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Individual Rationality in Topological Distance Games is Surprisingly Hard

Argyrios Deligkas, Eduard Eiben, Dušan Knop, Šimon Schierreich

TL;DR

This work analyzes the complexity of finding individually rational outcomes in topological distance games (TDGs), where agents are placed on a topology and their utilities depend on distances to others. It provides a comprehensive complexity landscape: IR-TDG is NP-hard in unrestricted settings, with a sharp dichotomy showing polynomial-time solvability when at most one agent has enemies, while hardness persists under multiple enmity arcs and even on path topologies. The study then maps the parameterized boundary: an XP algorithm exists for a fixed number of agents, but IR-TDG is W[1]-hard with respect to the number of agents in general; however, fixed-parameter tractability is achieved under certain structural restrictions, including bounded distance factor together with twin-width or shrub-depth of the topology, and, in some cases, path topologies with constrained enmity. ETH-based lower bounds further cement that substantial improvements over these bounds are unlikely in general. Overall, the results delineate concrete tractable regimes for IR in TDGs and highlight where efficient algorithms are achievable or provably unlikely.

Abstract

In the recently introduced topological distance games, strategic agents need to be assigned to a subset of vertices of a topology. In the assignment, the utility of an agent depends on both the agent's inherent utilities for other agents and its distance from them on the topology. We study the computational complexity of finding individually rational outcomes; this notion is widely assumed to be the very minimal stability requirement and requires that the utility of every agent in a solution is non-negative. We perform a comprehensive study of the problem's complexity, and we prove that even in very basic cases, deciding whether an individually rational solution exists is intractable. To reach at least some tractability, one needs to combine multiple restrictions of the input instance, including the number of agents and the topology and the influence of distant agents on the utility.

Individual Rationality in Topological Distance Games is Surprisingly Hard

TL;DR

This work analyzes the complexity of finding individually rational outcomes in topological distance games (TDGs), where agents are placed on a topology and their utilities depend on distances to others. It provides a comprehensive complexity landscape: IR-TDG is NP-hard in unrestricted settings, with a sharp dichotomy showing polynomial-time solvability when at most one agent has enemies, while hardness persists under multiple enmity arcs and even on path topologies. The study then maps the parameterized boundary: an XP algorithm exists for a fixed number of agents, but IR-TDG is W[1]-hard with respect to the number of agents in general; however, fixed-parameter tractability is achieved under certain structural restrictions, including bounded distance factor together with twin-width or shrub-depth of the topology, and, in some cases, path topologies with constrained enmity. ETH-based lower bounds further cement that substantial improvements over these bounds are unlikely in general. Overall, the results delineate concrete tractable regimes for IR in TDGs and highlight where efficient algorithms are achievable or provably unlikely.

Abstract

In the recently introduced topological distance games, strategic agents need to be assigned to a subset of vertices of a topology. In the assignment, the utility of an agent depends on both the agent's inherent utilities for other agents and its distance from them on the topology. We study the computational complexity of finding individually rational outcomes; this notion is widely assumed to be the very minimal stability requirement and requires that the utility of every agent in a solution is non-negative. We perform a comprehensive study of the problem's complexity, and we prove that even in very basic cases, deciding whether an individually rational solution exists is intractable. To reach at least some tractability, one needs to combine multiple restrictions of the input instance, including the number of agents and the topology and the influence of distant agents on the utility.
Paper Structure (11 sections, 15 theorems, 22 equations, 3 figures)

This paper contains 11 sections, 15 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Preliminaries
  5. Graph Theory
  6. Topological Distance Games
  7. Parameterized Complexity
  8. Exponential-Time Hypothesis.
  9. Unrestricted number of agents
  10. Parameter-Many Agents
  11. Conclusions

Key Result

Theorem 3.1

For every distance factor function $f$, it is NP-complete to decide the IR-Topological Distance Game problem even if the utilities are symmetric and every agent uses at most $2$ different utility values.

Figures (3)

  • Figure 1: A simplified overview of our results when the number of agents is part of the input. By const. comps. we mean that the topology is disconnected and consists of constant-size components.
  • Figure 2: The topology used in Theorem \ref{['thm:two_arcs_same']}.
  • Figure 3: A simplified overview of our results for the setting with parameter-many agents. All W[1]-complete combinations can be solved by an XP algorithm, which is asymptotically optimal under ETH (see \ref{['thm:XP_agents']}). We use $\operatorname{tww}$ to highlight that the result additionally requires the topology of bounded twin-width, and similarly $\operatorname{sd}$ represents topologies with bounded shrub-depth.

Theorems & Definitions (37)

  • Definition 1
  • proof
  • Theorem 3.1
  • proof
  • Claim 1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Claim 2
  • ...and 27 more