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

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.

• The paper introduces a novel two-graph model that distinguishes between feasibility and movement, enabling a more realistic approach to path discovery.
• The paper applies fixed-parameter tractability analyses and employs techniques like token jumping and layered auxiliary digraphs to handle diverse movement constraints.
• The paper reveals NP-hard and parameterized intractability results for cases like planar graphs and bounded pathwidth, emphasizing computational challenges in dynamic environments.

Separation of Feasibility and Movement: A Two-Graph Model for Solution Discovery

Motivation and Framework

The paper "Separating Feasibility and Movement in Solution Discovery: The Case of Path Discovery" (2604.27802) introduces a formal distinction between feasibility and movement in solution discovery via a directed, weighted two-graph model. Classical solution discovery models, such as token sliding and token jumping, conflate the graphs representing feasible solution spaces and those governing movement dynamics. However, many practical applications require a separation; for example, the feasibility of radio coverage in an area is constrained by geographic topology, whereas agent movement is restricted by the logistical infrastructure (e.g., roads). To address this, the authors define a problem graph $G$ specifying feasible combinatorial objects, and a movement graph $M$ specifying admissible relocations and associated costs. The initial configuration and a movement budget define the starting point and constraints. This abstraction enables asymmetry, heterogeneous transition costs, and more realistic modeling of practical systems, while subsuming traditional models as special cases.

Core Problems and Methodology

Two canonical problems are considered:

• Path Discovery: Identify a configuration containing a directed $s$-$t$ path in the problem graph, reachable from the initial configuration within the movement budget.
• Shortest Path Discovery: Identify such a configuration containing a shortest $s$-$t$ path.

The separation is crucial, as even in scenarios where the underlying optimization problem (finding a shortest path) is tractable in $G$, the corresponding discovery problem is often computationally hard due to movement constraints. The framework generalizes traditional token-sliding models (where $M=G$) and token-jumping (where $M$ is fully connected with unit costs). The authors systematically analyze parameterized complexity and exact algorithms for these problems.

Algorithmic Results

Fixed-Parameter Tractability

Several tractable cases are established:

• Token Jumping: When tokens may move freely, Path Discovery is solvable in polynomial time, since the order and identity of tokens are irrelevant and movement costs are uniform.
• Number of Tokens: Both Path Discovery and Shortest Path Discovery are fixed-parameter tractable (FPT) parameterized by the number of tokens $k$. The authors employ layered auxiliary digraphs and color-coding [AlonYusterZwick1995] to efficiently enumerate candidate paths and token assignments within $M$0.
• Bounded Solution Size: FPT algorithms are shown for parameters bounding the number of vertices on all simple $M$1-$M$2 paths, such as vertex integrity and treedepth of the underlying undirected graph, as well as for the distance from $M$3 to $M$4 in the positive-weight setting.
• Feedback Edge Set: If the feedback edge set number is small, the set of possible simple $M$5-$M$6 paths is bounded. Enumerative signature-based constructions and bipartite matching efficiently yield FPT algorithms.
• Treewidth: On graphs where the undirected union of $M$7 and $M$8 has bounded treewidth $M$9, the problems are in XP, solvable in $s$0 via dynamic programming on tree decompositions.

Hardness Results

The model admits significant intractability:

• Planar Graphs: Discovery remains NP-hard on planar graphs, even in the classical undirected token-sliding setting, via reductions from Circulating Orientation.
• Pathwidth and Cutwidth: The problems are XNLP-hard (and thus W[i]-hard for all $s$1) when parameterized by pathwidth or cutwidth, remaining hard for bounded-width graph classes.
• Directed Feedback Vertex Set: In directed two-graph models, parameterization by the directed feedback vertex set yields para-NP-hardness.
• Diameter and Budget: Allowing zero-weight edges in $s$2 or $s$3 creates para-NP-hard instances when parameterized by diameter or budget, respectively. Specifically, Shortest Path Discovery becomes hard for budget $s$4 if zero-weight movement edges are enabled.
• Treewidth Barrier: Since cutwidth bounds treewidth, the obtained XP algorithms for treewidth cannot be improved to FPT under standard complexity assumptions.

Implications and Discussion

The two-graph framework illuminates a critical distinction between solution feasibility and physical dynamics in combinatorial settings, bridging structural parameterized complexity and dynamic adaptation paradigms. The model captures asymmetric, heterogeneous, and weighted movement constraints, which are fundamental in applications such as robotics, logistics, and network repair.

Algorithmic tractability hinges on the interplay between movement restrictions and path structure. Parameters bounding path length or cycle complexity carve out polynomial and FPT regimes. Conversely, structural parameters like treewidth and cutwidth delineate regimes of intrinsic computational difficulty, even when classical solution construction is easy.

The theoretical implications extend to the study of other combinatorial discovery and reconfiguration problems, such as vertex cover, domination, matching, cut, and clustering, under realistic movement schemes. Practically, the results guide the design of adaptive algorithms for incremental deployment, resource reallocation, and repair in networked systems, where feasibility and movement must be treated independently.

Future Directions

Immediate avenues include extending the model to more diverse combinatorial objects, analyzing the threshold between tractable and intractable parameterizations, and generalizing to the classical reconfiguration setting. Further research can focus on kernelization, approximation, and real-time discovery under movement constraints. Investigations into algorithmic meta-theorems for solution discovery may uncover deeper structural relationships governing tractability. Empirical studies could validate the framework's applicability to real-world deployment and adaptation problems in AI-driven systems.

Conclusion

This work proposes a flexible, expressive two-graph model separating feasibility and movement in solution discovery. Detailed parameterized analyses reveal a nuanced landscape: tractability in several structural cases, contrasted with strong, broad-reaching hardness results for canonical parameters. The separation is not merely an abstraction; it enables modeling of realistic constraints and initiates a new direction in understanding dynamic solution discovery under heterogeneous movement protocols, with significant implications for combinatorial optimization and adaptive AI.