Reachability of Independent Sets and Vertex Covers Under Extended Reconfiguration Rules

TL;DR

This work investigates the computational complexity of Independent Set Reconfiguration ($ ext ISR$) and Vertex Cover Reconfiguration ($ ext VCR$) under extended rules $k$-Token Jumping and $k$-Token Sliding. By dissecting the problems across regimes where the number of simultaneous moves $k$ is tied to the initial solution size via a guaranteed value $ extmu$, the authors show NP-hardness for ISR with $k=|I|- extmu$ on sparse graphs, and NP-membership for $ extmu=O(\\log |I|)$ on certain classes, while establishing XP membership for VCR under the same parameterization. They also prove PSPACE-completeness for fixed $k$ on planar max-degree-3 graphs and on line/claw-free graphs, and extend the hardness to superconstant $k$ relative to the input size via reductions from MaxminISR and MinmaxVCR, respectively. The results map a comprehensive landscape of reconfiguration complexity, highlighting that extended rules can both complicate and sometimes complicate differently for ISR and VCR, with broad implications for graph-theoretic reconfiguration in sparse and restricted graph classes. The combination of intricate gadget-based reductions (notably Nondeterministic Constraint Logic) and compressed-reconfiguration graph techniques yields sharp boundaries between tractable and intractable instances, informing both theory and potential practical deployment in systems requiring concurrent reconfigurations.

Abstract

In reconfiguration problems, we are given two feasible solutions to a graph problem and asked whether one can be transformed into the other via a sequence of feasible intermediate solutions under a given reconfiguration rule. While earlier work focused on modifying a single element at a time, recent studies have started examining how different rules impact computational complexity. Motivated by recent progress, we study Independent Set Reconfiguration (ISR) and Vertex Cover Reconfiguration (VCR) under the $k$-Token Jumping ($k$-TJ) and $k$-Token Sliding ($k$-TS) models. In $k$-TJ, up to $k$ vertices may be replaced, while $k$-TS additionally requires a perfect matching between removed and added vertices. It is known that the complexity of ISR crucially depends on $k$, ranging from PSPACE-complete and NP-complete to polynomial-time solvable. In this paper, we further explore the gradient of computational complexity of the problems. We first show that ISR under $k$-TJ with $k = |I| - μ$ remains NP-hard when $μ$ is any fixed positive integer and the input graph is restricted to graphs of maximum degree 3 or planar graphs of maximum degree 4, where $|I|$ is the size of feasible solutions. In addition, we prove that the problem belongs to NP not only for $μ=O(1)$ but also for $μ= O(\log |I|)$. In contrast, we show that VCR under $k$-TJ is in XP when parameterized by $μ= |S| - k$, where $|S|$ is the size of feasible solutions. Furthermore, we establish the PSPACE-completeness of ISR and VCR under both $k$-TJ and $k$-TS on several graph classes, for fixed $k$ as well as superconstant $k$ relative to the size of feasible solutions.

Figures (9)

• Figure 1: Illustration of the construction of $G$ from a boolean formula $\phi'$ used in the proof of \ref{['ISR_NPcom_maxdeg3']}, where $\phi'=(x_1\lor \overline{x_2}\lor\overline{x_4})\land(\overline{x_1}\lor \overline{x_3}\lor x_4)\land (x_2\lor x_3\lor \overline{x_4})$. The blue marked tokens are on $I$, and the red marked tokens are on $J$.
• Figure 2: An illustration of replacing the crossing edges $u_1u_2$ and $v_1v_2$ with a crossover gadget. The gadget contains eight vertices and three tokens, and its maximum degree is $4$.
• Figure 3: Another drawing of $G$ (shown in \ref{['fig:ISRmaxdg3']}), corresponding to the Boolean formula $\phi'=(x_1\lor \overline{x_2}\lor\overline{x_4})\land(\overline{x_1}\lor \overline{x_3}\lor x_4)\land (x_2\lor x_3\lor \overline{x_4})$, on a grid with $11$ rows and $18$ columns. The intersection of a dotted horizontal line labeled $i \in \{1,\ldots,11\}$ and a dotted vertical line labeled $j \in \{1,\ldots,18\}$ represents the coordinate $(i,j)$. Thick lines represent the edges of the graph $G$, and all edge crossings occur only between edges belonging to distinct variable gadgets.
• Figure 4: An illustration of adding tokens to $I$ and $J$, yielding $I^*$ and $J^*$. (a) Vertices $u_1$ and $v_1$ belong to $I$ (red tokens), and $u_2$ and $v_2$ belong to $J$ (blue tokens). (b) In $I^*$, the new vertices $w_1$, $u'_2$, and $v'_2$ are added, while in $J^*$, the new vertices $w_3$, $u'_1$, and $v'_1$ are added. The two independent sets $I^*$ and $J^*$ remain disjoint.
• Figure 5: An illustration of how the crossover gadget works. The crossing edges $u_1u_2$ and $v_1v_2$, where one endpoint of each edge is occupied by a token, are replaced by a crossover gadget: (a) $u_1$ and $v_1$ are occupied; (b) $u_2$ and $v_1$ are occupied; (c) $u_1$ and $v_2$ are occupied; (b) $u_2$ and $v_2$ are occupied.

Theorems & Definitions (35)

• Theorem 2
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• Lemma 1
• Theorem 2