On the Complexity of Various Parameterizations of Common Induced Subgraph Isomorphism [article]

Faisal N. Abu-Khzam, Édouard Bonnet, Florian Sikora
2017 arXiv   pre-print
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 Clique, it is W[1]-hard parameterized by the size of the solution. Being NP-hard even on
more » ... s, 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 := vc(G_1)+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 k). We complement this result by showing that, unless the ETH fails, it cannot be solved in time 2^o(k 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 ⊆coNP/poly. Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters.
arXiv:1412.1261v2 fatcat:ownbquhznvcrnmz4uthnys5o3m