In the Exact Matching Problem (EM), we are given a graph equipped with a fixed coloring of its edges with two colors (red and blue), as well as a positive integer $k$. The task is then to decide whether the given graph contains a perfect matching exactly $k$ of whose edges have color red. EM generalizes several important algorithmic problems such as perfect matching and restricted minimum weight spanning tree problems. When introducing the problem in 1982, Papadimitriou and Yannakakis conjectured EM to be $\textbf{NP}$-complete. Later however, Mulmuley et al.~presented a randomized polynomial time algorithm for EM, which puts EM in $\textbf{RP}$. Given that to decide whether or not $\textbf{RP}=\textbf{P}$ represents a big open challenge in complexity theory, this makes it unlikely for EM to be $\textbf{NP}$-complete, and in fact indicates the possibility of a deterministic polynomial time algorithm. EM remains one of the few natural combinatorial problems in $\textbf{RP}$ which are not known to be contained in $\textbf{P}$, making it an interesting instance for testing the hypothesis $\textbf{RP}=\textbf{P}$. Despite EM being quite well-known, attempts to devise deterministic polynomial algorithms have remained illusive during the last 40 years and progress has been lacking even for very restrictive classes of input graphs. In this paper we finally push the frontier of positive results forward by proving that EM can be solved in deterministic polynomial time for input graphs of bounded independence number, and for bipartite input graphs of bounded bipartite independence number. This generalizes previous positive results for complete (bipartite) graphs which were the only known results for EM on dense graphs.
翻译:Exact 匹配问题 (EM) 中, 我们得到一张配有两种颜色( 红色和 蓝色) 的固定边缘颜色的图表, 以及正整整美元。 然后的任务是决定给定的图表是否完全匹配$k$, 其边际有色红色。 EM 概括了一些重要的算法问题, 例如完美匹配和限制最小重量, 包括树状问题 。 在1982 引入这个问题时, Papadimitriou 和 Yannakakakis 调试的 EM 是 $\ textbf{ NP} 完成的 。 但是, 以后, Mulmley 和 Al. 提供了一种随机化的混合时间算法。 将EM 放在 $\ textb{ { RP} 上。 鉴于要决定是否是 $ textbffffff f}{ p} 复杂理论中的一大挑战, 这让EM 无法成为普通的 entritime 数字 的 数字 数字 。 。 在 IM tal deal deal deal deal deal ral raltial ral ral ral ral 。