We study online algorithms for maximum cardinality matchings with edge arrivals in graphs of low degree. Buchbinder, Segev, and Tkach showed that no online algorithm for maximum cardinality fractional matchings can achieve a competitive ratio larger than $4/(9-\sqrt 5)\approx 0.5914$ even for graphs of maximum degree three. The negative result of Buchbinder et al. holds even when the graph is bipartite and edges are revealed according to vertex arrivals, i.e. once a vertex arrives, all edges are revealed that include the newly arrived vertex and one of the previously arrived vertices. In this work, we complement the negative result of Buchbinder et al. by providing an online algorithm for maximum cardinality fractional matchings with a competitive ratio at least $4/(9-\sqrt 5)\approx 0.5914$ for graphs of maximum degree three. We also demonstrate that no online algorithm for maximum cardinality integral matchings can have the competitive guarantee $0.5807$, establishing a gap between integral and fractional matchings for graphs of maximum degree three. Note that the work of Buchbinder et al. shows that for graphs of maximum degree two, there is no such gap between fractional and integral matchings, because for both of them the best achievable competitive ratio is $2/3$. Also, our results demonstrate that for graphs of maximum degree three best possible competitive ratios for fractional matchings are the same in the vertex arrival and in the edge arrival models.
We study the online stochastic matching problem. Consider a bipartite graph with offline vertices on one side, and with i.i.d.online vertices on the other side. The offline vertices and the distribution of online vertices are known to the algorithm beforehand. The realization of the online vertices, however, is revealed one at a time, upon which the algorithm immediately decides how to match it. For maximizing the cardinality of the matching, we give a $0.711$-competitive online algorithm, which improves the best previous ratio of $0.706$. When the offline vertices are weighted, we introduce a…
We provide online algorithms for secretary matching in general weighted graphs, under the well-studied models of vertex and edge arrivals. In both models, edges are associated with arbitrary weights that are unknown from the outset, and are revealed online. Under vertex arrival, vertices arrive online in a uniformly random order; upon the arrival of a vertex $v$, the weights of edges from $v$ to all previously arriving vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. Under edge arrival, edges arrive online in a uniformly random order; u…
Online bipartite matching with edge arrivals remained a major open question for a long time until a recent negative result by [Gamlath et al. FOCS 2019], who showed that no online policy is better than the straightforward greedy algorithm, i.e., no online algorithm has a worst-case competitive ratio better than $0.5$. In this work, we consider the bipartite matching problem with edge arrivals in a natural stochastic framework, i.e., Bayesian setting where each edge of the graph is independently realized according to a known probability distribution. We focus on a natural class of prune & greed…
Most prior work on online matching problems has been with the flexibility of keeping some vertices unmatched. We study three related online matching problems with the constraint of matching every vertex, i.e., with no rejections. We adopt a model in which vertices arrive in uniformly random order and the non-negative edge-weights are arbitrary. For the capacitated online bipartite matching problem, in which the vertices of one side of the graph are offline and those of the other side arrive online, we give a 4.62-competitive algorithm when the capacity of each offline vertex is 2. For the onli…
Motivated by display advertising on the internet, the online stochastic matching problem is proposed by Feldman, Mehta, Mirrokni, and Muthukrishnan (FOCS 2009). Consider a stochastic bipartite graph with offline vertices on one side and with i.i.d. online vertices on the other side. The algorithm knows the offline vertices and the distribution of the online vertices in advance. Upon the arrival of each online vertex, its type is realized and the algorithm immediately and irrevocably decides how to match it. In the vertex-weighted version of the problem, each offline vertex is associated with a…
We consider Online Minimum Bipartite Matching under the uniform metric. We show that Randomized Greedy achieves a competitive ratio equal to $(1+1/n) (H_{n+1}-1)$, which matches the lower bound. Comparing with the fact that RG achieves an optimal ratio of $Θ(\ln n)$ for the same problem but under the adversarial order, we find that the weaker arrival assumption of random order doesn't offer any extra algorithmic advantage for RG, or make the model strictly more tractable.
