Minimum Weight Cycles and Triangles: Equivalences and Algorithms
Minimum Weight Cycles and Triangles: Equivalences and Algorithms
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最小权重循环和三角形：等价物和算法
DOI：
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发表时间：
2011
期刊：
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通讯作者：
V. V. Williams
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文献类型：
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作者：
L. Roditty;V. V. Williams
我们考虑了在加权图中找到最小重量周期的基本算法问题。特别是，我们表明，在{1，...，M}中的边缘权重的无向n节图中的最小重量周期问题或在{-m，... ，m}，并且可以有效地降低不负循环为在{1，...，o（o（m）}中的权重，在theta（n）-node _undirected_ graph中找到最小重量_triangle_。粗略地说，我们的减少意味着以下令人惊讶的现象：最小循环具有任意数量的加权边缘的最小循环可以是``编码''，仅在大致相同的权重间隔内使用三个边缘！这解决了ITAI和Rodeh [Siam J. Computing 1978]在未加权图中的开创性工作中提出的长期开放问题。我们有效降低的直接结果是tilde {o}（Mn^{\ omega}）0）0）最小重量周期立即暗示A O（N^{3- \ delta}） - 时间算法（\ delta> 0） apsp。
We consider the fundamental algorithmic problem of finding a cycle of minimum weight in a weighted graph. In particular, we show that the minimum weight cycle problem in an undirected n-node graph with edge weights in {1,...,M} or in a directed n-node graph with edge weights in {-M,..., M} and no negative cycles can be efficiently reduced to finding a minimum weight _triangle_ in an Theta(n)-node _undirected_ graph with weights in {1,...,O(M)}. Roughly speaking, our reductions imply the following surprising phenomenon: a minimum cycle with an arbitrary number of weighted edges can be ``encoded'' using only three edges within roughly the same weight interval! This resolves a longstanding open problem posed in a seminal work by Itai and Rodeh [SIAM J. Computing 1978] on minimum cycle in unweighted graphs. A direct consequence of our efficient reductions are tilde{O}(Mn^{\omega})0) for minimum weight cycle immediately implies a O(n^{3-\delta})-time algorithm (\delta>0) for APSP.