Related papers: On Finding $\ell$-th Smallest Perfect Matchings
We reprove that all the matchings constructed during Edmonds' weighted perfect matching algorithm are optimal among those of the same cardinality (provided that certain mild restrictions are obeyed on the choices the algorithm makes). We…
Minimum Weight Cycle (MWC) is the problem of finding a simple cycle of minimum weight in a graph $G=(V,E)$. This is a fundamental graph problem with classical sequential algorithms that run in $\tilde{O}(n^3)$ and $\tilde{O}(mn)$ time where…
We propose a weighted common subgraph (WCS) matching algorithm to find the most similar subgraphs in two labeled weighted graphs. WCS matching, as a natural generalization of the equal-sized graph matching or subgraph matching, finds wide…
In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in $K_{3,3}$-minor-free graphs by building on Kasteleyn's scheme for planar graphs, and stated that this "opens up the possibility of obtaining an NC…
The $k$-ExactCover problem is a parameterized version of the ExactCover problem, in which we are given a universe $U$, a collection $S$ of subsets of $U$, and an integer $k$, and the task is to determine whether $U$ can be partitioned into…
We study the Maximum Weight Matching (MWM) problem for general graphs through the max-product Belief Propagation (BP) and related Linear Programming (LP). The BP approach provides distributed heuristics for finding the Maximum A Posteriori…
We consider the foundational problem of maintaining a $(1-\varepsilon)$-approximate maximum weight matching (MWM) in an $n$-node dynamic graph undergoing edge insertions and deletions. We provide a general reduction that reduces the problem…
In the Maximum Weight Independent Set of Rectangles problem (MWISR) we are given a weighted set of $n$ axis-parallel rectangles in the plane. The task is to find a subset of pairwise non-overlapping rectangles with the maximum possible…
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades, ever since the discovery of randomized NC matching algorithms [KUW85, MVV87]. Over…
Given a simple graph $G$, a weight function $w:E(G)\rightarrow \mathbb{N} \setminus \{0\}$, and an orientation $D$ of $G$, we define $\mu^-(D) = \max_{v \in V(G)} w_D^-(v)$, where $w^-_D(v) = \sum_{u\in N_D^{-}(v)}w(uv)$. We say that $D$ is…
We present deterministic distributed algorithms for computing approximate maximum cardinality matchings and approximate maximum weight matchings. Our algorithm for the unweighted case computes a matching whose size is at least $(1-\eps)$…
In 1982 Papadimitriou and Yannakakis introduced the Exact Matching problem, in which given a red and blue edge-colored graph $G$ and an integer $k$ one has to decide whether there exists a perfect matching in $G$ with exactly $k$ red edges.…
The minimum weight matching (MWM) and maximum likelihood decoding (MLD) are two widely used and distinct decoding strategies for quantum error correction. For a given syndrome, the MWM decoder finds the most probable physical error…
In a graph, a (perfect) matching cut is an edge cut that is a (perfect) matching. Matching Cut (MC), respectively, Perfect Matching Cut (PMC), is the problem of deciding whether a given graph has a matching cut, respectively, a perfect…
We consider the following stochastic matching problem on both weighted and unweighted graphs: A graph $G(V, E)$ along with a parameter $p \in (0, 1)$ is given in the input. Each edge of $G$ is realized independently with probability $p$.…
We consider the problem of counting matchings in planar graphs. While perfect matchings in planar graphs can be counted by a classical polynomial-time algorithm, the problem of counting all matchings (possibly containing unmatched vertices,…
We consider the problem of estimating the weight of a maximum weighted matching of a weighted graph $G(V,E)$ whose edges are revealed in a streaming fashion. We develop a reduction from the maximum weighted matching problem to the maximum…
Given a set of vertices $V$ with $|V| = n$, a weight vector $w \in (\mathbb{R}^+ \cup \{ 0 \})^{\binom{V}{2}}$, and a probability vector $x \in [0, 1]^{\binom{V}{2}}$ in the matching polytope, we study the quantity $\frac{E_{G}[…
We solve the Bin Packing problem in $O^*(2^k)$ time, where $k$ is the number of items less or equal to one third of the bin capacity. This parameter measures the distance from the polynomially solvable case of only large (i.e., greater than…
Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this paper, we introduce a general framework for solving problems on such graphs using matrix…