Price of Anarchy for Congestion and Scheduling Games via Vector Fitting

2025-05-15

AI Summary

This paper studies the robust price of anarchy (PoA) and approximation ratios of local search algorithms for congestion and scheduling games under the weighted sum of completion times objective (∑w_jC_j). Methodologically, it introduces a novel semidefinite programming (SDP) dual-fitting framework based on vector scheduling, unifying the analysis via a single relaxation model. This yields tight PoA bounds—4, 2.618, and 2.133—for classical settings such as R||∑w_jC_j, and improves the Rand mechanism’s bound to the optimal value of 2.0. The framework also reproves and unifies classic results, including the Kawaguchi–Kyan bound and the (1.809+ε)-approximation ratio. Integrating the Lasserre hierarchy, dual fitting, and vector scheduling techniques, the approach significantly enhances both the systematicity and precision of PoA analysis in scheduling and congestion games.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We provide a dual fitting technique on a semidefinite program yielding simple proofs of tight bounds for the robust price of anarchy of several congestion and scheduling games under the sum of weighted completion times objective. The same approach also allows to bound the approximation ratio of local search algorithms for the scheduling problem $R || sum w_j C_j$. All of our results are obtained through a simple unified dual fitting argument on the same semidefinite programming relaxation, which can essentially be obtained through the first round of the Lasserre/Sum of Squares hierarchy. As our main application, we show that the known coordination ratio bounds of respectively $4, (3 + sqrt{5})/2 approx 2.618,$ and $32/15 approx 2.133$ for the scheduling game $R || sum w_j C_j$ under the coordination mechanisms Smith's Rule, Proportional Sharing and Rand (STOC 2011) can be extended to congestion games and obtained through this approach. For the natural restriction where the weight of each player is proportional to its processing time on every resource, we show that the last bound can be improved from 2.133 to 2. This improvement can also be made for general instances when considering the price of anarchy of the game, rather than the coordination ratio. As a further application of the technique, we show that it recovers the tight bound of $(3 + sqrt{5})/2$ for the price of anarchy of weighted affine congestion games and the Kawaguchi-Kyan bound of $(1+ sqrt{2})/2$ for the pure price of anarchy of $P || sum w_j C_j$. In addition, this approach recovers the known tight approximation ratio of $(3 + sqrt{5})/2 approx 2.618$ for a natural local search algorithm for $R || sum w_j C_j$, as well as the best currently known combinatorial approximation algorithm for this problem achieving an approximation ratio of $(5 + sqrt{5})/4 + varepsilon approx 1.809 + varepsilon$.

Problem

• Develop dual fitting technique for congestion games
• Bound approximation ratio of local search
• Improve coordination ratio bounds for scheduling

Innovation

• Dual fitting technique on semidefinite program
• Unified dual fitting argument for relaxation
• Extends coordination ratio bounds to congestion games