最小权圈与三角形：等价性与算法
Minimum Weight Cycles and Triangles: Equivalences and Algorithms
摘要：
我们考虑的基本算法问题，找到一个圈的最小重量的加权图。特别地，我们证明了边权在{1，.，M}中，或者在具有{-M，...，M}且无负圈的三角形的最小权三角形的最小权三角形可以有效地简化为在Theta（n）-节点无向图中找到具有{1，.，O（M）}。粗略地说，我们的减少意味着以下令人惊讶的现象：一个最小的循环与任意数量的加权边缘可以“编码”使用只有三个边缘在大致相同的重量间隔！这解决了Itai和Rodeh [SIAM J. Computing 1978和STOC'77]提出的一个长期未决问题。我们的有效减少的一个直接后果是O（Mn^{omega}）-时间算法使用快速矩阵乘法（FMM）找到一个最小的重量循环在两个无向图整数权重从区间[1，M]和有向图整数权重从区间[-M，M]。后者似乎揭示了所有对最短路径（APSP）问题和有向图中的最小权重循环问题之间的强烈分离，因为Zwick [J. ACM 2002]的最快已知APSP算法的运行时间为O（M^{0.681}n^{2.575}）。相比之下，当仅允许组合算法时（即不使用FMM），最小权重循环的唯一已知解决方案是计算APSP。有趣的是，在这种情况下，这两个问题之间的任何分离都将是一个惊人的突破，正如Vassilevska W。和Williams [FOCS'10]，任何O（n^{3-eps}）时间算法（eps>0）的最小权重周期立即意味着O（n^{3-delta}）时间算法（delta>0）的APSP。
英文摘要：
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 by Itai and Rodeh [SIAM J. Computing 1978 and STOC'77]. A direct consequence of our efficient reductions are O (Mn^{omega})-time algorithms using fast matrix multiplication (FMM) for finding a minimum weight cycle in both undirected graphs with integral weights from the interval [1,M] and directed graphs with integral weights from the interval [-M,M]. The latter seems to reveal a strong separation between the all pairs shortest paths (APSP) problem and the minimum weight cycle problem in directed graphs as the fastest known APSP algorithm has a running time of O(M^{0.681}n^{2.575}) by Zwick [J. ACM 2002]. In contrast, when only combinatorial algorithms are allowed (that is, without FMM) the only known solution to minimum weight cycle is by computing APSP. Interestingly, any separation between the two problems in this case would be an amazing breakthrough as by a recent paper by Vassilevska W. and Williams [FOCS'10], any O(n^{3-eps})-time algorithm (eps>0) for minimum weight cycle immediately implies a O(n^{3-delta})-time algorithm (delta>0) for APSP.
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