Online bipartite matching with one-sided arrival and its variants have been extensively studied since the seminal work of Karp, Vazirani, and Vazirani (STOC 1990). Motivated by real-life applications with dynamic market structures, e.g. ride-sharing, two generalizations of the classical one-sided arrival model are proposed to allow non-bipartite graphs and to allow all vertices to arrive online. Namely, online matching with general vertex arrival is introduced by Wang and Wong (ICALP 2015), and fully online matching is introduced by Huang et al. (JACM 2020). In this paper, we study the fractio…
We consider the prophet inequality problem for (not necessarily bipartite) matching problems with independent edge values, under both edge arrivals and vertex arrivals. We show constant-factor prophet inequalities for the case where the online algorithm has only limited access to the value distributions through samples. First, we give a $16$-approximate prophet inequality for matching in general graphs under edge arrivals that uses only a single sample from each value distribution as prior information. Then, for bipartite matching and (one-sided) vertex arrivals, we show an improved bound of $…
The bipartite matching problem in the online and streaming settings has received a lot of attention recently. The classical vertex arrival setting, for which the celebrated Karp, Vazirani and Vazirani (KVV) algorithm achieves a $1-1/e$ approximation, is rather well understood: the $1-1/e$ approximation is optimal in both the online and semi-streaming setting, where the algorithm is constrained to use $n\cdot \log^{O(1)} n$ space. The more challenging the edge arrival model has seen significant progress recently in the online algorithms literature. For the strictly online model (no preemption) …
The Santa Claus problem is a fundamental problem in fair division: the goal is to partition a set of heterogeneous items among heterogeneous agents so as to maximize the minimum value of items received by any agent. In this paper, we study the online version of this problem where the items are not known in advance and have to be assigned to agents as they arrive over time. If the arrival order of items is arbitrary, then no good assignment rule exists in the worst case. However, we show that, if the arrival order is random, then for $n$ agents and any $\varepsilon > 0$, we can obtain a competi…
Cloud computing customers often submit repeating jobs and computation pipelines on \emph{approximately} regular schedules, with arrival and running times that exhibit variance. This pattern, typical of training tasks in machine learning, allows customers to partially predict future job requirements. We develop a model of cloud computing platforms that receive statements of work (SoWs) in an online fashion. The SoWs describe future jobs whose arrival times and durations are probabilistic, and whose utility to the submitting agents declines with completion time. The arrival and duration distribu…
Nearly three decades ago, Bar-Noy, Motwani and Naor showed that no online edge-coloring algorithm can edge color a graph optimally. Indeed, their work, titled "the greedy algorithm is optimal for on-line edge coloring", shows that the competitive ratio of $2$ of the naïve greedy algorithm is best possible online. However, their lower bound required bounded-degree graphs, of maximum degree $Δ= O(\log n)$, which prompted them to conjecture that better bounds are possible for higher-degree graphs. While progress has been made towards resolving this conjecture for restricted inputs and arrivals or…
We study the greedy-based online algorithm for edge-weighted matching with (one-sided) vertex arrivals in bipartite graphs, and edge arrivals in general graphs. This algorithm was first studied more than a decade ago by Korula and Pál for the bipartite case in the random-order model. While the weighted bipartite matching problem is solved in the random-order model, this is not the case in recent and exciting online models in which the online player is provided with a sample, and the arrival order is adversarial. The greedy-based algorithm is arguably the most natural and practical algorithm to…
We consider the online carpooling problem: given $n$ vertices, a sequence of edges arrive over time. When an edge $e_t = (u_t, v_t)$ arrives at time step $t$, the algorithm must orient the edge either as $v_t \rightarrow u_t$ or $u_t \rightarrow v_t$, with the objective of minimizing the maximum discrepancy of any vertex, i.e., the absolute difference between its in-degree and out-degree. Edges correspond to pairs of persons wanting to ride together, and orienting denotes designating the driver. The discrepancy objective then corresponds to every person driving close to their fair share of rid…
In this paper, we study max-weight stochastic matchings on online bipartite graphs under both vertex and edge arrivals. We focus on designing polynomial time approximation algorithms with respect to the online benchmark, which was first considered by Papadimitriou, Pollner, Saberi, and Wajc [EC'21]. In the vertex arrival version of the problem, the goal is to find an approximate max-weight matching of a given bipartite graph when the vertices in one part of the graph arrive online in a fixed order with independent chances of failure. Whenever a vertex arrives we should decide, irrevocably, whe…
We study a dynamic non-bipartite matching problem. There is a fixed set of agent types, and agents of a given type arrive and depart according to type-specific Poisson processes. Agent departures are not announced in advance. The value of a match is determined by the types of the matched agents. We present an online algorithm that is (1/8)-competitive with respect to the value of the optimal-in-hindsight policy, for arbitrary weighted graphs. Our algorithm treats agents heterogeneously, interpolating between immediate and delayed matching in order to thicken the market while still matching val…
A Nonparametric Framework for Online Stochastic Matching with Correlated Arrivals
Sloan School of Management, Massachusetts Institute of Technology（麻省理工学院斯隆管理学院）；Graduate School of Business and Data Science Institute, Columbia University（哥伦比亚大学商学院与数据科学研究所）
University of Bonn（波恩大学）；Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）
We introduce online learning with vector costs (\OLVCp) where in each time step $t \in \{1,\ldots, T\}$, we need to play an action $i \in \{1,\ldots,n\}$ that incurs an unknown vector cost in $[0,1]^{d}$. The goal of the online algorithm is to minimize the $\ell_p$ norm of the sum of its cost vectors. This captures the classical online learning setting for $d=1$, and is interesting for general $d$ because of applications like online scheduling where we want to balance the load between different machines (dimensions). We study \OLVCp in both stochastic and adversarial arrival settings, and give…
The fractional knapsack problem is one of the classical problems in combinatorial optimization, which is well understood in the offline setting. However, the corresponding online setting has been handled only briefly in the theoretical computer science literature so far, although it appears in several applications. Even the previously best known guarantee for the competitive ratio was worse than the best known for the integral problem in the popular random order model. We show that there is an algorithm for the online fractional knapsack problem that admits a competitive ratio of 4.39. Our res…
