On the Complexity of Various Parameterizations of Common Induced Subgraph Isomorphism

Abstract

In the Maximum Common Induced Subgraph problem (henceforth MCIS), given two graphs G1 and G2, one looks for a graph with the maximum number of vertices being both an induced subgraph of G1 and G2. 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 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 := vc(G1) + vc(G2) 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 ⊆ coNP/poly. Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters.

On the complexity of various parameterizations of common induced subgraph isomorphism

“…Thus, the 4k vertices of R∪ C can only be mapped to R∪C, such that for j ∈ [2], Rj is mapped to R j and Cj is mapped to C j . The edges r1 i r2…”

“…then c2 i has to be mapped to c 2 i ′ ). Hence, we can see the mapping from R ∪ C to R ∪ C as two permutations σ r and σ c on k elements, such that for j ∈ [2], for i ∈ [k], rj i is mapped to r j σr (i) and cj i is mapped to c j σc(i) . Then, the current partial mapping can be extended to a solution only if {(σ r (1), σ c (1)), .…”

“…. , (σ r (k), σ c (k))} is a clique in G. Indeed, ∀j ∈ [2], ∀i = i ′ ∈ [k], ṽ((i, i)(i ′ , i ′ ), j) can only be mapped to a potential v((σ r (i), σ c (i))(σ r (i ′ ), σ c (i ′ )), j) so that vertex has to exist, meaning that there should be an edge in G between (σ r (i), σ c (i)) and (σ r (i ′ ), σ c (i ′ )).…”

“…Indeed, to match a vertex u with a vertex v or a twin of v is equivalent. Thus, in time O * ((9k) k 2 k 2 ) we can enumerate all the solutions of MCIS where only vertices of I (1) could still be matched to vertices of I (2) . The optimal map of the independent sets can be done in polynomial time by matching the greatest number of vertices in each equivalent twin class (which is the size of the smaller of the two equivalent twin classes), where a twin class I To find a solution for MCCIS, the algorithm described in the above proof enumerates all possible maximal common induced subgraphs in time O * ((9k) k 2 k 2 ).…”