Finding Shortest Reconfiguration Sequences on Independent Set Polytopes

Abstract: We initiate the study of the shortest reconfiguration problem for independent sets under the adjacency relation derived from the independent set polytope. Given a graph and two independent sets, the problem asks for a shortest sequence transforming one into the other such that the subgraph induced by the symmetric difference of any two consecutive sets is connected. This is equivalent to finding a shortest path on the $1$-skeleton of the independent set polytope. We prove that the problem is NP-hard even on planar graphs of bounded degree, as well as on split graphs. Notably, the hardness for planar graphs of bounded degree still holds even when deciding whether the target can be reached in at most two steps. For split graphs, we further show the W[2]-hardness when parameterized by the number of steps, as well as the inapproximability of the optimal length. As a consequence, we prove that the length of a shortest path between two vertices of a 0/1 polytope in $\mathbb{R}^n$ described by $O(n)$ linear inequalities is hard to approximate within a factor of $(1-\varepsilon)\ln n$ for any constant $ε>0$, unless $P=NP$. On the positive side, we provide polynomial-time algorithms for block graphs, cographs, and bipartite chain graphs. Moreover, for paths and cycles, we show that the optimal length of the shortest reconfiguration sequence exactly matches a trivial upper bound.

• The paper establishes NP-completeness and inapproximability of the reconfiguration problem under polyhedral adjacency constraints.
• It presents efficient polynomial-time algorithms for special graph classes like block graphs, cographs, bipartite chain graphs, and trees.
• It links polyhedral structure to practical applications in multi-agent systems and highlights limitations in simplex-like pivot algorithms.

Shortest Reconfiguration Sequences on Independent Set Polytopes: Complexity and Algorithms

Problem Formulation and Polyhedral Context

The paper "Finding Shortest Reconfiguration Sequences on Independent Set Polytopes" (2604.24132) initiates the formal investigation of the shortest reconfiguration problem for independent sets under a polyhedral adjacency relation. Unlike standard reconfiguration models (TJ, TS, TAR), adjacency here is induced by the 1-skeleton of the independent set polytope: two independent sets $I$ and $J$ are adjacent if $G[I \Delta J]$ is connected, yielding a shortest path problem on the polytope's graph. This rule generalizes previous models and is motivated by both combinatorial geometry and applications in multi-agent systems.

The adjacency characterization rests on Chvátal's theorem: adjacency in the independent set polytope is equivalent to connectedness of the symmetric difference induced subgraph. Consequently, the reconfiguration graph embodies the combinatorial structure of the polytope, bringing the complexity of polyhedral navigation and simplex-like optimization into focus.

Hardness Results and Complexity Barriers

The authors establish NP-completeness for the SYMISR problem on planar graphs with bounded degree and degeneracy 2, even for sequences of length at most 2. This provides a sharp complexity threshold—reachability is trivial, yet shortest sequence computation is computationally resistant. Furthermore, SYMISR is shown NP-hard for split graphs, yielding W[2]-hardness parameterized by sequence length and robust inapproximability: the shortest reconfiguration sequence cannot be approximated within $(1-\epsilon)\log n$ for any constant $\epsilon > 0$ unless $P=NP$.

These results extend to the polyhedral context: for a 0/1 polytope in $\mathbb{R}^n$ described by $O(n)$ inequalities, shortest path computation in the skeleton admits no polynomial-time approximation within logarithmic factors. Additionally, for monotone paths (those strictly improving a linear objective), the same inapproximability lower bounds are derived, strengthening prior results for matching and hypergraphic polytopes.

Strong numerical results include:

• NP-completeness on planar graphs for $k=2$
• Inapproximability within $(1-\epsilon)\log n$ for split graph polytopes
• W[2]-hardness for parameterized sequence length

These contradict any prospect of PTAS or XP algorithms for the shortest reconfiguration sequence in these restricted classes.

Positive Results: Exact and Efficient Algorithms

In contrast, several graph classes admit polynomial-time algorithms:

• Block graphs: Using decomposition into rooted block graphs and dynamic programming over synchronized flips, the shortest sequence can be computed in $J$0 time.
• Cographs: By exploiting the recursive structure (disjoint union, join), the problem reduces to constant-length sequences (at most 2), yielding linear-time solvability.
• Bipartite chain graphs: The structure permits $J$1 algorithms based on chain orderings and connectivity conditions.
• Trees: As a subclass of block graphs, trees admit $J$2 algorithms via optimized DP tracking flips at root cliques.

For paths and cycles, the trivial upper bound (number of connected components in $J$3) matches the optimum. Thus, the problem is linear-time solvable and achieves tight bounds.

Implications: Polyhedral Algorithms, Reconfiguration, and Optimization

The polyhedral hardness results have significant implications:

• Simplex Method: No pivot rule can guarantee polynomial-time monotone paths within logarithmic approximation factors in polytopes with structured combinatorial formulations.
• Combinatorial Optimization: Shortest reconfiguration problems encapsulate barriers faced in pivot-based algorithms and polytope navigation.
• Multi-Agent Systems: The polyhedral adjacency rule models coordinated transformations beyond token-based single moves, relevant for MAS with global, connected state transitions.

Algorithmic tractability in special graph classes reveals deep structural connections: block graphs, cographs, and bipartite chain graphs are characterized by well-behaved polytopes or recursive graph operations, allowing efficient dynamic programming and decomposition.

Future Directions

Several open questions remain:

• Bounded-treewidth graphs: The complexity of SYMISR is unresolved here; prior work indicates reconfiguration hardness persists in restricted bandwidth/tree-depth settings.
• Approximation gaps: Tightness of trivial upper bounds versus optimality, especially outside paths and cycles.
• Extension of efficient algorithms: Broader classes such as bipartite or distance-hereditary graphs are promising for algorithmic advances.

Polyhedral reconfiguration intersects discrete geometry, algorithm design, and parameterized complexity, suggesting further research in polytope skeleton navigation, monotonicity constraints, and token-move abstractions.

Conclusion

This work rigorously characterizes the complexity landscape of shortest independent set reconfiguration under polyhedral adjacency. It delineates sharp NP-completeness and inapproximability boundaries in both sparse and dense cases, establishes polynomial-time solvability for several perfect graph subclasses, and connects polyhedral structure to algorithmic tractability. The implications extend to pivot algorithms, combinatorial optimization, and multi-agent transformations. Continuing exploration of bounded-width classes, tightness of bounds, and polyhedral algorithms promises further advances in combinatorial reconfiguration and polytope theory.