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arXiv 0912.2371cs.DScs.DM
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求子图和计数的快速算法

Faster Algorithms for Finding and Counting Subgraphs

Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, B. V. Raghavendra Rao, Saket Saurabh

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摘要：

In this paper we study a natural generalization of both { \sc $k$ -Path} and { \sc $k$ -Tree} problems, namely, the { \sc Subgraph Isomorphism} problem. In the { \sc Subgraph Isomorphism} problem we are given two graphs $F$ and $G$ on $k$ and $n$ vertices respectively as an input, and the question is whether there exists a subgraph of $G$ isomorphic to $F$ . We show that if the treewidth of $F$ is at most $t$ , then there is a randomized algorithm for the { \sc Subgraph Isomorphism} problem running in time $\cO^*(2^k n^{2t})$ . To do so, we associate a new multivariate {Homomorphism polynomial} of degree at most $k$ with the { \sc Subgraph Isomorphism} problem and construct an arithmetic circuit of size at most $n^{\cO(t)}$ for this polynomial. Using this polynomial, we also give a deterministic algorithm to count the number of homomorphisms from $F$ to $G$ that takes $n^{\cO(t)}$ time and uses polynomial space. For the counting version of the { \sc Subgraph Isomorphism} problem, where the objective is to count the number of distinct subgraphs of $G$ that are isomorphic to $F$ , we give a deterministic algorithm running in time and space $\cO^*({n \choose k/2}n^{2p})$ or ${n\choose k/2}n^{\cO(t \log k)}$ . We also give an algorithm running in time $\cO^{*}(2^{k}{n \choose k/2}n^{5p})$ and taking space polynomial in $n$ . Here $p$ and $t$ denote the pathwidth and the treewidth of $F$ , respectively. Thus our work not only improves on known results on { \sc Subgraph Isomorphism} but it also extends and generalize most of the known results on { \sc $k$ -Path} and { \sc $k$ -Tree}.

英文摘要：

In this paper we study a natural generalization of both {\sc $k$-Path} and {\sc $k$-Tree} problems, namely, the {\sc Subgraph Isomorphism} problem. In the {\sc Subgraph Isomorphism} problem we are given two graphs $F$ and $G$ on $k$ and $n$ vertices respectively as an input, and the question is whether there exists a subgraph of $G$ isomorphic to $F$. We show that if the treewidth of $F$ is at most $t$, then there is a randomized algorithm for the {\sc Subgraph Isomorphism} problem running in time $\cO^*(2^k n^{2t})$. To do so, we associate a new multivariate {Homomorphism polynomial} of degree at most $k$ with the {\sc Subgraph Isomorphism} problem and construct an arithmetic circuit of size at most $n^{\cO(t)}$ for this polynomial. Using this polynomial, we also give a deterministic algorithm to count the number of homomorphisms from $F$ to $G$ that takes $n^{\cO(t)}$ time and uses polynomial space. For the counting version of the {\sc Subgraph Isomorphism} problem, where the objective is to count the number of distinct subgraphs of $G$ that are isomorphic to $F$, we give a deterministic algorithm running in time and space $\cO^*({n \choose k/2}n^{2p})$ or ${n\choose k/2}n^{\cO(t \log k)}$. We also give an algorithm running in time $\cO^{*}(2^{k}{n \choose k/2}n^{5p})$ and taking space polynomial in $n$. Here $p$ and $t$ denote the pathwidth and the treewidth of $F$, respectively. Thus our work not only improves on known results on {\sc Subgraph Isomorphism} but it also extends and generalize most of the known results on {\sc $k$-Path} and {\sc $k$-Tree}.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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