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Optimal Online Bipartite Matching in Degree-2 Graphs

Amey Bhangale, Arghya Chakraborty, Prahladh Harsha

TL;DR

This work analyzes online bipartite matching when online vertices have degree at most 2, revealing a sharp separation between fractional and randomized integral formulations. The Water-Level algorithm achieves the fractional optimum of $0.75$, while the Half-Half algorithm is shown to be optimal for randomized integral matching with the constant $\eta = 1 - \sum_{i=1}^{\infty} \frac{1}{2^{2^i+i-1}} \approx 0.717772$, a bound that is tight via a Yao-based lower bound. The authors employ a nontrivial online primal–dual analysis, constructing $\gamma$-feasible duals and bounding $\Delta P$ against $\Delta D$, and they provide a detailed inductive dual-update mechanism for Half-Half. This results in a formal separation between fractional and randomized integral matching in degree-2 graphs and raises open questions about higher-degree graphs and rounding strategies. The findings have implications for understanding when fractional solutions can be reliably rounded to integral online solutions and for the design of optimal online algorithms under tight degree constraints.

Abstract

Online bipartite matching is a classical problem in online algorithms and we know that both the deterministic fractional and randomized integral online matchings achieve the same competitive ratio of $1-\frac{1}{e}$. In this work, we study classes of graphs where the online degree is restricted to $2$. As expected, one can achieve a competitive ratio of better than $1-\frac{1}{e}$ in both the deterministic fractional and randomized integral cases, but surprisingly, these ratios are not the same. It was already known that for fractional matching, a $0.75$ competitive ratio algorithm is optimal. We show that the folklore \textsc{Half-Half} algorithm achieves a competitive ratio of $η\approx 0.717772\dots$ and more surprisingly, show that this is optimal by giving a matching lower-bound. This yields a separation between the two problems: deterministic fractional and randomized integral, showing that it is impossible to obtain a perfect rounding scheme.

Optimal Online Bipartite Matching in Degree-2 Graphs

TL;DR

This work analyzes online bipartite matching when online vertices have degree at most 2, revealing a sharp separation between fractional and randomized integral formulations. The Water-Level algorithm achieves the fractional optimum of , while the Half-Half algorithm is shown to be optimal for randomized integral matching with the constant , a bound that is tight via a Yao-based lower bound. The authors employ a nontrivial online primal–dual analysis, constructing -feasible duals and bounding against , and they provide a detailed inductive dual-update mechanism for Half-Half. This results in a formal separation between fractional and randomized integral matching in degree-2 graphs and raises open questions about higher-degree graphs and rounding strategies. The findings have implications for understanding when fractional solutions can be reliably rounded to integral online solutions and for the design of optimal online algorithms under tight degree constraints.

Abstract

Online bipartite matching is a classical problem in online algorithms and we know that both the deterministic fractional and randomized integral online matchings achieve the same competitive ratio of . In this work, we study classes of graphs where the online degree is restricted to . As expected, one can achieve a competitive ratio of better than in both the deterministic fractional and randomized integral cases, but surprisingly, these ratios are not the same. It was already known that for fractional matching, a competitive ratio algorithm is optimal. We show that the folklore \textsc{Half-Half} algorithm achieves a competitive ratio of and more surprisingly, show that this is optimal by giving a matching lower-bound. This yields a separation between the two problems: deterministic fractional and randomized integral, showing that it is impossible to obtain a perfect rounding scheme.

Paper Structure

This paper contains 13 sections, 8 theorems, 31 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Brief overview
  3. Preliminaries
  4. Primal and Dual programs for bipartite matching
  5. Techniques
  6. Randomized Integral Matching
  7. Lower Bound
  8. Upper Bound
  9. Analysis of the half-half algorithm
  10. Fractional Matching
  11. Lower Bound
  12. Upper Bound
  13. Conclusion

Key Result

Theorem 1.1

For any graph, $G$, with $n$ online vertices and degree of online vertices at most $2$, there exists an online randomized integral matching algorithm with competitive ratio $\eta$, and no online algorithm can attain a competitive ratio better than $\eta$,

Figures (1)

  • Figure 1: The graph for the lower bound, with $8$ online and offline vertices with online degree bounded by $2$. The black edges correspond to online vertices in the first phase, the red ones correspond to the second phase and the green ones correspond to the third phase.

Theorems & Definitions (23)

  • Theorem 1.1: Integral Matching
  • Theorem 1.2: Fractional Matching
  • Lemma 2.1
  • proof
  • Claim 2.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • Claim 4.3
  • ...and 13 more