[Paper Review] Random-order Contention Resolution via Continuous Induction: Tightness for Bipartite Matching under Vertex Arrivals

Calum MacRury, Will Ma|arXiv (Cornell University)|Oct 16, 2023

Cooperative Communication and Network Coding4 citations

TL;DR

This paper introduces a continuous-time induction framework for designing Random-order Contention Resolution Schemes (RCRS) with exact selection probabilities via a time-discretized algorithm that samples past execution. It achieves a tight (1+e⁻²)/2 ≈ 0.567-selectable RCRS for bipartite graphs under vertex arrivals, and extends this to general graphs with odd girth g, showing convergence to the same bound as g→∞.

ABSTRACT

We introduce a new approach for designing Random-order Contention Resolution Schemes (RCRS) via exact solution in continuous time. Given a function $c(y):[0,1] ightarrow [0,1]$, we show how to select each element which arrives at time $y \in [0,1]$ with probability exactly $c(y)$. We provide a rigorous algorithmic framework for achieving this, which discretizes the time interval and also needs to sample its past execution to ensure these exact selection probabilities. We showcase our framework in the context of online contention resolution schemes for matching with random-order vertex arrivals. For bipartite graphs with two-sided arrivals, we design a $(1+e^{-2})/2 \approx 0.567$-selectable RCRS, which we also show to be tight. Next, we show that the presence of short odd-length cycles is the only barrier to attaining a (tight) $(1+e^{-2})/2$-selectable RCRS on general graphs. By generalizing our bipartite RCRS, we design an RCRS for graphs with odd-length girth $g$ which is $(1+e^{-2})/2$-selectable as $g ightarrow \infty$. This convergence happens very rapidly: for triangle-free graphs (i.e., $g \ge 5$), we attain a $121/240 + 7/16 e^2 \approx 0.563$-selectable RCRS. Finally, for general graphs we improve on the $8/15 \approx 0.533$-selectable RCRS of Fu et al. (ICALP, 2021) and design an RCRS which is at least $0.535$-selectable. Due to the reduction of Ezra et al. (EC, 2020), our bounds yield a $0.535$-competitive (respectively, $(1+e^{-2})/2$-competitive) algorithm for prophet secretary matching on general (respectively, bipartite) graphs under vertex arrivals.

Motivation & Objective

• To design a new framework for Random-order Contention Resolution Schemes (RCRS) with exact selection probabilities in continuous time.
• To achieve tight selectability bounds for online matching under random-order vertex arrivals, particularly in bipartite graphs.
• To identify structural graph properties—specifically odd-length girth—as the key barrier to achieving optimal selectability.
• To generalize the RCRS construction to general graphs and quantify convergence to the optimal bound as odd girth increases.
• To improve upon prior competitive ratios for prophet secretary matching via the reduction from RCRS to online algorithms.

Proposed method

• Propose a continuous-time induction framework that defines a selection function c(y) ∈ [0,1] for each arrival time y ∈ [0,1], ensuring exact selection probability c(y) for elements arriving at time y.
• Discretize the time interval [0,1] and use past execution sampling to maintain exact selection probabilities under random-order arrivals.
• Derive a differential equation system and use stochastic process analysis to verify that the selection probability at time t matches the desired target via induction.
• Construct a selection function c(y) for graphs with odd girth g ≥ 3, incorporating the upper incomplete gamma function to account for cycle structure.
• Use the reduction from RCRS to prophet secretary algorithms to derive competitive ratios for online matching under vertex arrivals.
• Apply concentration bounds and martingale arguments (e.g., Theorem 3 from [BM23a]) to prove convergence and high-probability bounds on the performance of the scheme.

Experimental results

Research questions

• RQ1 Can a continuous-time induction framework achieve exact selection probabilities in RCRS under random-order vertex arrivals?
• RQ2 What is the tightest possible selectability achievable for bipartite matching under two-sided vertex arrivals?
• RQ3 How does the presence of short odd-length cycles affect the maximum achievable selectability in general graphs?
• RQ4 To what extent does selectability converge to (1+e⁻²)/2 as the odd girth g → ∞?
• RQ5 Can the framework be extended to yield improved competitive ratios for prophet secretary matching on general graphs?

Key findings

• The proposed framework achieves a (1+e⁻²)/2 ≈ 0.567-selectable RCRS for bipartite graphs with two-sided vertex arrivals, and this bound is proven to be tight.
• For triangle-free graphs (odd girth g ≥ 5), the RCRS achieves a selectability of approximately 0.563, close to the optimal 0.567.
• As the odd girth g → ∞, the selectability of the RCRS converges to (1+e⁻²)/2, demonstrating that odd cycles are the only structural barrier to optimality.
• For general graphs, the method yields a 0.535-selectable RCRS, improving upon the prior 8/15 ≈ 0.533 bound from [FTW+21].
• Via the reduction from [EFGT22], the results yield a 0.535-competitive algorithm for prophet secretary matching on general graphs, and a (1+e⁻²)/2-competitive algorithm on bipartite graphs.