Assessing the Computational Complexity of Multi-Layer Subgraph Detection

Multi-layer graphs consist of several graphs (layers) over the same vertex set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the border of computational (in)tractability for the class of subgraph detection problems on multi-layer graphs, including fundamental problems such as maximum matching, finding certain clique relaxations (motivated by community detection), or path problems. Mostly encountering hardness results, sometimes even for two or three layers, we can also spot some islands of tractability

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

From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More

We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting $\text{OPT}$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t(\text{OPT})\text{poly}(N)$ time and outputs a solution of size $f(\text{OPT})$, for any functions $t$ and $f$ that are independent of $N$ (for Clique, we want $f(\text{OPT})=\omega(1)$)? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no $o(\text{OPT})$-FPT-approximation algorithm for Clique and no $f(\text{OPT})$-FPT-approximation algorithm for DomSet, for any function $f$ (e.g., this holds even if $f$ is the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even $(1 - \epsilon)$-satisfiable for some constant $\epsilon > 0$. Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and Maximum Induced Matching in bipartite graphs. Additionally, we rule out $k^{o(1)}$-FPT-approximation algorithm for Densest $k$-Subgraph although this ratio does not yet match the trivial $O(k)$-approximation algorithm.Comment: 43 pages. To appear in FOCS'1

Unit Interval Editing is Fixed-Parameter Tractable

Given a graph~$G$ and integers $k_1$, $k_2$, and~$k_3$, the unit interval editing problem asks whether $G$ can be transformed into a unit interval graph by at most $k_1$ vertex deletions, $k_2$ edge deletions, and $k_3$ edge additions. We give an algorithm solving this problem in time $2^{O(k\log k)}\cdot (n+m)$, where $k := k_1 + k_2 + k_3$, and $n, m$ denote respectively the numbers of vertices and edges of $G$. Therefore, it is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm implies the fixed-parameter tractability of the unit interval edge deletion problem, for which we also present a more efficient algorithm running in time $O(4^k \cdot (n + m))$. Another result is an $O(6^k \cdot (n + m))$-time algorithm for the unit interval vertex deletion problem, significantly improving the algorithm of van 't Hof and Villanger, which runs in time $O(6^k \cdot n^6)$.Comment: An extended abstract of this paper has appeared in the proceedings of ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an appendix is provided for a brief overview of related graph classe