Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Online $b$-Matching with Stochastic Rewards

Susanne Albers, Sebastian Schubert

TL;DR

The upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms, for $b$-matching with stochastic rewards, are developed and it is proved that the deterministic StochasticBalance algorithm achieves a competitive ratio of $1-1/e$, as server capacities increase, for arbitrary heterogeneous non-vanishing edge probabilities.

Abstract

The $b$-matching problem is an allocation problem where the vertices on the left-hand side of a bipartite graph, referred to as servers, may be matched multiple times. In the setting with stochastic rewards, an assignment between an incoming request and a server turns into a match with a given success probability. Mehta and Panigrahi (FOCS 2012) introduced online bipartite matching with stochastic rewards, where each vertex may be matched once. The framework is equally interesting in graphs with vertex capacities. In Internet advertising, for instance, the advertisers seek successful matches with a large number of users. We develop (tight) upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms, for $b$-matching with stochastic rewards. Our bounds hold for both offline benchmarks considered in the literature. As in prior work, we first consider vanishing probabilities. We show that no randomized online algorithm can achieve a competitive ratio greater than $1-1/e\approx 0.632$, even for identical vanishing probabilities and arbitrary uniform server capacities. Furthermore, we conduct a primal-dual analysis of the deterministic \textsc{StochasticBalance} algorithm. We prove that it achieves a competitive ratio of $1-1/e$, as server capacities increase, for arbitrary heterogeneous non-vanishing edge probabilities. This performance guarantee holds in a general setting where servers have individual capacities and for the vertex-weighted problem extension. To the best of our knowledge, this is the first result for \textsc{StochasticBalance} with arbitrary non-vanishing probabilities. We remark that our impossibility result implies in particular that, for the AdWords problem, no online algorithm can be better than $(1-1/e)$-competitive in the setting with stochastic rewards.

Online $b$-Matching with Stochastic Rewards

TL;DR

The upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms, for -matching with stochastic rewards, are developed and it is proved that the deterministic StochasticBalance algorithm achieves a competitive ratio of , as server capacities increase, for arbitrary heterogeneous non-vanishing edge probabilities.

Abstract

The -matching problem is an allocation problem where the vertices on the left-hand side of a bipartite graph, referred to as servers, may be matched multiple times. In the setting with stochastic rewards, an assignment between an incoming request and a server turns into a match with a given success probability. Mehta and Panigrahi (FOCS 2012) introduced online bipartite matching with stochastic rewards, where each vertex may be matched once. The framework is equally interesting in graphs with vertex capacities. In Internet advertising, for instance, the advertisers seek successful matches with a large number of users. We develop (tight) upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms, for -matching with stochastic rewards. Our bounds hold for both offline benchmarks considered in the literature. As in prior work, we first consider vanishing probabilities. We show that no randomized online algorithm can achieve a competitive ratio greater than , even for identical vanishing probabilities and arbitrary uniform server capacities. Furthermore, we conduct a primal-dual analysis of the deterministic \textsc{StochasticBalance} algorithm. We prove that it achieves a competitive ratio of , as server capacities increase, for arbitrary heterogeneous non-vanishing edge probabilities. This performance guarantee holds in a general setting where servers have individual capacities and for the vertex-weighted problem extension. To the best of our knowledge, this is the first result for \textsc{StochasticBalance} with arbitrary non-vanishing probabilities. We remark that our impossibility result implies in particular that, for the AdWords problem, no online algorithm can be better than -competitive in the setting with stochastic rewards.

Paper Structure

This paper contains 10 sections, 18 theorems, 55 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Contribution
  4. Upper Bound of $1-1/e$ for Vanishing Probabilities
  5. Expected Number of Matches by StochasticBalance on $G_n^b$
  6. Stochastic Benchmark on $G_n^b$
  7. Lower Bound: Competitiveness of StochasticBalance for $b\rightarrow \infty$
  8. Extensions of StochasticBalance
  9. Extension for Variable Server Capacities
  10. Extension for Vertex Weights

Key Result

Lemma 1

StochasticBalance is optimal for input graph $G_n^b$, for any combination of $n$ and $b$.

Figures (1)

  • Figure 1: The area of the gray rectangles is proportional to the value of $\hat{x}(s)$ when $s$ is assigned edges with probabilities $p_1, p_2, \ldots, p_j$.

Theorems & Definitions (33)

  • Lemma 1: cf., Lemma 12 of Mehta and Panigrahi MP
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 23 more