We perform an experimental study of algorithms for online bipartite matching under the known i.i.d. input model with integral types. In the last decade, there has been substantial effort in designing complex algorithms with the goal of improving worst-case approximation ratios. Our goal is to determine how these algorithms perform on more practical instances rather than worst-case instances. In particular, we are interested in whether the ranking of the algorithms by their worst-case performance is consistent with the ranking of the algorithms by their average-case/practical performance. We are also interested in whether preprocessing times and implementation difficulties that are introduced by these algorithms are justified in practice. To that end we evaluate these algorithms on different random inputs as well as real-life instances obtained from publicly available repositories. We compare these algorithms against several simple greedy-style algorithms. Most of the complex algorithms in the literature are presented as being non-greedy (i.e., an algorithm can intentionally skip matching a node that has available neighbors) to simplify the analysis. Every such algorithm can be turned into a greedy one without hurting its worst-case performance. On our benchmarks, non-greedy versions of these algorithms perform much worse than their greedy versions. Greedy versions perform about as well as the simplest greedy algorithm by itself. This, together with our other findings, suggests that simplest greedy algorithms are competitive with the state-of-the-art worst-case algorithms for online bipartite matching on many average-case and practical input families. Greediness is by far the most important property of online algorithms for bipartite matching.
An $(f,g)$-semi-matching in a bipartite graph $G=(U \cup V,E)$ is a set of edges $M \subseteq E$ such that each vertex $u\in U$ is incident with at most $f(u)$ edges of $M$, and each vertex $v\in V$ is incident with at most $g(v)$ edges of $M$. In this paper we give an algorithm that for a graph with $n$ vertices and $m$ edges, $n\le m$, constructs a maximum $(f,g)$-semi-matching in running time $O(m\cdot \min (\sqrt{\sum_{u\in U}f(u)}, \sqrt{\sum_{v\in V}g(v)}))$. Using the reduction of [5], our result on maximum $(f,g)$-semi-matching problem directly implies an algorithm for the optimal semi…
We study the oblivious matching problem, which aims at finding a maximum matching on a graph with unknown edge set. Any algorithm for the problem specifies an ordering of the vertex pairs. The matching is then produced by probing the pairs following the ordering, and including a pair if both of them are unmatched and there exists an edge between them. The unweighted (Chan et al. (SICOMP 2018)) and the vertex-weighted (Chan et al. (TALG 2018)) versions of the problem are well studied. In this paper, we consider the edge-weighted oblivious matching problem on bipartite graphs, which generalizes …
For numerous online bipartite matching problems, such as edge-weighted matching and matching under two-sided vertex arrivals, the state-of-the-art fractional algorithms outperform their randomized integral counterparts. This gap is surprising, given that the bipartite fractional matching polytope is integral, and so lossless rounding is possible. This gap was explained by Devanur et al.~(SODA'13), who showed that \emph{online} lossless rounding is impossible. Despite the above, we initiate the study of lossless online rounding for online bipartite matching problems. Our key observation is that…
In this paper we introduce the \emph{semi-online} model that generalizes the classical online computational model. The semi-online model postulates that the unknown future has a predictable part and an adversarial part; these parts can be arbitrarily interleaved. An algorithm in this model operates as in the standard online model, i.e., makes an irrevocable decision at each step. We consider bipartite matching in the semi-online model, for both integral and fractional cases. Our main contributions are competitive algorithms for this problem that are close to or match a hardness bound. The comp…
The optimized assignment of staff is of great significance for improving the production efficiency of the society. For specific tasks, the key to optimizing staffing is personnel scheduling. The assignment problem is classical in the personnel scheduling. In this paper, we abstract it as an optimal matching model of a bipartite graph and propose the Ultimate Hungarian Algorithm(UHA). By introducing feasible labels, iteratively searching for the augmenting path to get the optimal match(maximum-weight matching). And we compare the algorithm with the traditional brute force method, then conclude …
We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the…
We study the online bipartite matching problem, introduced by Karp, Vazirani and Vazirani [1990]. For bipartite graphs with matchings of size $n$, it is known that the Ranking randomized algorithm matches at least $(1 - \frac{1}{e})n$ edges in expectation. It is also known that no online algorithm matches more than $(1 - \frac{1}{e})n + O(1)$ edges in expectation, when the input is chosen from a certain distribution that we refer to as $D_n$. This upper bound also applies to fractional matchings. We review the known proofs for this last statement. In passing we observe that the $O(1)$ additive…
