Agent Arrangement Problem

TL;DR

This paper introduces the concept of arrangements and transfers in graph pairs, analyzing their properties and computational complexity, including an efficient algorithm for a decision problem and proving another problem is co-NP complete.

Contribution

It defines the notion of t-equivalence and the almighty property for graph pairs, providing an algorithm for t-equivalence and establishing the complexity of testing if a pair is almighty.

Findings

Developed an O(|E(G_A)| + (|V(G_M)| + |E(G_A)|)|V(G_A)|) algorithm for t-equivalence.

Proved that deciding if a pair of graphs is almighty is co-NP complete.

Abstract

An {\em arrangement} of an ordered pair $(G_A, G_M)$ of graphs is defined as a function $f$ from $V(G_A)$ to $V(G_M)$ such that, for each vertex $c$ of $G_M$, the vertex-set $f^{-1}(c)$ of $G_A$ either is $\emptyset$ (the case when $c \not\in f(V(G_A))$) or induces a connected subgraph of $G_A$ and that the family $\{f^{-1}(y) : y \in V(G_M), f^{-1}(y) \neq \emptyset\}$ is a partition of $V(G_A)$. Let $f$ be an arrangement of $(G_A, G_M)$, let $pq$ be an edge of $G_M$ and let $U$ be a subset of $f^{-1}(p)$ such that each of the three graphs $G_A[U]$, $G_A[f^{-1}(p)\setminus U]$ and $G_A[f^{-1}(q)\cup U]$ is ether connected or $\emptyset$ and that $\big(f^{-1}(p)\cup f^{-1}(q) \big) \setminus U \neq \emptyset$. A {\em transfer} of $U$ from $p$ to $q$ is defined as the modification $f^{\prime}$ of $f$ such that $f^{\prime}(x):=f(x)$ for every $ x \notin U$ and $f^{\prime}(u):=q$ for every $u \in U$. Two arrangements $f$ and $g$ of $(G_A, G_M)$ are called {\em t-equivalent} if they can be transformed into each other by a finite sequence of transfers. An ordered pair $(G_A, G_M)$ of graphs is called {\em almighty} if every two arrangements of the pair $(G_A, G_M)$ are t-equivalent. In this study, we consider the following two decision problems. [{\bf (P1)}]{For a given pair of arrangements $f$ and $g$ of a given ordered pair $(G_A,G_M)$ of graphs, decide whether $f$ is t-equivalent to $g$ or not.} [{\bf (P2)}]{For a given ordered pair $(G_A,G_M)$ of graphs, decide whether the pair $(G_A,G_M)$ is almighty or not.} We show an $\Od(|E(G_A)|+(|V(G_M)|+|E(G_A)|)|V(G_A)|)$-time algorithm for {\bf (P1)}, and prove the $\co\np$-completeness of {\bf (P2)}.