arXiv subjects

Philip Wellnitz

Publications and source records attributed to Philip Wellnitz.

22 records · Page 2Linked to original sources

arXiv2020

Counting Small Induced Subgraphs Satisfying Monotone Properties

Given a graph property $Φ$, the problem $\#\mathsf{IndSub}(Φ)$ 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 $Φ$. The search for explicit criteria on $Φ$ ensuring that $\#\mathsf{IndSub}(Φ)$ 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 Dörfler 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 $Φ$, the problem $\#\mathsf{IndSub}(Φ)$ 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.

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arXiv2019

Counting Answers to Existential Questions

Conjunctive queries select and are expected to return certain tuples from a relational database. We study the potentially easier problem of counting all selected tuples, rather than enumerating them. In particular, we are interested in the problem's parameterized and data complexity, where the query is considered to be small or fixed, and the database is considered to be large. We identify two structural parameters for conjunctive queries that capture their inherent complexity: The dominating star size and the linked matching number. If the dominating star size of a conjunctive query is large, then we show that counting solution tuples to the query is at least as hard as counting dominating sets, which yields a fine-grained complexity lower bound under the Strong Exponential Time Hypothesis as well as a #W[2]-hardness result. Moreover, if the linked matching number of a conjunctive query is large, then we show that the structure of the query is so rich that arbitrary queries up to a certain size can be encoded into it; this essentially establishes #A[2]-completeness. Using ideas stemming from Lovász, we lift complexity results from the class of conjunctive queries to arbitrary existential or universal formulas that might contain inequalities and negations on constraints over the free variables. As a consequence, we obtain a complexity classification that generalizes previous results of Chen, Durand, and Mengel (ToCS 2015; ICDT 2015; PODS 2016) for conjunctive queries and of Curticapean and Marx (FOCS 2014) for the subgraph counting problem. Our proof also relies on graph minors, and we show a strengthening of the Excluded-Grid-Theorem which might be of independent interest: If the linked matching number is large, then not only can we find a large grid somewhere in the graph, but we can find a large grid whose diagonal has disjoint paths leading into an assumed node-well-linked set.

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arXiv2019

Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness

We study the problem #IndSub(P) of counting all induced subgraphs of size k in a graph G that satisfy the property P. This problem was introduced by Jerrum and Meeks and shown to be #W[1]-hard when parameterized by k for some families of properties P including, among others, connectivity [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two of the authors introduced a novel technique for the complexity analysis of #IndSub(P), inspired by the "topological approach to evasiveness" of Kahn, Saks and Sturtevant [FOCS 83] and the framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], allowing them to prove hardness of a wide range of properties P. In this work, we refine this technique for graph properties that are non-trivial on edge-transitive graphs with a prime power number of edges. In particular, we fully classify the case of monotone bipartite graph properties: It is shown that, given any graph property P that is closed under the removal of vertices and edges, and that is non-trivial for bipartite graphs, the problem #IndSub(P) is #W[1]-hard and cannot be solved in time f(k)*n^{o(k)} for any computable function f, unless the Exponential Time Hypothesis fails. This holds true even if the input graph is restricted to be bipartite and counting is done modulo a fixed prime. A similar result is shown for properties that are closed under the removal of edges only.

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arXiv2018

Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Tree-adjoining grammars are a generalization of context-free grammars that are well suited to model human languages and are thus popular in computational linguistics. In the tree-adjoining grammar recognition problem, given a grammar $Γ$ and a string $s$ of length $n$, the task is to decide whether $s$ can be obtained from $Γ$. Rajasekaran and Yooseph's parser (JCSS'98) solves this problem in time $O(n^{2ω})$, where $ω< 2.373$ is the matrix multiplication exponent. The best algorithms avoiding fast matrix multiplication take time $O(n^6)$. The first evidence for hardness was given by Satta (J. Comp. Linguist.'94): For a more general parsing problem, any algorithm that avoids fast matrix multiplication and is significantly faster than $O(|Γ| n^6)$ in the case of $|Γ| = Θ(n^{12})$ would imply a breakthrough for Boolean matrix multiplication. Following an approach by Abboud et al. (FOCS'15) for context-free grammar recognition, in this paper we resolve many of the disadvantages of the previous lower bound. We show that, even on constant-size grammars, any improvement on Rajasekaran and Yooseph's parser would imply a breakthrough for the $k$-Clique problem. This establishes tree-adjoining grammar parsing as a practically relevant problem with the unusual running time of $n^{2ω}$, up to lower order factors.

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