On (In)approximability of MaxMin Independent Set Reconfiguration

Abstract: In the Independent Set Reconfiguration problem under the Token Addition/Removal rule, given a graph $G$ and two independent sets $I$ and $J$ of $G$, we want to transform $I$ into $J$ by adding and removing vertices, such that all the sets throughout the process are independent sets. Its approximate version called MaxMin Independent Set Reconfiguration aims to maximise the minimum size of the independent sets in the process above. We study the (in)approximability of this problem for general graphs as well as restricted graph classes. Firstly, on general graphs, we obtain a polynomial-time $(n / \log n)$-factor approximation algorithm, complementing the $\mathsf{PSPACE}$-hardness of $n^{Ω(1)}$-factor approximation due to Hirahara and Ohsaka [STOC 2024, ICALP 2024] and the $\mathsf{NP}$-hardness of $n^{1-\varepsilon}$-factor approximation due to Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno [TCS 2011]. Secondly, we present a polynomial-time approximation algorithm for degenerate graphs as well as $\mathsf{FPT}$-approximation schemes for bounded-treewidth graphs and $H$-minor-free graphs. Lastly, we extend the above inapproximability results to bounded-degree graphs, graphs of bandwidth $n^{\frac{1}{2}+Θ(1)}$, and bipartite graphs.

• The paper introduces the first nontrivial approximation algorithm for MMISR, achieving a polynomial-time (n/log n)-factor for general graphs.
• It develops specialized algorithms for k-degenerate, bounded-treewidth, and H-minor-free graphs, providing explicit reconfiguration sequences with proven guarantees.
• The study strengthens inapproximability bounds for MMISR across various graph classes, linking problem hardness to structural graph parameters and well-known complexity assumptions.

(In)approximability of MaxMin Independent Set Reconfiguration: Comprehensive Analysis

Problem Formulation and Context

The paper presents a thorough investigation into the approximability and inapproximability boundaries of the MaxMin Independent Set Reconfiguration (MMISR) problem under the Token Addition/Removal reconfiguration rule. MMISR asks, given two independent sets $I$ and $J$ of a graph $G$, for a sequence of intermediate independent sets transforming $I$ to $J$, maximizing the minimum size kept throughout the process. This problem targets both practical algorithmic challenges and theoretical complexity questions that arise in combinatorial reconfiguration frameworks.

Combinatorial reconfiguration, particularly for the independent set (IS) problem, has been recognized for its $\mathsf{PSPACE}$-completeness across several reconfiguration rules, even in planar and bounded bandwidth graphs. MMISR further differs from shortest-sequence reconfiguration approximability, which offers minimal leverage due to the underlying structural hardness of the original IS Reconfiguration (ISR) problem.

Algorithmic Results and Approximation Schemes

The paper delivers four salient algorithmic contributions:

1. General graph approximation: It establishes a polynomial-time $(n/\log n)$-factor approximation for MMISR, producing explicit $(I,J)$-reconfiguration sequences. This is the first nontrivial approximation algorithm for MMISR, matching the known $\mathsf{NP}$-hardness threshold for $n^{1-\epsilon}$-factor approximation and complementing $J$0 inapproximability from $J$1-hardness. The construction leverages auxiliary graphs partitioned by vertex subsets, and a pigeonhole argument guaranteeing a large enough intersection with the partition.
2. Degenerate graphs: For $J$2-degenerate graphs, a polynomial-time algorithm achieves an approximation factor of $J$3. The core technique incrementally expands the intersection between $J$4 and $J$5 using degree-aware local moves, with provable minimum size guarantees per step driven by degeneracy constraints.
3. Bounded-treewidth and $J$6-minor-free graphs: For graphs of treewidth at most $J$7, a polynomial-time algorithm provides an approximation of $J$8 where $J$9 is the input independent set size. Further, for fixed treewidth and $G$0-minor-free graphs, FPT-approximation schemes (FPT-AS) are constructed, achieving $G$1-factor in FPT time, benefitting from balanced separators and structural decompositions (Baker’s technique extended to minors).
4. Algorithmic outputs: Notably, all algorithms return explicit reconfiguration sequences, not just minimum sizes, which ensures practical applicability for system updating scenarios.

Hardness and Inapproximability Landscapes

The paper strengthens known inapproximability boundaries for MMISR across classes:

• Bounded-degree graphs: MMISR on graphs of maximum degree $G$2 is $G$3-hard to approximate within a factor of $G$4, which quantitatively improves over previously established $G$5-hardness for $G$6-factor approximation.
• Bandwidth and treewidth: MMISR on graphs of bandwidth $G$7 ($G$8) is $G$9-hard to approximate within a factor of $I$0, showing parameterization by bandwidth/treewidth (when superconstant) does not admit poly-factor approximation unless $I$1.
• Bipartite graphs: MMISR on bipartite graphs inherits general graph hardness, cannot be approximated within a $I$2 factor (for any $I$3), assuming the Small Set Expansion Hypothesis (SSEH) and $I$4. This result ties MMISR hardness in bipartite graphs directly to the known barriers for balanced biclique approximation.

The reductions rely on gap-preserving constructions mapping instances of MIS or Balanced Biclique to MMISR, utilizing spectral properties (Ramanujan graphs for bounded-degree inapproximability), clique and biclique gadgets for bandwidth, and structural equivalence arguments for bipartite classes.

Theoretical Implications and Open Directions

The analysis delineates a landscape where known approximation factors for MMISR are tightly bounded between the degeneracy and its square root for bounded-degree/degeneracy, and where superconstant treewidth or bandwidth render polynomial-time approximation infeasible. The paper highlights the obstacle where static IS approximation methods do not extend to MMISR due to the requirement of maintaining large sets throughout the entire reconfiguration sequence, not just at endpoints.

Implications include:

• Algorithmic design: Future algorithmic strategies must be reconfiguration-aware, possibly integrating sequence-level optimization rather than leveraging endpoint-IS approximation alone.
• Parameterized complexity: MMISR admits FPT-AS for fixed treewidth and $I$5-minor-free cases, encouraging parameterized and graph structural techniques, but superconstant parameters quickly break tractability.
• Complexity theory: The results underscore the limits of approximation for reconfiguration problems, and tie MMISR hardness to some of the strongest complexity assumptions (e.g., $I$6-completeness, SSEH).

Notable directions for further research involve seeking (E)PTAS algorithms for MMISR on superconstant treewidth ($I$7), and refining algorithms for special sparse classes (e.g., nowhere dense).

Conclusion

This paper advances the theoretical framework for MMISR with new approximation algorithms, sharp inapproximability thresholds for multiple graph parameters, and explores the interface between combinatorial reconfiguration, graph structural complexity, and algorithmic design. It provides both constructive and hardness results, clarifies parameterized and structural boundaries, and sets the stage for further study in reconfiguration algorithms dependent on sequence-wide constraints and specialized graph classes.

Citation: "On (In)approximability of MaxMin Independent Set Reconfiguration" (2604.26714)