On the Complexity of Various Parameterizations of Common Induced Subgraph Isomorphism

TL;DR

This paper investigates the parameterized complexity of the Maximum Common Induced Subgraph problem, establishing tight bounds and kernelization limits, especially when parameterized by the sum of vertex cover numbers of the input graphs.

Contribution

It provides a tight lower bound for the fixed-parameter algorithm based on vertex cover sum and shows the non-existence of polynomial kernels under standard complexity assumptions.

Findings

No $2^{o(k \, \log k)}$ time algorithm unless ETH fails.

The problem is W[1]-hard when parameterized by solution size.

MCIS does not admit a polynomial kernel unless NP ⊆ coNP/poly.

Abstract

In the Maximum Common Induced Subgraph problem (henceforth MCIS), given two graphs $G_1$ and $G_2$, one looks for a graph with the maximum number of vertices being both an induced subgraph of $G_1$ and $G_2$. MCIS is among the most studied classical NP-hard problems. It remains NP-hard on many graph classes including forests. In this paper, we study the parameterized complexity of MCIS. As a generalization of \textsc{Clique}, it is W[1]-hard parameterized by the size of the solution. Being NP-hard even on forests, most structural parameterizations are intractable. One has to go as far as parameterizing by the size of the minimum vertex cover to get some tractability. Indeed, when parameterized by $k := \text{vc}(G_1)+\text{vc}(G_2)$ the sum of the vertex cover number of the two input graphs, the problem was shown to be fixed-parameter tractable, with an algorithm running in time $2^{O(k \log k)}$. We complement this result by showing that, unless the ETH fails, it cannot be solved in time $2^{o(k \log k)}$. This kind of tight lower bound has been shown for a few problems and parameters but, to the best of our knowledge, not for the vertex cover number. We also show that MCIS does not have a polynomial kernel when parameterized by $k$, unless $NP \subseteq \mathsf{coNP}/poly$. Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters.