On the complexity of various parameterizations of common induced subgraph isomorphism

Faisal N. Abu-Khzam, Édouard Bonnet, Florian Sikora

Abstract: In the Maximum Common Induced Subgraph problem (henceforth MCIS), given two graphs G1<math><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub></math> and G2<math><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub></math>, one looks for a graph with the maximum number of vertices being both an induced subgraph of G1<math><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub></math> and G2<math><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub></math>. MCIS is among the most studied classical NP<math><mi mathvariant="sans-serif" is="true">NP</mi></math>-hard problems. It remains NP<math><mi mathvariant="sans-serif" is="true">NP</mi></math>-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]<math><mi mathvariant="sans-serif" is="true">W</mi><mo stretchy="false" is="true">[</mo><mn mathvariant="sans-serif" is="true">1</mn><mo stretchy="false" is="true">]</mo></math>-hard parameterized by the size of the solution. Being NP<math><mi mathvariant="sans-serif" is="true">NP</mi></math>-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)<math><mi is="true">k</mi><mo is="true">:</mo><mo is="true">=</mo><mtext is="true">vc</mtext><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub><mo stretchy="false" is="true">)</mo><mo is="true">+</mo><mtext is="true">vc</mtext><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mo stretchy="false" is="true">)</mo></math> 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 2O(klog⁡k)<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></mrow></msup></math>. We complement this result by showing that, unless the ETH fails, it cannot be solved in time 2o(klog⁡k)<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">o</mi><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></mrow></msup></math>. 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<math><mrow is="true"><mi mathvariant="sans-serif" is="true">NP</mi></mrow><mo is="true">⊆</mo><mrow is="true"><mi mathvariant="sans-serif" is="true">coNP</mi></mrow><mo stretchy="false" is="true">/</mo><mrow is="true"><mi mathvariant="italic" is="true">poly</mi></mrow></math>. Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters.