Distance Preservation Games

TL;DR

This work introduces distance preservation games (DPGs), where agents declare ideal pairwise distances and place themselves on the unit interval to preserve these distances. It provides a comprehensive complexity landscape for two core objectives: jump stability (Nash-equilibrium-like profiles) and welfare optimality (maximizing total utility), revealing NP-hardness in general but tractable results for natural restrictions such as symmetric and acyclic DPGs. A greedy 1/2-approximation and special-case FPTAS/0.879-approximation algorithms are developed to approximate welfare, along with a universal PoA upper bound of 2 that links stability to efficiency. The findings identify both hard barriers and practical algorithms, offering a foundation for distance-aware placement models in continuous topologies and suggesting avenues for extension to higher dimensions and additional constraints.

Abstract

We introduce and analyze distance preservation games (DPGs). In DPGs, agents express ideal distances to other agents and need to choose locations in the unit interval while preserving their ideal distances as closely as possible. We analyze the existence and computation of location profiles that are jump stable (i.e., no agent can benefit by moving to another location) or welfare optimal for DPGs, respectively. Specifically, we prove that there are DPGs without jump stable location profiles and identify important cases where such outcomes always exist and can be computed efficiently. Similarly, we show that finding welfare optimal location profiles is NP-complete and present approximation algorithms for finding solutions with social welfare close to optimal. Finally, we prove that DPGs have a price of anarchy of at most $2$.

Figures (3)

• Figure 1: The preference graph of the DPG in Example 1
• Figure 2: The preference graph of the DPG of Example 2. The edges are bidirectional and colorcoded to ease readability. Blue edges indicate an ideal distance of $0$, red edges of $1$, and green edges of ${1}/{k}$.
• Figure 3: Preference graph of the DPG in the proof of \ref{['thm:POA']}

Theorems & Definitions (40)

• Example 1
• Proposition 1
• proof
• Theorem 1
• proof : Proof Sketch
• Theorem 2
• proof
• Theorem 3
• proof : Proof Sketch
• Example 2