Counting Problems in Parameterized Complexity

This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity. After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large graphs, and counting perfect matchings and Hamiltonian cycles in well-structured graphs. While this survey presupposes familiarity with parameterized algorithms and complexity, we aim at explaining all relevant notions from counting complexity in a self-contained way

Data-Compression for Parametrized Counting Problems on Sparse Graphs

We study the concept of compactor, which may be seen as a counting-analogue of kernelization in counting parameterized complexity. For a function F:Sigma^* -> N and a parameterization kappa: Sigma^* -> N, a compactor (P,M) consists of a polynomial-time computable function P, called condenser, and a computable function M, called extractor, such that F=M o P, and the condensing P(x) of x has length at most s(kappa(x)), for any input x in Sigma^*. If s is a polynomial function, then the compactor is said to be of polynomial-size. Although the study on counting-analogue of kernelization is not unprecedented, it has received little attention so far. We study a family of vertex-certified counting problems on graphs that are MSOL-expressible; that is, for an MSOL-formula phi with one free set variable to be interpreted as a vertex subset, we want to count all A subseteq V(G) where |A|=k and (G,A) models phi. In this paper, we prove that every vertex-certified counting problems on graphs that is MSOL-expressible and treewidth modulable, when parameterized by k, admits a polynomial-size compactor on H-topological-minor-free graphs with condensing time O(k^2n^2) and decoding time 2^{O(k)}. This implies the existence of an FPT-algorithm of running time O(n^2 k^2)+2^{O(k)}. All aforementioned complexities are under the Uniform Cost Measure (UCM) model where numbers can be stored in constant space and arithmetic operations can be done in constant time

Finding and counting vertex-colored subtrees

The problems studied in this article originate from the Graph Motif problem introduced by Lacroix et al. in the context of biological networks. The problem is to decide if a vertex-colored graph has a connected subgraph whose colors equal a given multiset of colors $M$. It is a graph pattern-matching problem variant, where the structure of the occurrence of the pattern is not of interest but the only requirement is the connectedness. Using an algebraic framework recently introduced by Koutis et al., we obtain new FPT algorithms for Graph Motif and variants, with improved running times. We also obtain results on the counting versions of this problem, proving that the counting problem is FPT if M is a set, but becomes W[1]-hard if M is a multiset with two colors. Finally, we present an experimental evaluation of this approach on real datasets, showing that its performance compares favorably with existing software.Comment: Conference version in International Symposium on Mathematical Foundations of Computer Science (MFCS), Brno : Czech Republic (2010) Journal Version in Algorithmic

Counting induced subgraphs: a topological approach to #W[1]-hardness

We investigate the problem $\#\mathsf{IndSub}(\Phi)$ of counting all induced subgraphs of size $k$ in a graph $G$ that satisfy a given property $\Phi$. This continues the work of Jerrum and Meeks who proved the problem to be $\#\mathrm{W[1]}$-hard for some families of properties which include, among others, (dis)connectedness [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties $\Phi$, the problem $\#\mathsf{IndSub}(\Phi)$ is hard for $\#\mathrm{W[1]}$ if the reduced Euler characteristic of the associated simplicial (graph) complex of $\Phi$ is non-zero. This observation links $\#\mathsf{IndSub}(\Phi)$ to Karp's famous Evasiveness Conjecture, as every graph complex with non-vanishing reduced Euler characteristic is known to be evasive. Applying tools from the "topological approach to evasiveness" which was introduced in the seminal paper of Khan, Saks and Sturtevant [FOCS 83], we prove that $\#\mathsf{IndSub}(\Phi)$ is $\#\mathrm{W[1]}$-hard for every monotone property $\Phi$ that does not hold on the Hamilton cycle as well as for some monotone properties that hold on the Hamilton cycle such as being triangle-free or not $k$-edge-connected for $k > 2$. Moreover, we show that for those properties $\#\mathsf{IndSub}(\Phi)$ can not be solved in time $f(k)\cdot n^{o(k)}$ for any computable function $f$ unless the Exponential Time Hypothesis (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that $\#\mathsf{IndSub}(\Phi)$ is $\#\mathrm{W[1]}$-hard if $\Phi$ is any non-trivial modularity constraint on the number of edges with respect to some prime $q$ or if $\Phi$ enforces the presence of a fixed isolated subgraph

Polynomial-delay Enumeration Kernelizations for Cuts of Bounded Degree

Enumeration kernelization was first proposed by Creignou et al. [TOCS 2017] and was later refined by Golovach et al. [JCSS 2022] into two different variants: fully-polynomial enumeration kernelization and polynomial-delay enumeration kernelization. In this paper, we consider the d-CUT problem from the perspective of (polynomial-delay) enumeration kenrelization. Given an undirected graph G = (V, E), a cut F = E(A, B) is a d-cut of G if every u in A has at most d neighbors in B and every v in B has at most d neighbors in A. Checking the existence of a d-cut in a graph is a well-known NP-hard problem and is well-studied in parameterized complexity [Algorithmica 2021, IWOCA 2021]. This problem also generalizes a well-studied problem MATCHING CUT (set d = 1) that has been a central problem in the literature of polynomial-delay enumeration kernelization. In this paper, we study three different enumeration variants of this problem, ENUM d-CUT, ENUM MIN-d-CUT and ENUM MAX-d-CUT that intends to enumerate all the d-cuts, all the minimal d-cuts and all the maximal d-cuts respectively. We consider various structural parameters of the input and provide polynomial-delay enumeration kernels for ENUM d-CUT and ENUM MAX-d-CUT and fully-polynomial enumeration kernels of polynomial size for ENUM MIN-d-CUT.Comment: 25 page

Some hard families of parameterised counting problems

We consider parameterised subgraph-counting problems of the following form: given a graph G, how many k-tuples of its vertices have a given property? A number of such problems are known to be #W[1]-complete; here we substantially generalise some of these existing results by proving hardness for two large families of such problems. We demonstrate that it is #W[1]-hard to count the number of k-vertex subgraphs having any property where the number of distinct edge-densities of labelled subgraphs that satisfy the property is o(k^2). In the special case that the property in question depends only on the number of edges in the subgraph, we give a strengthening of this result which leads to our second family of hard problems.Comment: A few more minor changes. This version to appear in the ACM Transactions on Computation Theor