有界亏格二部图完美匹配的空间复杂度
Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs
摘要：
研究了嵌入在常亏格曲面（可定向或不可定向）上的二部图的完美匹配问题的空间复杂度。我们证明了判定这类图是否具有（1）完美匹配和（2）唯一完美匹配的问题属于对数空间复杂性类\SPL。由于\SPL \包含在对数空间计数类$\oplusŁ$中（实际上在所有$k\geq 2$的\modk \中），\CeqL和\PL，我们的上限也将上述匹配问题放在这些计数类中。我们还表明，搜索版本，计算完美匹配，这类图是在$\FL^{\SPL}$。我们的研究结果扩展了这些问题的上界在二部平面图已知的。作为我们的主要技术成果，我们设计了一个对数空间可计算和多项式有界的权重函数，它隔离了嵌入在常亏格表面上的二分图中的最小权重完美匹配。我们使用代数拓扑的结果来证明权函数的正确性。
英文摘要：
We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the problems of deciding whether such graphs have (1) a perfect matching or not and (2) a unique perfect matching or not, are in the logspace complexity class \SPL. Since \SPL\ is contained in the logspace counting classes $\oplusŁ$ (in fact in \modk\ for all $k\geq 2$), \CeqL, and \PL, our upper bound places the above-mentioned matching problems in these counting classes as well. We also show that the search version, computing a perfect matching, for this class of graphs is in $\FL^{\SPL}$. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier. As our main technical result, we design a logspace computable and polynomially bounded weight function which isolates a minimum weight perfect matching in bipartite graphs embedded on surfaces of constant genus. We use results from algebraic topology for proving the correctness of the weight function.
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