When Stochastic Rewards Reduce to Deterministic Rewards in Online Bipartite Matching

Rajan Udwani

Home/Archive/arXiv

arXiv · 2308.09767

When Stochastic Rewards Reduce to Deterministic Rewards in Online Bipartite Matching

Rajan Udwani

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the problem of vertex-weighted online bipartite matching with stochastic rewards where matches may fail with some known probability and the decision maker has to adapt to the sequential realization of these outcomes. Recent works have studied several special cases of this problem and it was known that the (randomized) Perturbed Greedy algorithm due to Aggarwal et al. (SODA, 2011) achieves the best possible competitive ratio guarantee of $(1-1/e)$ in some cases. We give a simple proof of these results by reducing (special cases of) the stochastic rewards problem to the deterministic setting of online bipartite matching (Karp, Vazirani, Vazirani (STOC, 1990)). More broadly, our approach gives conditions under which it suffices to analyze the competitive ratio of an algorithm for the simpler setting of deterministic rewards in order to obtain a competitive ratio guarantee for stochastic rewards. The simplicity of our approach reveals that the Perturbed Greedy algorithm has a competitive ratio of $(1-1/e)$ even in certain settings with correlated rewards, where no results were previously known. Finally, we show that without any special assumptions, the Perturbed Greedy algorithm has a competitive ratio strictly less than $(1-1/e)$ for vertex-weighted online matching with stochastic rewards.

Explore related subjects

cs.DS

Keep this discovery

BibTeX RIS

Rajan Udwani. 2023-08-18. When Stochastic Rewards Reduce to Deterministic Rewards in Online Bipartite Matching. https://arxiv.org/abs/2308.09767

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

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

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↗