A matching $M$ is a $\mathscr{P}$-matching if the subgraph induced by the endpoints of the edges of $M$ satisfies property $\mathscr{P}$. As examples, for appropriate choices of $\mathscr{P}$, the problems Induced Matching, Uniquely Restricted Matching, Connected Matching and Disconnected Matching arise. For many of these problems, finding a maximum $\mathscr{P}$-matching is a knowingly NP-Hard problem, with few exceptions, such as connected matchings, which has the same time complexity as usual Maximum Matching problem. The weighted variant of Maximum Matching has been studied for decades, with many applications, including the well-known Assignment problem. Motivated by this fact, in addition to some recent researches in weighted versions of acyclic and induced matchings, we study the Maximum Weight Connected Matching. In this problem, we want to find a matching $M$ such that the endpoint vertices of its edges induce a connected subgraph and the sum of the edge weights of $M$ is maximum. Unlike the unweighted Connected Matching problem, which is in P for general graphs, we show that Maximum Weight Connected Matching is NP-Hard even for bounded diameter bipartite graphs, starlike graphs, planar bipartite, and bounded degree planar graphs, while solvable in linear time for trees and subcubic graphs. When we restrict edge weights to be non negative only, we show that the problem turns to be polynomially solvable for chordal graphs, while it remains NP-Hard for most of the cases when weights can be negative. Our final contributions are on parameterized complexity. On the positive side, we present a single exponential time algorithm when parameterized by treewidth. In terms of kernelization, we show that, even when restricted to binary weights, Weighted Connected Matching does not admit a polynomial kernel when parameterized by vertex cover under standard complexity-theoretical hypotheses.
对于n个申请者的集合A和m个项目的集合I，我们考虑计算申请者与项目的匹配的问题，即，函数M将A映射到I;这里我们假设每个申请人$x \in A$提供I中项目的偏好列表。如果p在其偏好列表中位于比q更高的位置，则我们说申请人$x \in A$偏好项目p而不是项目q，并且如果x偏好M（x）而不是M '（x），则我们说x偏好匹配的M而不是匹配的M'。对于给定的匹配问题A、I和偏好列表，如果偏好M而不是M '的申请者的数量大于偏好M'而不是M的申请者的数量，则我们说M比M'更受欢迎，并且如果没有其他匹配比M更受欢迎，则M被称为受欢迎匹配。这里，我们考虑A被划分为$A_{1},A_{2},...,A_{k}$，并且每个$A_{i}$被分配权重$w_{i}>0$，使得w_{1} > w_{2} >..> w_{k} >0 $. For such a matching problem, we say that M is more popular than M' if the total weight of applicants preferring M over M' is larger than that of applicants preferring M' over M, and we call M an k-weighted popular matching if there is no…
Balanced spanning forests of graphs: Hardness and solving methods
KU Leuven（荷语鲁汶大学）；Research Centre for Operations Research and Business Statistics (ORSTAT)（运筹学与商业统计研究中心）；Institute of Computing, University of Campinas（坎皮纳斯大学计算研究所）
Finding popular branchings in vertex-weighted digraphs
流行匹配作为稳定匹配的一个宽松概念，近年来得到了广泛的研究。通过将流行匹配的概念应用于有向图中的分支，Kavitha et al.（2020年）推出热门分支机构。在有向图$G=(V_G,E_G)$中，每个顶点都对其引入的边具有优先级。对于$G$中的分支$B_1$和$B_2$，如果$v$更喜欢$B_1$的传入边而不是$B_2$的传入边，则顶点$v\in V_G$更喜欢$B_1$而不是$B_2$，其中具有任意传入边比没有传入边更好，并且如果偏好$B_1$的顶点的数量大于偏好$B_2$的顶点的数量，则$B_1$比$B_2$更受欢迎。如果没有分支比$B$更流行，则分支$B$称为流行分支。Kavitha等人（2020）提出了一种算法，当每个顶点的偏好由严格的偏序给出时，该算法可以找到一个流行的分支。利用加权树形图对偶的经典定理证明了该算法的有效性。在本文中，我们将流行分支推广到顶点加权有向图中的加权流行分支，其方式与Mestre（2014）的加权流行匹配相同。我们给出了一个寻找加权流行分支的算法，它扩展了Kavitha等人的算法，当每个顶点的偏好由一个全预序给出且权值满足一定条件时，我们的算法包括详细的程序所产生的顶点权重，其有效性是通过扩展的参数的对偶加权树形图。
