The parameterised complexity of counting even and odd induced subgraphs

We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that each of the problems admits an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory

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

Counting Small Induced Subgraphs Satisfying Monotone Properties

Given a graph property $\Phi$, the problem $\#\mathsf{IndSub}(\Phi)$ asks, on input a graph $G$ and a positive integer $k$, to compute the number of induced subgraphs of size $k$ in $G$ that satisfy $\Phi$. The search for explicit criteria on $\Phi$ ensuring that $\#\mathsf{IndSub}(\Phi)$ is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property $\Phi$, the problem $\#\mathsf{IndSub}(\Phi)$ cannot be solved in time $f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})}$ for any function $f$, unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a $\#\mathsf{W}[1]$-completeness result

Counting Small Induced Subgraphs Satisfying Monotone Properties

Given a graph property $\Phi$, the problem $\#\mathsf{IndSub}(\Phi)$ asks, on input a graph $G$ and a positive integer $k$, to compute the number of induced subgraphs of size $k$ in $G$ that satisfy $\Phi$. The search for explicit criteria on $\Phi$ ensuring that $\#\mathsf{IndSub}(\Phi)$ is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property $\Phi$, the problem $\#\mathsf{IndSub}(\Phi)$ cannot be solved in time $f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})}$ for any function $f$, unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a $\#\mathsf{W}[1]$-completeness result.Comment: 33 pages, 2 figure

The simple, little and slow things count : on parameterized counting complexity

In this thesis, we study the parameterized complexity of counting problems, as introduced by Flum and Grohe. This area mainly involves questions of the following kind: On inputs x with a parameter k, can we solve a given counting problem in time f(k)*|x|^c for a function f that depends only on k? In the positive case, we call the problem fixed-parameter tractable (fpt). Otherwise, we try to prove its #W[1]-hardness, which is the parameterized analogue of #P-hardness. We introduce a general technique that bridges parameterized counting complexity and the so-called Holant framework. We then apply this technique to the problem of counting perfect matchings (or equivalently, the permanent) subject to structural parameters of the input graph G: On the algorithmic side, we introduce a new tractable structural parameter, namely, the minimal size of an excluded single-crossing minor of G. We complement this by showing that counting perfect matchings is #W[1]-hard when parameterized by the size of an arbitrary excluded minor. Then we turn our attention to counting general subgraphs H other than perfect matchings in a host graph G. Instead of imposing structural parameters on G, we parameterize by the size of H, giving rise to the problems #Sub(C) for fixed graph classes C: For inputs H and G with H in C, we wish to count H-copies in G. Here, C could be the class of matchings, cycles, paths, or any other recursively enumerable class. We give a full dichotomy for these problems: Either #Sub(C) has a polynomial-time algorithm or it is #W[1]-complete. Assuming that FPT and #W[1] do not coincide, we can thus precisely identify the graph classes C for which the subgraph counting problem #Sub(C) admits polynomial-time algorithms. Furthermore, we obtain an unexpected application of our extensions to the Holant framework: We show that, given two unweighted graphs, it is C=P-complete to decide whether they have the same number of perfect matchings. Finally, we prove conditional lower bounds for counting problems under the counting exponential-time hypothesis #ETH. This hypothesis, introduced by Dell et al., asserts that the satisfying assignments to n-variable formulas in 3-CNF cannot be counted in time 2^o(n). Building upon this, we introduce a general technique that allows to derive tight lower bounds for other counting problems, such as counting perfect matchings, the Tutte polynomial, and the matching polynomial.Die vorliegende Arbeit befasst sich mit der parametrisierten Komplexität von Zählproblemen, einem von Flum und Grohe gegründeten Gebiet, in welchem Fragen der folgenden Art betrachtet werden: Können gegebene Probleme auf Eingaben x mit Parameter k in Zeit f(k)*|x|^c gelöst werden, wobei f eine Funktion ist, die nur von k abhängt? Im positiven Falle bezeichnen wir das Problem als parametrisierbar (FPT). Andernfalls versuchen wir typischerweise, dessen #W[1]-Härte zu beweisen - diese lässt sich vereinfachend als ein parametrisiertes Äquivalent der #P-Härte auffassen. Wir führen zunächst eine allgemeine Technik ein, welche die parametrisierte Zählkomplexität mit dem sogenannten Holant-Rahmenwerk verbindet. Anschließend setzen wir diese zum Zählen perfekter Paarungen (oder äquivalent, zur Auswertung der Permanente) unter strukturellen Parametern des Eingabegraphens G ein: Wir zeigen, dass das Zählen perfekter Paarungen parametrisierbar ist durch die minimale Größe eines ausgeschlossenen Minors von G, der höchstens eine Kreuzung besitzt. Dieses algorithmische Resultat komplementieren wir durch die #W[1]-Härte des Zählens perfekter Paarungen, wenn die minimale Größe eines beliebigen ausgeschlossenen Minors als Parameter betrachtet wird. Anschließend widmen wir uns dem Zählen beliebiger Subgraphen H in Graphen G. Anstelle von strukturellen Parametern betrachten wir die Größe von H als Parameter und erhalten hierdurch die Probleme #Sub(C) für feste Graphklassen C: Auf Eingaben H und G mit H in C gilt es, die H-Kopien in G zu zählen. Hierbei kann C die Klasse der Paarungen, Zyklen, Pfade, oder eine beliebige andere Klasse von Graphen darstellen. Wir zeigen eine vollständige Dichotomie für diese Probleme: Das Problem #Sub(C) ist entweder in P oder #W[1]-hart. Unter der gängigen Annahme, dass FPT und #W[1] nicht zusammenfallen, erhalten wir somit eine vollständige Klassifikation der Polynomialzeit-lösbaren Probleme #Sub(C). Weiterhin erhalten wir eine unerwartete Anwendung unserer Erweiterungen des Holant-Rahmenwerks: Wir zeigen die C=P-Vollständigkeit der Frage, ob die Anzahlen perfekter Paarungen in zwei gegebenen ungewichteten Graphen übereinstimmen. Schlussendlich zeigen wir bedingte untere Schranken für Zählprobleme unter der Zählversion der Exponentialzeithypothese #ETH, eingeführt durch Dell et al. Diese postuliert, dass die erfüllenden Belegungen in 3-KNF-Formeln mit n Variablen nicht in Zeit 2^o(n) gezählt werden können. Darauf aufbauend führen wir eine allgemeine Technik ein, die es ermöglicht, scharfe untere Schranken für andere Zählprobleme zu erhalten: Dies umfasst das Zählen perfekter Paarungen, das Tutte-Polynom und das Paarungs-Polynom