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A Smoothed FPTAS for Equilibria in Congestion Games

Yiannis Giannakopoulos

TL;DR

This work establishes a smoothed analysis-based FPTAS for computing approximate equilibria in congestion games by showing that, under independent $\phi$-smooth perturbations of resource costs, a $1+\varepsilon$-improving dynamics converges to a $1+\varepsilon$-PNE in strongly polynomial expected time. The approach combines Rosenthal’s potential-based progress with a novel probabilistic lemma on order statistics of $\phi$-smooth random variables, yielding tight bounds on the expected number of improving moves. The results cover general congestion games and several structured models—step-function, polynomial, and cost-sharing—and extend to network congestion scenarios without dependence on strategy-set cardinality. Together, the two core ideas provide a unified, tunable framework that bridges worst-case hardness and practical efficiency for approximate equilibria in large networked systems.

Abstract

We present a fully polynomial-time approximation scheme (FPTAS) for computing equilibria in congestion games, under smoothed running-time analysis. More precisely, we prove that if the resource costs of a congestion game are randomly perturbed by independent noises, whose density is at most $φ$, then any sequence of $(1+\varepsilon)$-improving dynamics will reach an $(1+\varepsilon)$-approximate pure Nash equilibrium (PNE) after an expected number of steps which is strongly polynomial in $\frac{1}{\varepsilon}$, $φ$, and the size of the game's description. Our results establish a sharp contrast to the traditional worst-case analysis setting, where it is known that better-response dynamics take exponentially long to converge to $α$-approximate PNE, for any constant factor $α\geq 1$. As a matter of fact, computing $α$-approximate PNE in congestion games is PLS-hard. We demonstrate how our analysis can be applied to various different models of congestion games including general, step-function, and polynomial cost, as well as fair cost-sharing games (where the resource costs are decreasing). It is important to note that our bounds do not depend explicitly on the cardinality of the players' strategy sets, and thus the smoothed FPTAS is readily applicable to network congestion games as well.

A Smoothed FPTAS for Equilibria in Congestion Games

TL;DR

This work establishes a smoothed analysis-based FPTAS for computing approximate equilibria in congestion games by showing that, under independent -smooth perturbations of resource costs, a -improving dynamics converges to a -PNE in strongly polynomial expected time. The approach combines Rosenthal’s potential-based progress with a novel probabilistic lemma on order statistics of -smooth random variables, yielding tight bounds on the expected number of improving moves. The results cover general congestion games and several structured models—step-function, polynomial, and cost-sharing—and extend to network congestion scenarios without dependence on strategy-set cardinality. Together, the two core ideas provide a unified, tunable framework that bridges worst-case hardness and practical efficiency for approximate equilibria in large networked systems.

Abstract

We present a fully polynomial-time approximation scheme (FPTAS) for computing equilibria in congestion games, under smoothed running-time analysis. More precisely, we prove that if the resource costs of a congestion game are randomly perturbed by independent noises, whose density is at most , then any sequence of -improving dynamics will reach an -approximate pure Nash equilibrium (PNE) after an expected number of steps which is strongly polynomial in , , and the size of the game's description. Our results establish a sharp contrast to the traditional worst-case analysis setting, where it is known that better-response dynamics take exponentially long to converge to -approximate PNE, for any constant factor . As a matter of fact, computing -approximate PNE in congestion games is PLS-hard. We demonstrate how our analysis can be applied to various different models of congestion games including general, step-function, and polynomial cost, as well as fair cost-sharing games (where the resource costs are decreasing). It is important to note that our bounds do not depend explicitly on the cardinality of the players' strategy sets, and thus the smoothed FPTAS is readily applicable to network congestion games as well.
Paper Structure (12 sections, 2 theorems, 37 equations, 1 algorithm)

This paper contains 12 sections, 2 theorems, 37 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results and Techniques
  3. Model and Notation
  4. Congestion Games
  5. Smoothed Congestion Games
  6. The Smoothed FPTAS
  7. The Key Probabilistic Lemma
  8. General Congestion Games
  9. Special Congestion Game Models
  10. Step-Function Games
  11. Polynomial Games
  12. Cost-Sharing Games

Key Result

Theorem 1

In $\phi$-smooth congestion games, $(1+\varepsilon)$-BRD always find a $(1+\varepsilon)$-PNE, after an expected number of iterations which is strongly polynomial in $\frac{1}{\varepsilon}$, $\phi$, and the description of the game. This holds for congestion games, and even under a succinct network representation of all models (item:main-FPTAS-general)--(item:main-FPTAS-cost-sharing). More precisel

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 1
  • proof