Selfish, Local and Online Scheduling via Vector Fitting

TL;DR

This paper introduces a unified dual fitting approach using semidefinite programming to analyze and bound the efficiency of various scheduling and congestion games, improving known bounds and extending results to broader settings.

Contribution

It presents a novel dual fitting technique that simplifies proofs and extends bounds for the price of anarchy and approximation ratios in scheduling and congestion games.

Findings

Extended coordination ratio bounds to congestion games.

Improved bounds for the price of anarchy in weighted affine congestion games.

Analyzed local search and online algorithms, achieving tight bounds and competitive ratios.

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 and the competitive ratio of online 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 this 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 $P || \sum w_j C_j$. Moreover, this approach can analyze a simple local search algorithm for $R || \sum w_j C_j$, the best currently known combinatorial approximation algorithm for this problem achieving an approximation ratio of $(5 + \sqrt{5})/4 + \varepsilon$ and an online greedy algorithm which is $4$-competitive.