\rho)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)\cap S|\in \sigma$ for every $u\in S$, and $|N(v)\cap S|\in \rho$ for every $v\not\in S$. The problem of finding a $(\sigma,\rho)$-set (of a certain size) unifies standard problems such as Independent Set, Dominating Set, Independent Dominating Set, and many others. For all pairs of finite or cofinite sets $(\sigma,\rho)$, we determine (under standard complexity assumptions) the best possible value $c_{\sigma,\rho}$ such that there is an algorithm that counts $(\sigma,\rho)$-sets in time $c_{\sigma,\rho}^{\sf tw}\cdot n^{O(1)}$ (if a tree decomposition of width ${\sf tw}$ is given in the input). For example, for the Exact Independent Dominating Set problem (also known as Perfect Code) corresponding to $\sigma=\{0\}$ and $\rho=\{1\}$, we improve the $3^{\sf tw}\cdot n^{O(1)}$ algorithm of [van Rooij, 2020] to $2^{\sf tw}\cdot n^{O(1)}$. Despite the unusually delicate definition of $c_{\sigma,\rho}$, we show that our algorithms are most likely optimal, i.e., for any pair $(\sigma, \rho)$ of finite or cofinite sets where the problem is non-trivial, and any $\varepsilon>0$, a $(c_{\sigma,\rho}-\varepsilon)^{\sf tw}\cdot n^{O(1)}$-algorithm counting the number of $(\sigma,\rho)$-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets $\sigma$ and $\rho$, our lower bounds also extend to the decision version, showing that our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets $\sigma,\rho$ of non-negative integers, a $(\sigma,\rho)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)\cap S|\in \sigma$ for every $u\in S$, and $|N(v)\cap S|\in \rho$ for every $v\not\in S$. The problem of finding a $(\sigma,\rho)$-set (of a certain size) unifies standard problems such as \textsc{Independent Set}, \textsc{Dominating Set}, \textsc{Independent Dominating Set}, and many others. For almost all pairs of finite or cofinite sets $(\sigma,\rho)$, we determine (under standard complexity assumptions) the best possible value $c_{\sigma,\rho}$ such that there is an algorithm that counts $(\sigma,\rho)$-sets in time c_{\sigma,\rho}^\tw\cdot n^{\O(1)} (if a tree decomposition of width \tw is given in the input). Let \sigMax denote the largest element of $\sigma$ if $\sigma$ is finite, or the largest missing integer $+1$ if $\sigma$ is cofinite; \rhoMax is defined analogously for $\rho$. Surprisingly, $c_{\sigma,\rho}$ is often significantly smaller than the natural bound \sigMax+\rhoMax+2 achieved by existing algorithms [van Rooij, 2020]. Toward defining $c_{\sigma,\rho}$, we say that $(\sigma, \rho)$ is \mname-structured if there is a pair $(\alpha,\beta)$ such that every integer in $\sigma$ equals $\alpha$ mod \mname, and every integer in $\rho$ equals $\beta$ mod \mname. Then, setting \begin{itemize} \item c_{\sigma,\rho}=\max\{\sigMax,\rhoMax\}+1 if $(\sigma,\rho)$ is \mname-structured for some \mname \ge 3, or 2-structured with \sigMax\neq \rhoMax, or 2-structured with \sigMax=\rhoMax being odd, \item c_{\sigma,\rho}=\max\{\sigMax,\rhoMax\}+2 if $(\sigma,\rho)$ is 2-structured, but not \mname-structured for any \mname \ge 3, and \sigMax=\rhoMax is even, and \item c_{\sigma,\rho}=\sigMax+\rhoMax+2 if $(\sigma,\rho)$ is not \mname-structured for any \mname\ge 2, \end{itemize} we provide algorithms counting $(\sigma,\rho)$-sets in time c_{\sigma,\rho}^\tw\cdot n^{\O(1)}. For example, for the \textsc{Exact Independent Dominating Set} problem (also known as \textsc{Perfect Code}) corresponding to $\sigma=\{0\}$ and $\rho=\{1\}$, this improves the 3^\tw\cdot n^{\O(1)} algorithm of van Rooij to 2^\tw\cdot n^{\O(1)}. Despite the unusually delicate definition of $c_{\sigma,\rho}$, we show that our algorithms are most likely optimal, i.e., for any pair $(\sigma, \rho)$ of finite or cofinite sets where the problem is non-trivial (except those having cofinite $\sigma$ with $\rho=\mathbb Z_{\ge0}$), and any $\varepsilon>0$, a (c_{\sigma,\rho}-\varepsilon)^\tw\cdot n^{\O(1)}-algorithm counting the number of $(\sigma,\rho)$-sets would violate the Counting Strong Exponential-Time Hypothesis (\#SETH). For finite sets $\sigma$ and $\rho$, our lower bounds also extend to the decision version, showing that our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets