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Stability in Distance Preservation Games on Graphs

Argyrios Deligkas, Eduard Eiben, Tiger-Lily Goldsmith, Dušan Knop, Šimon Schierreich

TL;DR

A comprehensive study of the (parameterized) complexity of the problem in three different dimensions: the topology of the graph, the number of agents, and the structure of preferences of the agents.

Abstract

We introduce a new class of network allocation games called graphical distance preservation games. Here, we are given a graph, called a topology, and a set of agents that need to be allocated to its vertices. Moreover, every agent has an ideal (and possibly different) distance in which to be from some of the other agents. Given an allocation of agents, each one of them suffers a cost that is the sum of the differences from the ideal distance for each agent in their subset. The goal is to decide whether there is a stable allocation of the agents, i.e., no agent would like to deviate from their location. Specifically, we consider three different stability notions: envy-freeness, swap stability, and jump stability. We perform a comprehensive study of the (parameterized) complexity of the problem in three different dimensions: the topology of the graph, the number of agents, and the structure of preferences of the agents.

Stability in Distance Preservation Games on Graphs

TL;DR

A comprehensive study of the (parameterized) complexity of the problem in three different dimensions: the topology of the graph, the number of agents, and the structure of preferences of the agents.

Abstract

We introduce a new class of network allocation games called graphical distance preservation games. Here, we are given a graph, called a topology, and a set of agents that need to be allocated to its vertices. Moreover, every agent has an ideal (and possibly different) distance in which to be from some of the other agents. Given an allocation of agents, each one of them suffers a cost that is the sum of the differences from the ideal distance for each agent in their subset. The goal is to decide whether there is a stable allocation of the agents, i.e., no agent would like to deviate from their location. Specifically, we consider three different stability notions: envy-freeness, swap stability, and jump stability. We perform a comprehensive study of the (parameterized) complexity of the problem in three different dimensions: the topology of the graph, the number of agents, and the structure of preferences of the agents.
Paper Structure (13 sections, 24 theorems, 11 equations, 3 figures, 1 algorithm)

This paper contains 13 sections, 24 theorems, 11 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. The Model
  5. Envy-Freeness
  6. Restricted Topology
  7. Bounded Number of Agents
  8. Restricted Preferences
  9. Jump and Swap Stability
  10. Discussion
  11. "At most" and "at least" distances.
  12. Jump and Swap Stability.
  13. Maximize Social Welfare.

Key Result

Proposition 1

A distance preservation game $\Gamma = (A,(M_a)_{a\in A},(\operatorname{d}_a)_{a\in A},G)$ admits an envy-free and jump stable allocation $\pi$ if and only if the distance preservation game $\Gamma' = (A',(M'_a)_{a\in A'},(\operatorname{d}'_a)_{a\in A'},G)$, where $A' = A \cup \{ x_i \mid i \in [|V|

Figures (3)

  • Figure 1: An example of a topology with allocated agents (on the left) and a preference graph (on the right). Observe that $M_a = \{b,c\}$, $M_b = \{a\}$, and $M_c = \emptyset$, that is, agent $c$ is indifferent. The preferences are clearly not symmetric.
  • Figure 2: An illustration of the topology used in the proof of \ref{['thm:EF:pNPh:vc']}. Each set of vertices with yellow (light gray) background is of size $B=\frac{3N-k}{2}$ and the set $M$ with green (dark gray) background is of size $k$. Clearly, the vertices $c_1$ and $c_2$ form a vertex cover of this graph of size two.
  • Figure 3: An illustration of the construction used to prove \ref{['thm:EF:NPh:path']}. An arrow from an agent $a$ to an agent $b$ represents that $b\in M_a$.

Theorems & Definitions (60)

  • Definition 1
  • Definition 2
  • proof
  • Definition 3
  • Proposition 1
  • proof
  • Example 1
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 50 more