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The Complexity of Optimizing Atomic Congestion

Cornelius Brand, Robert Ganian, Subrahmanyam Kalyanasundaram, Fionn Mc Inerney

TL;DR

This work identifies the exact boundaries of tractability for the problem through the lens of the parameterized complexity paradigm, and obtains a set of results which demonstrate that the structural parameters which control the computational (in)tractability of the problem are not vertex-separator based in nature, but rather based on edge separators.

Abstract

Atomic congestion games are a classic topic in network design, routing, and algorithmic game theory, and are capable of modeling congestion and flow optimization tasks in various application areas. While both the price of anarchy for such games as well as the computational complexity of computing their Nash equilibria are by now well-understood, the computational complexity of computing a system-optimal set of strategies -- that is, a centrally planned routing that minimizes the average cost of agents -- is severely understudied in the literature. We close this gap by identifying the exact boundaries of tractability for the problem through the lens of the parameterized complexity paradigm. After showing that the problem remains highly intractable even on extremely simple networks, we obtain a set of results which demonstrate that the structural parameters which control the computational (in)tractability of the problem are not vertex-separator based in nature (such as, e.g., treewidth), but rather based on edge separators. We conclude by extending our analysis towards the (even more challenging) min-max variant of the problem.

The Complexity of Optimizing Atomic Congestion

TL;DR

This work identifies the exact boundaries of tractability for the problem through the lens of the parameterized complexity paradigm, and obtains a set of results which demonstrate that the structural parameters which control the computational (in)tractability of the problem are not vertex-separator based in nature, but rather based on edge separators.

Abstract

Atomic congestion games are a classic topic in network design, routing, and algorithmic game theory, and are capable of modeling congestion and flow optimization tasks in various application areas. While both the price of anarchy for such games as well as the computational complexity of computing their Nash equilibria are by now well-understood, the computational complexity of computing a system-optimal set of strategies -- that is, a centrally planned routing that minimizes the average cost of agents -- is severely understudied in the literature. We close this gap by identifying the exact boundaries of tractability for the problem through the lens of the parameterized complexity paradigm. After showing that the problem remains highly intractable even on extremely simple networks, we obtain a set of results which demonstrate that the structural parameters which control the computational (in)tractability of the problem are not vertex-separator based in nature (such as, e.g., treewidth), but rather based on edge separators. We conclude by extending our analysis towards the (even more challenging) min-max variant of the problem.
Paper Structure (11 sections, 10 theorems, 3 equations, 4 figures)

This paper contains 11 sections, 10 theorems, 3 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our Contributions.
  3. Preliminaries
  4. Formal Problem Definition.
  5. Parameterized Complexity Theory.
  6. Structural Parameters.
  7. The Surprising Difficulty of Solving Atomic Congestion Games
  8. A Fixed-Parameter Algorithm for SOAC
  9. The Min-Max Atomic Congestion Problem
  10. Concluding Remarks
  11. Acknowledgements

Key Result

Lemma 1

Every graph $G$ with slim treecut width $k$ admits a spanning tree $T$ over some supergraph $G'$ of $G$ such that $(G',T)$ has edge-cut width at most $3(k+1)^2$. Moreover, such a pair $(G',T)$ can be computed in time $2^{k^{{\mathcal{O}}(1)}}\cdot |V(G)|^4$.

Figures (4)

  • Figure 1: A mind map of our results on computing system-optimal strategies in congestion games. The formal problem definition as well as a discussion of the considered parameters is provided in Section \ref{['sec:prelims']}; here, tw stands for treewidth, deg stands for maximum degree, td stands for treedepth, (s)tcw stands for (slim) treecut width, and DAGs stands for directed acyclic graphs.
  • Figure 2: A pictorial view of the tractability of SOAC when capacities are bounded. An arc from a parameter $x$ to a parameter $y$ indicates that $x$ is dominated by $y$, i.e., a bound on $x$ implies a bound on $y$ but the opposite does not hold.
  • Figure 3: The arc gadget replacing each undirected edge $uv$ in the reductions from the proofs of Thms. \ref{['hard-td']} and \ref{['W1-tcw']} (left), and the digraph constructed in the proof of Thm. \ref{['hard-dags']} (right).
  • Figure 4: The digraph constructed in the proof of Thm. \ref{['hard-tw-delta']}.

Theorems & Definitions (19)

  • Definition 1
  • Lemma 1: Prop. 27 and Thm. 30, GanianK22
  • Lemma 2: Prop. 22, Prop. 26, and Thm. 30, GanianK22
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 9 more