[Paper Review] Integrality Gap of the Configuration LP for the Restricted Max-Min Fair Allocation

Siu-Wing Cheng, Yuchen Mao|arXiv (Cornell University)|Jul 11, 2018

Optimization and Search Problems13 references3 citations

TL;DR

This paper improves the upper bound on the integrality gap of the configuration LP for the restricted max-min fair allocation problem from 4 to 23/6 ≈ 3.833 by leveraging a dual-based analysis that amortizes edge values using scaled-down resource values. The approach uses a constructive matching algorithm on a hypergraph derived from configurations, ensuring a polynomial-time approximation within 23/6 + δ of the optimal value for any δ > 0.

ABSTRACT

The max-min fair allocation problem seeks an allocation of resources to players that maximizes the minimum total value obtained by any player. Each player $p$ has a non-negative value $v_{pr}$ on resource $r$. In the restricted case, we have $v_{pr}\in \{v_r, 0\}$. That is, a resource $r$ is worth value $v_r$ for the players who desire it and value 0 for the other players. In this paper, we consider the configuration LP, a linear programming relaxation for the restricted problem. The integrality gap of the configuration LP is at least $2$. Asadpour, Feige, and Saberi proved an upper bound of $4$. We improve the upper bound to $23/6$ using the dual of the configuration LP. Since the configuration LP can be solved to any desired accuracy $δ$ in polynomial time, our result leads to a polynomial-time algorithm which estimates the optimal value within a factor of $23/6+δ$.

Motivation & Objective

• To close the gap between the known lower bound of 2 and the previous upper bound of 4 for the integrality gap of the configuration LP in the restricted max-min fair allocation problem.
• To develop a tighter upper bound on the integrality gap using a dual-based analysis of the configuration LP.
• To design a polynomial-time algorithm that approximates the optimal allocation within a factor of 23/6 + δ for any δ > 0.
• To overcome limitations in prior constructions that relied on worst-case edge value assumptions by introducing amortized dual values through resource scaling.

Proposed method

• Formulate the configuration LP for the restricted max-min fair allocation problem, where each player receives a subset of resources with total value at least T.
• Introduce the dual of the configuration LP and scale down the values of large resources to reduce the dual cost of edges containing many high-value resources.
• Construct a hypergraph where nodes represent players and hyperedges represent configurations; use a blocking sequence of thin edges to guide the matching process.
• Apply a dual-based analysis to bound the total dual value of resources covered by blockers and addable edges, ensuring it remains below 1 − 4λ/3.
• Use a signature vector to prove finite termination of the matching construction algorithm, relying on lexicographic decrease in the sequence of blocker sets.
• Construct a perfect matching in the hypergraph using iterative Build and Contract operations, ensuring each player receives at least λ worth of resources.

Experimental results

Research questions

• RQ1 What is the tightest possible upper bound on the integrality gap of the configuration LP for the restricted max-min fair allocation problem?
• RQ2 Can a dual-based analysis improve upon the 4-approximation bound established by Asadpour, Feige, and Saberi?
• RQ3 How can resource value scaling in the dual LP help amortize the cost of high-value configurations and edges?
• RQ4 Is the construction algorithm for the matching process finite and polynomial-time, given the current constraints?
• RQ5 Can the analysis be refined to further reduce the integrality gap beyond 23/6?

Key findings

• The integrality gap of the configuration LP for the restricted max-min fair allocation problem is at most 23/6, improving the previous upper bound of 4.
• The dual LP analysis with scaled resource values enables a tighter amortized bound on the total dual cost of blocking and addable edges.
• The construction algorithm terminates in finite time, bounded by O(e^√m) steps, due to lexicographic decrease in a signature vector of blocker set sizes.
• A perfect matching in the hypergraph ensures each player receives at least λ worth of resources, leading to a valid (23/6 + δ)-approximation.
• The result implies a polynomial-time algorithm that approximates the optimal allocation within a factor of 23/6 + δ for any δ > 0.
• The analysis identifies potential improvements, such as smarter edge selection and relaxed disjointness constraints in the matching construction.