[Paper Review] Making Three Out of Two: Three-Way Online Correlated Selection

Yong-Ho Shin, Hyung-Chan An|arXiv (Cornell University)|Jul 6, 2021

Optimization and Search Problems33 references4 citations

TL;DR

This paper presents a novel three-way online correlated selection (3-OCS) algorithm that extends two-way OCS by using two cascaded instances of two-way OCS to achieve negative correlation in choices from triples of elements. The method achieves a 0.5093-competitive ratio for edge-weighted online bipartite matching, improving upon the prior 0.5086 bound by leveraging a surrogate distribution to approximate the complex output behavior of the cascaded OCS.

ABSTRACT

Two-way online correlated selection (two-way OCS) is an online algorithm that, at each timestep, takes a pair of elements from the ground set and irrevocably chooses one of the two elements, while ensuring negative correlation in the algorithm's choices. Whilst OCS was initially invented by Fahrbach, Huang, Tao, and Zadimoghaddam to solve the edge-weighted online bipartite matching problem, it is an interesting technique on its own due to its capability of introducing a powerful algorithmic tool, namely negative correlation, to online algorithms. As such, Fahrbach et al. posed two tantalizing open questions in their paper, one of which was the following: Can we obtain n-way OCS for n>2, in which the algorithm can be given n>2 elements to choose from at each timestep? In this paper, we affirmatively answer this open question by presenting a three-way OCS. Our algorithm uses two-way OCS as its building block and is simple to describe; however, as it internally runs two instances of two-way OCS, one of which is fed with the output of the other, the final output probability distribution becomes highly elusive. We tackle this difficulty by approximating the output distribution of OCS by a flat, less correlated function and using it as a safe "surrogate" of the real distribution. Our three-way OCS also yields a 0.5093-competitive algorithm for edge-weighted online matching, demonstrating its usefulness.

Motivation & Objective

• To resolve an open question posed by Fahrbach et al. on the existence of n-way OCS for n > 2.
• To design a simple yet effective three-way OCS that maintains strong negative correlation in online selection from triples.
• To demonstrate the utility of 3-OCS by improving the competitive ratio for edge-weighted online bipartite matching.
• To develop a method for approximating complex output distributions of cascaded OCS using flatter, symmetric surrogate distributions.

Proposed method

• The algorithm selects two elements uniformly at random from a triple and feeds them to the first two-way OCS.
• The output of the first OCS is combined with the third element (left out of the first selection) and fed into a second two-way OCS.
• The final output is the result of the second OCS, ensuring negative correlation across choices.
• The method uses a surrogate distribution that is flatter and symmetric to approximate the true output distribution of the cascaded OCS.
• It leverages the unimodal and symmetric nature of two-way OCS output to bound the probability of repeated non-selection of a common element.
• A factor-revealing linear program is formulated with bounded variables to compute the competitive ratio, using truncation at k_max = 25 and ℓ_max = 25.

Experimental results

Research questions

• RQ1 Can a three-way online correlated selection algorithm be constructed that maintains negative correlation across choices from triples?
• RQ2 How can the complex output distribution of cascaded two-way OCS be approximated to enable analysis?
• RQ3 What competitive ratio can be achieved for edge-weighted online bipartite matching using the proposed 3-OCS?
• RQ4 Is the 3-OCS method capable of improving upon the prior 0.5086 competitive ratio in edge-weighted online matching?
• RQ5 Can the surrogate distribution technique be formally justified to bound the probability of non-selection under cascaded OCS?

Key findings

• The proposed 3-OCS achieves a competitive ratio of 0.50930725 for edge-weighted online bipartite matching, improving upon the prior 0.5086 bound.
• The algorithm is constructed using two cascaded instances of two-way OCS, with the second OCS receiving the output of the first and the unselected element.
• The analysis uses a surrogate distribution that is flatter and symmetric to approximate the true distribution of selection counts, enabling tractable bounds.
• The competitive ratio is derived via a factor-revealing LP with k_max = 25 and ℓ_max = 25, yielding Γ = 0.50930725.
• The method successfully addresses the challenge of non-Markovian dependencies in cascaded OCS by exploiting unimodal symmetry and convexity approximations.
• The result confirms the feasibility of extending OCS beyond two-way selection and establishes a new benchmark for online matching algorithms.