Approximating the maximum clique minor and some subgraph homeomorphism problems

Abstract

We consider the “minor” and “homeomorphic” analogues of the maximum clique problem, i.e., the problems of determining the largest h such that the input graph (on n vertices) has a minor isomorphic to K h or a subgraph homeomorphic to K h , respectively, as well as the problem of finding the corresponding subgraphs. We term them as the maximum clique minor problem and the maximum homeomorphic clique problem, respectively. We observe that a known result of Kostochka and Thomason supplies an O ( n ) bound on the approximation factor for the maximum clique minor problem achievable in polynomial time. We also provide an independent proof of nearly the same approximation factor with explicit polynomial-time estimation, by exploiting the minor separator theorem of Plotkin et al. Next, we show that another known result of Bollobás and Thomason and of Komlós and Szemerédi provides an O ( n ) bound on the approximation factor for the maximum homeomorphic clique achievable in polynomial time. On the other hand, we show an Ω ( n 1 / 2 − O ( 1 / ( log n ) γ ) ) lower bound (for some constant γ , unless NP ⊆ ZPTIME ( 2 ( log n ) O ( 1 ) ) ) on the best approximation factor achievable efficiently for the maximum homeomorphic clique problem, nearly matching our upper bound. Finally, we derive an interesting trade-off between approximability and subexponential time for the problem of subgraph homeomorphism where the guest graph has maximum degree not exceeding three and low treewidth.