关于奇红双色完美匹配多面体的复杂性
On the Complexity of the Odd-Red Bipartite Perfect Matching Polytope
AI总结：
研究奇红双色完美匹配问题的多面体结构，发现其具有复杂的面结构，需使用大系数约束，推翻传统放松方法，并指出双模整数程序的多面体也具有复杂结构。
摘要：
奇红双色完美匹配问题要求在给定的红蓝边着色双色图中找到包含奇数个红边的完美匹配。尽管该问题属于P类，但其多面体结构仍不清楚。最近的进展表明，精确双色完美匹配多面体的扩展复杂性在一般情况下是指数级的。有趣的是，这一结果也适用于更简单的奇红双色完美匹配问题。对于该问题，他们引入了指数级规模的放松，并留出是否为精确描述的问题。除了证明该描述不精确且难以分离外，我们还证明了红奇双色完美匹配多面体具有复杂的面结构：任何精确描述都需要具有大且多样系数的约束。这推翻了基于系数全部在{0,±1}中的传统放松方法，如上述提到的，表明可能需要显著偏离以往方法才能获得精确描述。更一般地，我们得出结论，对应于双模整数程序的多面体也具有复杂的面结构。
英文摘要：
The odd-red bipartite perfect matching problem asks to find a perfect matching containing an odd number of red edges in a given red-blue edge-colored bipartite graph. While this problem lies in $\mathsf{P}$, its polyhedral structure remains elusive, despite renewed attention to achieving better polyhedral understanding, nurtured by recent advances from two complementary angles. Apart from being a special case of bimodular integer programs, whose polyhedral structure is also badly understood, it is related to one of the most notorious open derandomization questions in theoretical computer science: whether there is a deterministic efficient algorithm for the exact bipartite perfect matching problem, which asks to find a perfect matching with exactly $k$ red edges. Recent progress towards deterministic algorithms for this problem crucially relies on a good polyhedral understanding. Motivated by this, Jia, Svensson, and Yuan show that the extension complexity of the exact bipartite perfect matching polytope is exponential in general. Interestingly, their result is true even for the easier odd-red bipartite perfect matching problem. For this problem, they introduce an exponential-size relaxation and leave open whether it is an exact description. Apart from showing that this description is not exact and even hard to separate over, we show, more importantly, that the red-odd bipartite perfect matching polytope exhibits complex facet structure: any exact description needs constraints with large and diverse coefficients. This rules out classical relaxations based on constraints with all coefficients in $\{0,\pm1\}$, such as the above-mentioned one, and suggests that significant deviations from prior approaches may be needed to obtain an exact description. More generally, we obtain that also polytopes corresponding to bimodular integer programs have complex facet structure.
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