Approximating the maximum clique minor and some subgraph homeomorphism problems

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 (sqrt(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 (sqrt(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.