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A Nearly Optimal Deterministic Algorithm for Online Transportation Problem

Tsubasa Harada, Toshiya Itoh

TL;DR

The paper introduces Subtree-Decomposition (SD), a deterministic online algorithm for the online transportation problem (OTR) with $k$ servers across $m$ sites, achieving a competitive ratio of at most $8m-5$ and per-request time $O(m)$. The approach hinges on a T-strong competitive framework on tree metrics and a reduction that embeds general metrics into tree metrics via MST-based embeddings, enabling transfer of tree-metric guarantees to general instances. The SD algorithm is shown to be MPFS and, on power-of-two tree metrics, is T-strongly $(3k-3)$-competitive, which combined with MPFS transfer yields the overall $(8m-5)$-competitive result for OTR and a $(4k-3)$ bound for online metric matching with unit capacities. This work narrows the gap toward the $2m-1$ lower bound, offers a scalable deterministic online method, and provides a general reduction technique for designing tree-metric-based online algorithms for broader metric-space optimization problems.

Abstract

For the online transportation problem with $m$ server sites, it has long been known that the competitive ratio of any deterministic algorithm is at least $2m-1$. Kalyanasundaram and Pruhs conjectured in 1998 that a deterministic $(2m-1)$-competitive algorithm exists for this problem, a conjecture that has remained open for over two decades. In this paper, we propose a new deterministic algorithm named Subtree-Decomposition for the online transportation problem and show that it achieves a competitive ratio of at most $8m-5$. This is the first $O(m)$-competitive deterministic algorithm, coming close to the lower bound of $2m-1$ within a constant factor.

A Nearly Optimal Deterministic Algorithm for Online Transportation Problem

TL;DR

The paper introduces Subtree-Decomposition (SD), a deterministic online algorithm for the online transportation problem (OTR) with servers across sites, achieving a competitive ratio of at most and per-request time . The approach hinges on a T-strong competitive framework on tree metrics and a reduction that embeds general metrics into tree metrics via MST-based embeddings, enabling transfer of tree-metric guarantees to general instances. The SD algorithm is shown to be MPFS and, on power-of-two tree metrics, is T-strongly -competitive, which combined with MPFS transfer yields the overall -competitive result for OTR and a bound for online metric matching with unit capacities. This work narrows the gap toward the lower bound, offers a scalable deterministic online method, and provides a general reduction technique for designing tree-metric-based online algorithms for broader metric-space optimization problems.

Abstract

For the online transportation problem with server sites, it has long been known that the competitive ratio of any deterministic algorithm is at least . Kalyanasundaram and Pruhs conjectured in 1998 that a deterministic -competitive algorithm exists for this problem, a conjecture that has remained open for over two decades. In this paper, we propose a new deterministic algorithm named Subtree-Decomposition for the online transportation problem and show that it achieves a competitive ratio of at most . This is the first -competitive deterministic algorithm, coming close to the lower bound of within a constant factor.
Paper Structure (21 sections, 26 theorems, 43 equations, 1 figure, 1 algorithm)

This paper contains 21 sections, 26 theorems, 43 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Our contributions
  4. Our techniques
  5. Related Work
  6. Preliminaries
  7. Definition of Problems
  8. Notation and Terminology
  9. Technical Lemmas from Prior Work
  10. T-strong competitive ratio for Tree Metrics
  11. New Algorithm: Subtree-Decomposition
  12. Notation for Graphs and Trees
  13. Definition of Subtree-Decomposition
  14. Hybrid Algorithm
  15. Hybrid Algorithm of Subtree-Decomposition
  16. ...and 6 more sections

Key Result

Theorem 5

If there exists an $\alpha$-competitive algorithm $\mathcal{A}$ for $\mathrm{OMM}_S$, then there exists a $(2\alpha +1)$-competitive algorithm $\mathcal{B}$ for $\mathrm{OMM}$.

Figures (1)

  • Figure : Subtree-Decomposition (denoted by $\mathcal{A}^*$)

Theorems & Definitions (43)

  • Conjecture 1: Kalyanasundaram and Pruhs KalP1998network
  • Remark 2
  • Definition 3: Power-of-two Weighted Tree
  • Definition 4: Power-of-two Tree Metric
  • Theorem 5: Meyerson et al. MNP2006
  • Definition 6: MPFS Algorithm HIM2023
  • Remark 7
  • Theorem 8: Harada et al. HIM2023
  • Theorem 9
  • Definition 10: T-strong competitive ratio
  • ...and 33 more