Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2008.09260

Greedy Approaches to Online Stochastic Matching

Allan Borodin·Calum MacRury·Akash Rakheja
View original source ↗Read source document ↗Export metadata ↓

Abstract

Within the context of stochastic probing with commitment, we consider the online stochastic matching problem; that is, the one-sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist based on edge probabilities that become known when an online vertex arrives. If a probed edge exists, it must be used in the matching (if possible). We consider the competitiveness of online algorithms in both the adversarial order model (AOM) and the random order model (ROM). More specifically, we consider a bipartite stochastic graph $G = (U,V,E)$ where $U$ is the set of offline vertices, $V$ is the set of online vertices and $G$ has edge probabilities $(p_{e})_{e \in E}$ and edge weights $(w_{e})_{e \in E}$. Additionally, $G$ has probing constraints $(\scr{C}_{v})_{v \in V}$, where $\scr{C}_v$ indicates which sequences of edges adjacent to an online vertex $v$ can be probed. We assume that $U$ is known in advance, and that $\scr{C}_v$, together with the edge probabilities and weights adjacent to an online vertex are only revealed when the online vertex arrives. This model generalizes the various settings of the classical bipartite matching problem, and so our main contribution is in making progress towards understanding which classical results extend to the stochastic probing model.

Explore related subjects

cs.DSmath.CO

Keep this discovery

BibTeXRIS

Allan Borodin, Calum MacRury, Akash Rakheja. 2020-08-21. Greedy Approaches to Online Stochastic Matching. https://arxiv.org/abs/2008.09260

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