The approximate degree of a Boolean function is the least degree of a real multilinear polynomial approximating it in the $\ell_\infty$-norm over the Boolean hypercube. We show that the approximate degree of the Bipartite Perfect Matching function, which is the indicator over all bipartite graphs having a perfect matching, is $\widetildeΘ(n^{3/2})$.
The upper bound is obtained by fully characterizing the unique multilinear polynomial representing the Boolean dual of the perfect matching function, over the reals. Crucially, we show that this polynomial has very small $\ell_1$-norm -- only exponential in $Θ(n \log n)$. The lower bound follows by bounding the spectral sensitivity of the perfect matching function, which is the spectral radius of its cut-graph on the hypercube \cite{aaronson2020degree, huang2019induced}. We show that the spectral sensitivity of perfect matching is exactly $Θ(n^{3/2})$.
We obtain complete characterizations of the Unique Bipartite Perfect Matching function, and of its Boolean dual, using multilinear polynomials over the reals. Building on previous results, we show that, surprisingly, the dual description is sparse and has low $\ell_1$-norm -- only exponential in $Θ(n \log n)$, and this result extends even to other families of matching-related functions. Our approach relies on the Möbius numbers in the matching-covered lattice, and a key ingredient in our proof is Möbius' inversion formula. These polynomial representations yield complexity-theoretic results. Fo…
Given a graph with edges colored red or blue and an integer $k$, the exact perfect matching problem asks if there exists a perfect matching with exactly $k$ red edges. There exists a randomized polylogarithmic-time parallel algorithm to solve this problem, dating back to the eighties, but no deterministic polynomial-time algorithm is known, even for bipartite graphs. In this paper we show that there is no sub-exponential sized linear program that can describe the convex hull of exact matchings in bipartite graphs. In fact, we prove something stronger, that there is no sub-exponential sized lin…
A complexity dichotomy for Matching Cut in (bipartite) graphs of fixed diameter
在图中，匹配割是匹配的边割。匹配割是判定给定图是否有匹配割的问题，即使仅限于二分图，也是NP完全的。证明了直径为2的图的匹配割是多项式可解的。本文证明了对任意固定的整数$d\ge 3$，匹配割在直径为$d$的图类中是NP-完全的.这解决了Borowiecki和Jesse-Józefczyk在[Matching cutsets in graphs of diameter $2$，Theoretical计算机科学407（2008）574-582]中提出的公开问题。然后，我们表明，对于任何固定的整数$d\ge 4$，匹配割是NP-完全的，即使限制在类的二部图的直径$d$。作为对硬度结果的补充，我们证明了匹配割在直径不超过3的二部图类中是多项式时间可解的，并提出了一种新的求解直径为$2$的图类中匹配割的简单多项式时间算法.
Matching minors are a specialisation of minors fit for the study of graph with perfect matchings. The notion of matching minors has been used to give a structural description of bipartite graphs on which the number of perfect matchings can becomputed efficiently, based on a result of Little, by McCuaig et al. in 1999.In this paper we generalise basic ideas from the graph minor series by Robertson and Seymour to the setting of bipartite graphs with perfect matchings. We introducea version of Erdos-Posa property for matching minors and find a direct link between this property and planarity. From…
We study the classical weighted perfect matchings problem for bipartite graphs or sometimes referred to as the assignment problem, i.e., given a weighted bipartite graph $G = (U\cup V,E)$ with weights $w : E \rightarrow \mathcal{R}$ we are interested to find the maximum matching in $G$ with the minimum/maximum weight. In this work we present a new and arguably simpler analysis of one of the earliest techniques developed for solving the assignment problem, namely the auction algorithm. Using our analysis technique we present tighter and improved bounds on the runtime complexity for finding an a…
The Total Matching Polytope generalizes the Stable Set Polytope and the Matching Polytope. In this paper, we give the perfect formulation for Trees and we derive two new families of valid inequalities, the balanced biclique inequalities which are always facet-defining and the non-balanced lifted biclique inequalities obtained by a lifting procedure, which are facet-defining for bipartite graphs. Finally, we give a complete description for Complete Bipartite Graphs.
