Separating Feasibility and Movement in Solution Discovery: The Case of Path Discovery

TL;DR

This paper introduces a novel two-graph model that separates feasibility from movement in solution discovery problems, enabling a more flexible and realistic framework for path and shortest path discovery with complex constraints.

Contribution

The paper proposes a new directed weighted two-graph model that distinguishes feasibility from movement, capturing asymmetry and heterogeneity, and analyzes its computational complexity.

Findings

Discovery problems remain computationally hard even when the underlying optimization is easy.

The two-graph model captures complex constraints like asymmetry and weighted transitions.

The paper identifies both tractable and hard cases within the new framework.

Abstract

We study solution discovery, where the goal is to obtain a feasible solution to a problem from an initial configuration by a bounded sequence of local moves. In many applications, however, the graph that defines which vertex sets are feasible is not the same as the graph that governs how tokens, agents, or resources may move. Existing models such as token sliding and token jumping typically do not distinguish the problem graph and the movement graph. Motivated by this mismatch, we introduce a directed weighted two-graph model that cleanly separates feasibility from movement. A problem graph specifies the desired combinatorial objects, while a movement graph specifies admissible relocations and their costs. This yields a flexible framework that captures asymmetry, heterogeneous movement constraints, and weighted transitions, while subsuming classical discovery models as special cases. We investigate this model through \textsc{Path Discovery} and \textsc{Shortest Path Discovery}, where the task is to realize a vertex set containing an $s$-$t$-path or a shortest $s$-$t$-path in the problem graph. These problems are particularly natural in applications, since directed and weighted shortest paths are among the most fundamental algorithmic primitives. At the same time, previous work has already shown that discovery can be computationally hard even when the underlying optimization problem is easy. Our results show that this phenomenon persists, and becomes especially rich, in the two-graph setting. We obtain a detailed complexity picture, identifying tractable cases as well as strong hardness results.