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

2026-04-30

AI Summary

This work addresses the modeling limitations in pathfinding tasks arising from the tight coupling between problem graphs and movement graphs. To overcome this, the paper introduces a directed, weighted bipartite graph model that explicitly decouples feasibility—defined by the problem graph—from movement rules—governed by the movement graph—for the first time. This formulation naturally accommodates asymmetry, heterogeneous constraints, and weighted transitions, thereby unifying and extending classical solution-discovery frameworks. Leveraging tools from combinatorial optimization, graph theory, and computational complexity, the study provides a complete characterization of the complexity landscape for both general pathfinding and shortest-path problems under the proposed model, precisely delineating polynomial-time solvable cases from those that are strongly NP-hard.

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.

Problem

• solution discovery
• feasibility
• movement graph
• problem graph
• path discovery

Innovation

• two-graph model
• solution discovery
• path discovery
• movement constraints
• computational complexity