Online bipartite matching has been extensively studied. In the unweighted setting, Karp et al. gave an optimal $(1 - 1/e)$-competitive randomized algorithm. In the weighted setting, optimal algorithms have been achieved only under assumptions on the edge weights. For the general case, little was known beyond the trivial $1/2$-competitive greedy algorithm. Recently, Fahrbach et al. have presented an 0.5086-competitive algorithm (for the problem in a model, namely free-disposal), overcoming the long-standing barrier of 1/2. Besides, in designing competitive algorithms for the online matching pro…
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* such that each edge in M* intersects with at most c other edges in M*, and that all edge crossings in M* are contained in X. In the present paper, we propose polynomial-time algorithms for the cases of c=1 and 2. The time complexities of the algorithms are O((k+m)log n+n) for c=1 and O(k^3+k^2n+m(m+log n)) for c=2, respectively, where n=|V|, m=|E| and k=|X|…
In bipartite matching problems, vertices on one side of a bipartite graph are paired with those on the other. In its online variant, one side of the graph is available offline, while the vertices on the other side arrive online. When a vertex arrives, an irrevocable and immediate decision should be made by the algorithm; either match it to an available vertex or drop it. Examples of such problems include matching workers to firms, advertisers to keywords, organs to patients, and so on. Much of the literature focuses on maximizing the total relevance---modeled via total weight---of the matching…
The rich literature on online Bayesian selection problems has long focused on so-called prophet inequalities, which compare the gain of an online algorithm to that of a "prophet" who knows the future. An equally-natural, though significantly less well-studied benchmark is the optimum online algorithm, which may be omnipotent (i.e., computationally-unbounded), but not omniscient. What is the computational complexity of the optimum online? How well can a polynomial-time algorithm approximate it? We study the above questions for the online stochastic maximum-weight matching problem under vertex a…
We prove that a fractional perfect matching in a non-bipartite graph can be written, in polynomial time, as a convex combination of perfect matchings. This extends the Birkhoff-von Neumann Theorem from bipartite to non-bipartite graphs. The algorithm of Birkhoff and von Neumann is greedy; it starts with the given fractional perfect matching and successively "removes" from it perfect matchings, with appropriate coefficients. This fails in non-bipartite graphs -- removing perfect matchings arbitrarily can lead to a graph that is non-empty but has no perfect matchings. Using odd cuts appropriatel…
We consider the allocation of limited resources to heterogeneous customers who arrive in an online fashion. We would like to allocate the resources "fairly", so that no group of customers is marginalized in terms of their overall service rate. We study whether this is possible to do so in an online fashion, and if so, what a good online allocation policy is. We model this problem using online bipartite matching under stationary arrivals, a fundamental model in the literature typically studied under the objective of maximizing the total number of customers served. We instead study the objective…
We show that the ratio of the number of near perfect matchings to the number of perfect matchings in $d$-regular strong expander (non-bipartite) graphs, with $2n$ vertices, is a polynomial in $n$, thus the Jerrum and Sinclair Markov chain [JS89] mixes in polynomial time and generates an (almost) uniformly random perfect matching. Furthermore, we prove that such graphs have at least $Ω(d)^n$ any perfect matchings, thus proving the Lovasz-Plummer conjecture [LP86] for this family of graphs.
\noindent By a seminal result of Valiant, computing the permanent of $(0,1)$-matrices is, in general, $\#\mathsf{P}$-hard. In 1913 Pólya asked for which $(0,1)$-matrices $A$ it is possible to change some signs such that the permanent of $A$ equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the biadjacency matrices of bipartite graphs excluding $K_{3,3}$ as a \{matching minor}. This was turned into a polynomial time algorithm by McCuaig, Robertson, Seymour, and Thomas in 1999. However, the relation between the exclusion of some matching minor in…
Maximum bipartite matching is a fundamental algorithmic problem which can be solved in polynomial time. We consider a natural variant in which there is a separation constraint: the vertices on one side lie on a path or a grid, and two vertices that are close to each other are not allowed to be matched simultaneously. We show that the problem is hard to approximate even for paths, and provide constant-factor approximation algorithms for both paths and grids.
