MaxMin Independent Set Reconfiguration

• The paper introduces MMISR as an optimization problem that seeks to maximize the minimum size of intermediate independent sets under the TAR rule, proving both NP- and PSPACE-hardness in various graph classes.
• Methodologies involve block partitioning, degeneracy-based elimination, and FPT-approximation schemes on bounded-treewidth and H-minor-free graphs to efficiently construct reconfiguration sequences.
• Results bridge MMISR with fixed-cardinality models like Token Jumping and Token Sliding, elucidating their structural equivalence and informing both hardness reductions and approximation strategies.

MaxMin Independent Set Reconfiguration is the optimization version of Independent Set Reconfiguration under the Token Addition/Removal rule: given a graph $G$ and two independent sets $I,J$, one seeks a reconfiguration sequence from $I$ to $J$ that maximizes the minimum size of an intermediate independent set (Hoang et al., 29 Apr 2026). Closely related formulations appear as TAR-reachability with a threshold $k$, where every intermediate independent set must have size at least $k$ (Bonsma, 2014), and as normalized maxmin objectives of the form $\min_i |I^{(i)}|/(\alpha(G)-1)$ used in gap-preserving hardness reductions (Ohsaka, 2022). The topic now has a two-sided theory: strong NP- and PSPACE-hardness of approximation on general and sparse graph classes coexist with the first non-trivial approximation algorithms on general graphs, polynomial-time approximation on degenerate graphs, and FPT-approximation schemes on bounded-treewidth and $H$-minor-free graphs (Hoang et al., 29 Apr 2026, Hirahara et al., 2023).

1. Formal problem and equivalent threshold views

Under TAR, a reconfiguration sequence from $I$ to $J$ is a sequence

$I,J$0

such that $I,J$1, $I,J$2, each $I,J$3 is an independent set, and each step adds or removes vertices. In the contemporary MMISR formulation, the value of a sequence is

$I,J$4

and the optimization target is

$I,J$5

(Hoang et al., 29 Apr 2026). The same paper notes that sequence length is not part of the objective, so steps that add or remove multiple vertices can be refined into unit TAR steps without changing the minimum size guarantee.

Earlier TAR work framed the same phenomenon as a threshold reachability problem. For $I,J$6, TAR-Reachability asks whether two independent sets $I,J$7 can be transformed into one another by adding or removing one vertex at a time while maintaining $I,J$8 throughout (Bonsma, 2014). In that language, MMISR asks for the largest threshold $I,J$9 for which $I$0 and $I$1 remain connected in the thresholded TAR graph.

A convenient structural notation comes from the TAR graph of independent sets. For independence as a robust $I$2-set parameter, $I$3 denotes the full TAR graph, and $I$4 its induced subgraph on independent sets of size at least $I$5 (Curtis et al., 2024). Thus, deciding whether $I$6 is exactly a connectivity question in $I$7.

A related one-source optimization problem, Opt-ISR, asks for a maximum independent set reachable from a given $I$8 while maintaining a lower bound $I$9 throughout (Ito et al., 2018). This is not the same as MMISR, because MMISR has prescribed endpoints $J$0 and $J$1; however, both problems optimize over TAR-feasible trajectories under a minimum-size constraint. Some hardness reductions also normalize the objective by $J$2,

$J$3

which reflects the usual TAR threshold between maximum independent sets (Ohsaka, 2022). This suggests that normalized and unnormalized formulations induce the same ordering of paths within a fixed instance.

2. Relation to TJ, TS, and other reconfiguration models

MMISR is usually studied under TAR, where cardinality may vary subject to a lower bound. By contrast, Token Jumping and Token Sliding are fixed-cardinality models. In the $J$4-TJ graph, vertices are independent sets of size $J$5, and an edge exchanges exactly one vertex; in TS, the exchanged vertices must also be adjacent in the base graph (Curtis et al., 2024, Bartier et al., 2024). For independent sets as a sub $J$6-set parameter, there is a precise equivalence: two size-$J$7 independent sets are connected in the $J$8-TJ graph iff they are connected in $J$9 (Curtis et al., 2024). This makes TJ a fixed-cardinality shadow of threshold TAR.

This distinction matters algorithmically. In $k$0-free graphs, Token Sliding and Token Jumping are PSPACE-complete unless $k$1 is a path, the claw, or a subdivision of the claw; under TS, fork-free graphs admit a polynomial-time algorithm, and TS on maximum independent sets in fork-free graphs is also polynomial-time (Bartier et al., 2024). These results concern fixed-size reconfiguration, not the maxmin TAR objective.

A further generalization replaces single-token moves by generalized token jumping. For Independent Set, any two size-$k$2 independent sets in a connected graph can be reconfigured under $k$3-Token Jumping in length $k$4 and time $k$5, whereas $k$6-Token Jumping is PSPACE-complete even on split graphs (Křišťan et al., 2024). Here all configurations have size exactly $k$7, so this is a strict fixed-cardinality regime. A common misconception is to identify such universal reachability results with TAR-based MMISR; they address a different feasibility space.

