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Online Deterministic Minimum Cost Bipartite Matching with Delays on a Line

Tung-Wei Kuo

TL;DR

This work addresses online bipartite matching with delays (MBPMD) on a line, where requests and servers arrive over time and delays incur costs. It introduces a deterministic online algorithm based on moving virtual servers within a time-augmented (TA) plane and leverages a Robust Matching framework to bound total cost via gamma-net-costs. The main result is a $\tilde{O}(m^{0.5})$-competitive algorithm, with a simplified variant achieving $O(\sqrt{m}\log^2 m)$-competitiveness for the line metric, representing a substantial improvement over prior deterministic bounds. The approach combines offline/online matchings, real and virtual augmenting paths, and carefully crafted dual updates to relate online performance to the offline optimum, with potential implications for deterministic online matching in structured metrics.

Abstract

We study the online minimum cost bipartite perfect matching with delays problem. In this problem, $m$ servers and $m$ requests arrive over time, and an online algorithm can delay the matching between servers and requests by paying the delay cost. The objective is to minimize the total distance and delay cost. When servers and requests lie in a known metric space, there is a randomized $O(\log n)$-competitive algorithm, where $n$ is the size of the metric space. When the metric space is unknown a priori, Azar and Jacob-Fanani proposed a deterministic $O\left(\frac{1}εm^{\log\left(\frac{3+ε}{2}\right)}\right)$-competitive algorithm for any fixed $ε> 0$. This competitive ratio is tight when $n = 1$ and becomes $O(m^{0.59})$ for sufficiently small $ε$. In this paper, we improve upon the result of Azar and Jacob-Fanani for the case where servers and requests are on the real line, providing a deterministic $\tilde{O}(m^{0.5})$-competitive algorithm. Our algorithm is based on the Robust Matching (RM) algorithm proposed by Raghvendra for the minimum cost bipartite perfect matching problem. In this problem, delay is not allowed, and all servers arrive in the beginning. When a request arrives, the RM algorithm immediately matches the request to a free server based on the request's minimum $t$-net-cost augmenting path, where $t > 1$ is a constant. In our algorithm, we delay the matching of a request until its waiting time exceeds its minimum $t$-net-cost divided by $t$.

Online Deterministic Minimum Cost Bipartite Matching with Delays on a Line

TL;DR

This work addresses online bipartite matching with delays (MBPMD) on a line, where requests and servers arrive over time and delays incur costs. It introduces a deterministic online algorithm based on moving virtual servers within a time-augmented (TA) plane and leverages a Robust Matching framework to bound total cost via gamma-net-costs. The main result is a -competitive algorithm, with a simplified variant achieving -competitiveness for the line metric, representing a substantial improvement over prior deterministic bounds. The approach combines offline/online matchings, real and virtual augmenting paths, and carefully crafted dual updates to relate online performance to the offline optimum, with potential implications for deterministic online matching in structured metrics.

Abstract

We study the online minimum cost bipartite perfect matching with delays problem. In this problem, servers and requests arrive over time, and an online algorithm can delay the matching between servers and requests by paying the delay cost. The objective is to minimize the total distance and delay cost. When servers and requests lie in a known metric space, there is a randomized -competitive algorithm, where is the size of the metric space. When the metric space is unknown a priori, Azar and Jacob-Fanani proposed a deterministic -competitive algorithm for any fixed . This competitive ratio is tight when and becomes for sufficiently small . In this paper, we improve upon the result of Azar and Jacob-Fanani for the case where servers and requests are on the real line, providing a deterministic -competitive algorithm. Our algorithm is based on the Robust Matching (RM) algorithm proposed by Raghvendra for the minimum cost bipartite perfect matching problem. In this problem, delay is not allowed, and all servers arrive in the beginning. When a request arrives, the RM algorithm immediately matches the request to a free server based on the request's minimum -net-cost augmenting path, where is a constant. In our algorithm, we delay the matching of a request until its waiting time exceeds its minimum -net-cost divided by .
Paper Structure (42 sections, 20 theorems, 88 equations, 2 figures)

This paper contains 42 sections, 20 theorems, 88 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Other Related Work
  3. Poisson Arrival Processes.
  4. Preliminaries
  5. An Online Matching Algorithm with Moving Virtual Servers
  6. Moving Virtual Servers
  7. The VRM Algorithm
  8. Computing Minimum Augmenting Paths
  9. Timings for Updating $P_i$ and $\widetilde{P}_i$
  10. Updating the Dual Variables
  11. Upper Bounding Distance in the TA Plane and Nonnegativity of $\gamma$-Net-Costs
  12. A Simplified Algorithm and Analysis
  13. Implications of Lemma \ref{['lemma: vlerev']}
  14. Proof of Theorem \ref{['thrm: main']}
  15. Proof of Lemma \ref{['lemma: vlerev']}
  16. ...and 27 more sections

Key Result

Theorem 1.1

There is a deterministic $O(\sqrt{m}\log^2 m)$-competitive algorithm for the MBPMD problem on a line.

Figures (2)

  • Figure 1: An example of the partition of $P$ and the construction of $P'$ where $cr = 3$ and $\mathop{\mathrm{sgn}}\limits(s_1^X, r_{cr}^X) = 1$.
  • Figure 2: An example of the partition of $P$ and the construction of $P'$ where $cr = 2$ and $\mathop{\mathrm{sgn}}\limits(s_1^X, r_{cr}^X) = -1$.

Theorems & Definitions (48)

  • Theorem 1.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma A.1
  • proof
  • Lemma A.2
  • ...and 38 more