TY - JOUR

T1 - Approximating the maximum clique minor and some subgraph homeomorphism problems

AU - Alon, Noga

AU - Lingas, Andrzej

AU - Wahlen, Martin

PY - 2007/4/20

Y1 - 2007/4/20

N2 - 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 Kh or a subgraph homeomorphic to Kh, 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 (n1 / 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.

AB - 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 Kh or a subgraph homeomorphic to Kh, 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 (n1 / 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.

KW - Approximation

KW - Graph homeomorphism

KW - Graph minors

UR - http://www.scopus.com/inward/record.url?scp=33947373382&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2006.12.021

DO - 10.1016/j.tcs.2006.12.021

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AN - SCOPUS:33947373382

SN - 0304-3975

VL - 374

SP - 149

EP - 158

JO - Theoretical Computer Science

JF - Theoretical Computer Science

IS - 1-3

ER -