3. Hardness and inapproximability

The approximation landscape begins with reductions from Maximum Independent Set. MMISR is NP-hard to approximate within a factor $k$8 for any $k$9 on $k$0-vertex graphs, and it is also PSPACE-hard to approximate within factor $k$1 on general graphs (Hoang et al., 29 Apr 2026). On bounded-degree graphs, Ohsaka’s gap-preserving framework and the later PCRP-based proof imply PSPACE-hardness of approximating Maxmin Independent Set Reconfiguration on bounded-degree graphs within a factor $k$2 for some universal $k$3 (Hirahara et al., 2023).

The hardness framework passes through Maxmin CSP Reconfiguration. In the TAR model, Maxmin Independent Set Reconfiguration is explicitly defined in normalized form by

$k$4

and a gap-preserving reduction from Gap Nondeterministic Constraint Logic to Gap Independent Set Reconfiguration yields constant-factor PSPACE-hardness under the Reconfiguration Inapproximability Hypothesis (Ohsaka, 2022). The later PCRP characterization of PSPACE confirms RIH and turns this conditional picture into an unconditional one under $k$5 (Hirahara et al., 2023).

A further refinement comes from alphabet reduction. The paper on alphabet reduction for reconfiguration problems proves that there exist universal constants $k$6 and $k$7 such that arbitrary-gap $k$8-CSP reconfiguration reduces to a $k$9 vs. $\min_i |I^{(i)}|/(\alpha(G)-1)$0 gap version of Binary CSP Reconfiguration with alphabet size $\min_i |I^{(i)}|/(\alpha(G)-1)$1, and then to Independent Set Reconfiguration via earlier gap-preserving reductions (Ohsaka, 2024). The same source states that optimization versions of Independent Set Reconfiguration are PSPACE-hard to approximate within a factor $\min_i |I^{(i)}|/(\alpha(G)-1)$2 under RIH; combined with the PCRP result, this supplies a uniform constant-gap hardness template (Hirahara et al., 2023, Ohsaka, 2024).

The 2026 approximability paper extends this landscape to several restricted classes. Its explicit hardness statements are summarized below (Hoang et al., 29 Apr 2026).

Graph class	Hardness statement	Assumption
General $\min_i |I^{(i)}|/(\alpha(G)-1)$3-vertex graphs	NP-hard within $\min_i |I^{(i)}|/(\alpha(G)-1)$4; PSPACE-hard within $\min_i |I^{(i)}|/(\alpha(G)-1)$5	$\min_i |I^{(i)}|/(\alpha(G)-1)$6, $\min_i |I^{(i)}|/(\alpha(G)-1)$7
Maximum degree $\min_i |I^{(i)}|/(\alpha(G)-1)$8	NP-hard within $\min_i |I^{(i)}|/(\alpha(G)-1)$9	randomized reductions
Bandwidth $H$0	NP-hard within $H$1	$H$2
Bipartite graphs	no $H$3-approximation	SSEH and $H$4

4. Approximation and exact algorithms

The first non-trivial approximation algorithm on general graphs achieves a polynomial-time $H$5-factor approximation (Hoang et al., 29 Apr 2026). The core device is a $H$6-sequence: a sequence of independent sets $H$7 with $H$8, $H$9, $I$0, and $I$1 independent for all $I$2. Partitioning $I$3 into $I$4 blocks yields a $I$5-sequence with

$I$6

and an auxiliary graph on block-contained independent sets makes the construction polynomial-time.

For $I$7-degenerate graphs, there is a polynomial-time algorithm producing a reconfiguration sequence $I$8 such that

$I$9

(Hoang et al., 29 Apr 2026). The algorithm maintains three sets $J$0, where $J$1 accumulates aligned vertices and $J$2, $J$3 track unresolved parts. Each iteration uses a low-degree vertex in the bipartite graph $J$4 to move one vertex into $J$5 while deleting at most $J$6 conflicting vertices.

On bounded-treewidth graphs, the same paper gives both a polynomial-time additive guarantee and an FPT-approximation scheme (Hoang et al., 29 Apr 2026). If $J$7 and $J$8, then given a tree decomposition of width $J$9, one can compute a sequence with

$I,J$00

Moreover, for every $I,J$01, there is an algorithm running in time $I,J$02 that achieves

$I,J$03

The extension to $I,J$04-minor-free graphs uses the Demaine–Hajiaghayi–Kawarabayashi partition theorem and yields an FPT-AS parameterized by $I,J$05 (Hoang et al., 29 Apr 2026).

Adjacent exact results clarify the boundary between optimization and decision. On chordal graphs, Opt-ISR is solvable in linear time: if $I,J$06 is a maximal independent set of size $I,J$07, it is the unique optimum; otherwise any maximum independent set is reachable under the $I,J$08-rule (Ito et al., 2018). On cographs, TAR-Reachability is decidable in $I,J$09, and when a threshold-$I,J$10 sequence exists, the length of a shortest reconfiguration sequence is at most $I,J$11 (Bonsma, 2014). In the TAR-graph language, even-hole-free graphs satisfy a particularly clean threshold statement: $I,J$12 is connected (Curtis et al., 2024).

