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The maximum bipartite matching problem is among the most fundamental and well-studied problems in combinatorial optimization. A beautiful and celebrated combinatorial algorithm of Hopcroft and Karp (1973) shows that maximum bipartite…

Data Structures and Algorithms · Computer Science 2023-12-21 Julia Chuzhoy , Sanjeev Khanna

This work presents a descending-price-auction algorithm to obtain the maximum market-clearing price vector (MCP) in unit-demand matching markets with m items by exploiting the combinatorial structure. With a shrewd choice of goods for which…

Computer Science and Game Theory · Computer Science 2017-11-07 Shih-Tang Su , Jacob D. Abernethy , Grant Schoenebeck , Vijay G. Subramanian

In this paper, we introduce a novel, non-recursive, maximal matching algorithm for double auctions, which aims to maximize the amount of commodities to be traded. It differs from the usual equilibrium matching, which clears a market at the…

Computer Science and Game Theory · Computer Science 2013-04-12 Jinzhong Niu , Simon Parsons

In this addendum, we show that the switching algorithm QPS-SERENA can be converted R(QPS-SERENA), an algorithm for computing approximate Maximum Weight Matching (MWM). Empirically, R(QPS-SERENA) computes $(1-\epsilon)$-MWM within linear…

Data Structures and Algorithms · Computer Science 2017-11-10 Long Gong , Jun , Xu

We provide new tradeoffs between approximation and running time for the decremental all-pairs shortest paths (APSP) problem. For undirected graphs with $m$ edges and $n$ nodes undergoing edge deletions, we provide four new approximate…

Data Structures and Algorithms · Computer Science 2024-04-30 Michal Dory , Sebastian Forster , Yasamin Nazari , Tijn de Vos

Let $G$ be an undirected bipartite graph with positive integer weights on the edges. We refine the existing decomposition theorem originally proposed by Kao et al., for computing maximum weight bipartite matching. We apply it to design an…

Discrete Mathematics · Computer Science 2019-12-11 Shibsankar Das

We design an additive approximation scheme for estimating the cost of the min-weight bipartite matching problem: given a bipartite graph with non-negative edge costs and $\varepsilon > 0$, our algorithm estimates the cost of matching all…

Data Structures and Algorithms · Computer Science 2023-11-14 Lorenzo Beretta , Aviad Rubinstein

Maximum bipartite matching (MBM) is a fundamental problem in combinatorial optimization with a long and rich history. A classic result of Hopcroft and Karp (1973) provides an $O(m \sqrt{n})$-time algorithm for the problem, where $n$ and $m$…

Data Structures and Algorithms · Computer Science 2024-06-03 Julia Chuzhoy , Sanjeev Khanna

We design a generic method for reducing the task of finding weighted matchings to that of finding short augmenting paths in unweighted graphs. This method enables us to provide efficient implementations for approximating weighted matchings…

Data Structures and Algorithms · Computer Science 2018-11-08 Buddhima Gamlath , Sagar Kale , Slobodan Mitrović , Ola Svensson

We study the Maximum Cardinality Matching (MCM) and the Maximum Weight Matching (MWM) problems, on trees and on some special classes of graphs, in the Online Preemptive and the Incremental Dynamic Graph models. In the {\em Online…

Data Structures and Algorithms · Computer Science 2018-01-23 Sumedh Tirodkar , Sundar Vishwanathan

We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph $G(A\cup B,E)$ with edge weights of $0$ or $1$. Let…

Computational Geometry · Computer Science 2019-03-26 Nathaniel Lahn , Sharath Raghvendra

Maximum cardinality matching in bipartite graphs is an important and well-studied problem. The fully dynamic version, in which edges are inserted and deleted over time has also been the subject of much attention. Existing algorithms for…

Data Structures and Algorithms · Computer Science 2015-08-18 Aaron Bernstein , Cliff Stein

Let c denote a non-negative constant. Suppose that we are given an edge-weighted bipartite graph G=(V,E) with its 2-layered drawing and a family X of intersecting edge pairs. We consider the problem of finding a maximum weighted matching M*…

Data Structures and Algorithms · Computer Science 2019-09-17 Kazuya Haraguchi , Kotaro Torii , Motomu Endo

In the dynamic approximate maximum bipartite matching problem we are given bipartite graph $G$ undergoing updates and our goal is to maintain a matching of $G$ which is large compared the maximum matching size $\mu(G)$. We define a dynamic…

Data Structures and Algorithms · Computer Science 2023-07-13 Joakim Blikstad , Peter Kiss

We discuss combinatorial algorithms for finding a maximum weight $f$-factor on an arbitrary multigraph, for given integral weights of magnitude at most $W$. For simple bipartite graphs the best-known time bound is $O(n^{2/3}\, m\, \log nW)$…

Data Structures and Algorithms · Computer Science 2020-10-05 Harold Gabow

We provide an algorithm that maintains, against an adaptive adversary, a $(1-\varepsilon)$-approximate maximum matching in $n$-node $m$-edge general (not necessarily bipartite) undirected graph undergoing edge deletions with high…

Data Structures and Algorithms · Computer Science 2024-12-05 Jiale Chen , Aaron Sidford , Ta-Wei Tu

Two general algorithms based on opportunity costs are given for approximating a revenue-maximizing set of bids an auctioneer should accept, in a combinatorial auction in which each bidder offers a price for some subset of the available…

Computational Engineering, Finance, and Science · Computer Science 2007-05-23 Karhan Akcoglu , James Aspnes , Bhaskar DasGupta , Ming-Yang Kao

Auctions are widely used in exchanges to match buy and sell requests. Once the buyers and sellers place their requests, the exchange determines how these requests are to be matched. The two most popular objectives used while determining the…

Data Structures and Algorithms · Computer Science 2024-03-06 Mohit Garg , Suneel Sarswat

In this paper we consider graph algorithms in models of computation where the space usage (random accessible storage, in addition to the read only input) is sublinear in the number of edges $m$ and the access to input data is constrained.…

Data Structures and Algorithms · Computer Science 2015-04-21 Kook Jin Ahn , Sudipto Guha

In this paper we present linear time approximation schemes for several generalized matching problems on nonbipartite graphs. Our results include $O_\epsilon(m)$-time algorithms for $(1-\epsilon)$-maximum weight $f$-factor and…

Data Structures and Algorithms · Computer Science 2020-05-11 Dawei Huang , Seth Pettie