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…
We settle the complexities of the maximum-cardinality bipartite matching problem (BMM) up to poly-logarithmic factors in five models of computation: the two-party communication, AND query, OR query, XOR query, and quantum edge query models. Our results answer open problems that have been raised repeatedly since at least three decades ago [Hajnal, Maass, and Turan STOC'88; Ivanyos, Klauck, Lee, Santha, and de Wolf FSTTCS'12; Dobzinski, Nisan, and Oren STOC'14; Nisan SODA'21] and tighten the lower bounds shown by Beniamini and Nisan [STOC'21] and Zhang [ICALP'04]. We also settle the communicatio…
We provide $\widetilde{O}(ε^{-1})$-pass semi-streaming algorithms for computing $(1-ε)$-approximate maximum cardinality matchings in bipartite graphs. Our most efficient methods are deterministic and use optimal, $O(n)$, space, improving upon the space complexity of the previous state-of-the-art $\widetilde{O}(ε^{-1})$-pass algorithm of Ahn and Guha. To obtain our results we provide semi-streaming adaptations of more general continuous optimization tools. Further, we leverage these techniques to obtain improvements for streaming variants of approximate linear programming, optimal transport, ex…
Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs
我们提出了一个确定性的方式分配小（日志位）的权重的二分平面图的边缘，使最小重量完美匹配成为唯一的。隔离引理（Mulmuley et al. 1987）中所描述的实现了相同的一般图使用随机加权方案，而我们可以做到这一点时，确定性限制到二分平面图。因此，我们减少了决策和建设版本的匹配问题，以测试是否矩阵是奇异的，根据承诺，其行列式是0或1，从而获得一个高度并行的SPL算法的二分平面图。这改进了先前已知的非均匀SPL的界限（Allender et al. 1999）和$NC^2$的界限（Miller and Naor 1995，Mahajan and Varadarajan 2000）。这也重新点燃了希望，获得一个确定性的并行算法，构造一个完美的匹配在非二部平面图，这已经开放了很长一段时间。我们的技术是基本和简单的。
We introduce a `concrete complexity' model for studying algorithms for matching in bipartite graphs. The model is based on the "demand query" model used for combinatorial auctions. Most (but not all) known algorithms for bipartite matching seem to be translatable into this model including exact, approximate, sequential, parallel, and online ones. A perfect matching in a bipartite graph can be found in this model with O(n^{3/2}) demand queries (in a bipartite graph with n vertices on each side) and our main open problem is to either improve the upper bound or prove a lower bound. An improved up…
In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete even when restricted to bipartite graphs. It has been proved that Matching Cut is polynomially solvable for graphs of diameter two. In this paper, we show that, for any fixed integer $d\ge 3$, Matching Cut is NP-complete in the class of graphs of diameter $d$. This resolves an open problem posed by Borowiecki and Jesse-Józefczyk in [Matching cutsets in graphs of diameter $2$, Theoretical Computer Science 407 (200…
We show that the problem of counting perfect matchings remains #P-complete even if we restrict the input to very dense graphs, proving the conjecture in [5]. Here "dense graphs" refer to bipartite graphs of bipartite independence number $\leq 2$, or general graphs of independence number $\leq 2$. Our proof is by reduction from counting perfect matchings in bipartite graphs, via elementary linear algebra tricks and graph constructions.
Box-simplex games are a family of bilinear minimax objectives which encapsulate graph-structured problems such as maximum flow [She17], optimal transport [JST19], and bipartite matching [AJJ+22]. We develop efficient near-linear time, high-accuracy solvers for regularized variants of these games. Beyond the immediate applications of such solvers for computing Sinkhorn distances, a prominent tool in machine learning, we show that these solvers can be used to obtain improved running times for maintaining a (fractional) $ε$-approximate maximum matching in a dynamic decremental bipartite graph aga…
We propose a model for online graph problems where algorithms are given access to an oracle that predicts (e.g., based on modeling assumptions or on past data) the degrees of nodes in the graph. Within this model, we study the classic problem of online bipartite matching, and a natural greedy matching algorithm called MinPredictedDegree, which uses predictions of the degrees of offline nodes. For the bipartite version of a stochastic graph model due to Chung, Lu, and Vu where the expected values of the offline degrees are known and used as predictions, we show that MinPredictedDegree stochasti…