The main algorithmic guarantees are succinctly summarized below [(Hoang et al., 29 Apr 2026); (Ito et al., 2018); (Bonsma, 2014)].

Setting	Guarantee
General graphs	polynomial-time $I,J$13-approximation
$I,J$14-degenerate graphs	$I,J$15
Treewidth $I,J$16	additive loss $I,J$17; also FPT-AS
$I,J$18-minor-free graphs	FPT-AS
Chordal Opt-ISR	linear-time exact algorithm
Cographs, TAR-Reachability	$I,J$19 decision; shortest sequence length $I,J$20

5. Methodological foundations

Two methodological strands dominate the subject. The algorithmic strand is structural. General-graph approximation uses block partitioning and an auxiliary compatibility graph on local independent sets (Hoang et al., 29 Apr 2026). Degeneracy approximation exploits low-degree elimination inside $I,J$21 (Hoang et al., 29 Apr 2026). Treewidth algorithms use $I,J$22-balanced separators and recurse on separated subinstances; the $I,J$23-minor-free approximation scheme then layers Baker-type decomposition over the treewidth scheme (Hoang et al., 29 Apr 2026). The FPT exact subroutine for bounded treewidth relies on an equivalence with Token Jumping on equal-size sets and on a recent FPT algorithm for ISR-TJ on $I,J$24-degenerate graphs, as recorded in the 2026 paper (Hoang et al., 29 Apr 2026).

The hardness strand is PCP-like. “Probabilistically Checkable Reconfiguration Proofs” encode PSPACE computations as exponentially long paths of locally checkable proofs whose adjacent configurations differ in at most one bit; the verifier uses $I,J$25 randomness and $I,J$26 queries (Hirahara et al., 2023). This yields a new characterization of PSPACE and a proof that Gap$I,J$27 $I,J$28-CSP Reconfiguration over alphabet size $I,J$29 is PSPACE-complete, from which constant-factor hardness for Maxmin Independent Set Reconfiguration on bounded-degree graphs follows via earlier reductions (Hirahara et al., 2023).

A second hardness ingredient is alphabet reduction. The reconfiguration analogue of Dinur-style alphabet reduction replaces large alphabets by a universal alphabet size $I,J$30 while preserving perfect completeness and degrading the soundness gap only by a constant factor (Ohsaka, 2024). Its central combinatorial input is a reconfigurability property of Hadamard codes: one can move between two codewords while staying within distance at most $I,J$31 of one endpoint codeword and farther than $I,J$32 from every other codeword, with $I,J$33 (Ohsaka, 2024). Combined with gap amplification and earlier NCL-to-Independent-Set gadget reductions, this produces uniform constant-gap hardness for optimization versions of Independent Set Reconfiguration (Ohsaka, 2022, Ohsaka, 2024).

6. Open directions, boundaries, and related extremal questions

Several current frontiers are explicit. For MMISR itself, two open directions are emphasized in the 2026 study: whether there is an EPTAS for superconstant treewidth, for example $I,J$34, and how to close the gap between the $I,J$35 approximation guarantee on $I,J$36-degenerate graphs and the $I,J$37 inapproximability inherited from bounded-degree hardness (Hoang et al., 29 Apr 2026).

A separate but easily conflated line of work studies extremal reconfiguration distance rather than maxmin size. For fixed-cardinality TJ, let $I,J$38 be the maximum diameter of a connected component of the $I,J$39-configuration graph over all $I,J$40-vertex graphs. Then $I,J$41, $I,J$42 with lower bound $I,J$43, and more generally

$I,J$44

for fixed $I,J$45; for some linear $I,J$46, the diameter is exponential (Bousquet et al., 2023). This is a different extremal problem: it maximizes the shortest path length between reconfigurable size-$I,J$47 configurations, whereas MMISR maximizes the minimum cardinality along a TAR path. A plausible implication is that path quality and path length are largely orthogonal parameters.

The surrounding literature also shows that fixed-size and threshold formulations continue to interact in subtle ways. In the $I,J$48-set framework, TAR and TJ are equivalent up to a one-unit threshold shift (Curtis et al., 2024). In $I,J$49-free graphs, fixed-cardinality TS/TJ admit a near-complete PSPACE-vs-P classification, but extending such graph-class classifications directly to TAR-based MMISR remains open (Bartier et al., 2024). Generalized token jumping demonstrates that modest relaxations of the move rule can collapse fixed-cardinality reachability from PSPACE-complete to universally feasible on connected graphs, again without directly settling TAR optimization (Křišťan et al., 2024).

Taken together, these results place MaxMin Independent Set Reconfiguration as a mature optimization problem with multiple formal avatars, a robust hardness theory grounded in reconfiguration PCPs and alphabet reduction, and a growing algorithmic theory shaped by sparsity, separators, and minor structure.