A dynamic bipartite matching model is given by a bipartite matching graph which determines the possible matchings between the various types of supply and demand items. Both supply and demand items arrive to the system according to a stochastic process. Matched pairs leave the system and the others wait in the queues, which induces a holding cost. We model this problem as a Markov Decision Process and study the discounted cost and the average cost problem. We fully characterize the optimal matching policy for complete matching graphs and for the N -shaped matching graph. In the former case, the…
A major step in the graph minors theory of Robertson and Seymour is the transition from the Grid Theorem which, in some sense uniquely, describes areas of large treewidth within a graph, to a notion of local flatness of these areas in form of the existence of a large flat wall within any huge grid of an H-minor free graph. In this paper, we prove a matching theoretic analogue of the Flat Wall Theorem for bipartite graphs excluding a fixed matching minor. Our result builds on a a tight relationship between structural digraph theory and matching theory and allows us to deduce a Flat Wall Theorem…
Two-sided popular matchings in bipartite graphs with forbidden/forced elements and weights
二部图中的双边流行匹配是婚姻环境中稳定匹配的一个众所周知的推广，当偏好列表不完整时，它们尤其相关。在这种情况下，稳定匹配的基数可以小到最大匹配大小的一半。流行的匹配允许更大规模的分配，同时仍然保证一定的公平性条件。事实上，稳定匹配是最小大小的流行匹配，并且在给定实例中，最大大小的流行匹配可以是（ny）稳定匹配的大小的两倍。流行匹配的结构似乎比稳定匹配的结构更复杂，目前也更难理解。在本文中，我们专注于流行的匹配相关的三个优化问题。首先，我们对带有禁止和强制元素的流行匹配问题的复杂性进行了粒度分析，从而补充了[Cseh和Kavitha，2016]的结果。特别是，我们表明，决定是否存在一个流行的匹配（或没有）两个给定的边缘是NP-难。这意味着找到一个流行的匹配的最大值（或）。最小）权重是NP-难的，即使所有权重都是非负的，也不能近似到1/2（分别是任何因素）。[Cseh and Kavitha，2016]中的分解定理可以用于对具有非负权重的最大加权流行匹配问题进行1/2近似，从而完全解决这些问题的复杂性。
A bipartite graph B is called a brace if it is connected and every matching of size at most two in B is contained in some perfect matching of B and a cycle C in B is called conformal if B-V(C) has a perfect matching. We show that there do not exist two disjoint alternating paths that form a cross over a conformal cycle C in a brace B if and only if one can reduce B, by an application of a matching theoretic analogue of small clique sums, to a planar brace H in which C bounds a face. We then utilise this result and provide a polynomial time algorithm which solves the 2-linkage problem for alter…
Prophet inequality for bipartite matching: merits of being simple and non adaptive
考虑二部图中匹配的贝叶斯在线选择问题，即，带边到达的在线加权匹配问题，其中在线算法知道权值的分布，对应于[Kleinberg and Wienberg STOC 12]模型中两个拟阵的交集。我们考虑一类简单的非自适应顶点添加剂的政策，分配静态价格的所有顶点在图中，并接受每个边，只有当它的重量超过边缘的端点的价格之和。我们表明存在一个顶点添加剂的政策，预期的回报至少有三分之一的先知的回报和目前的梯度下降型算法，迅速收敛到所需的顶点价格的向量。这在两个方面改进了[Kleinberg and Wienberg STOC 12]中关于两个拟阵交集的自适应在线策略：我们的策略是非自适应的，并且具有更好的$3$近似保证，而不是以前的$5.82$对先知的保证。我们给出了一个互补的下限$2.25$的任何在线算法中的二分匹配设置。
