Asymptotically Optimal Inapproximability of Maxmin $k$-Cut Reconfiguration

TL;DR

The paper studies Maxmin $k$-Cut Reconfiguration, asking to maximize the worst-case fraction of bichromatic edges along a recoloring sequence between two $k$-colorings. It establishes asymptotically tight hardness and algorithmic guarantees: it is $ extsf{PSPACE}$-hard to approximate within $1-\Theta\left(\frac{1}{k}\right)$ for all $k\ge 2$, while a deterministic algorithm achieves a $\left(1-\frac{2}{k}\right)$-factor, matching the hardness up to constant factors in the asymptotic regime. The hardness relies on a novel stripe-based encoding of 2-colorings into $k\times k$ grids and a trio of probabilistic verifiers (Stripe, Consistency, Edge) to enforce consistency of encodings via gap-preserving reductions from Gap$_{1-\varepsilon_c,1-\varepsilon_s}$ 2-Cut Reconfiguration. The algorithmic contribution derives from a random-reconfiguration approach through a random $k$-coloring and a derandomization by conditional expectations, combined with a degree-based decomposition and Chernoff-type concentration to achieve the $1-\frac{2}{k}$ guarantee. Collectively, these results close the approximability gap for Maxmin $k$-Cut Reconfiguration and illustrate a PCP-like hardness landscape for reconfiguration problems, with implications for both theory and potential practical planning of reconfiguration sequences.

Abstract

$k$-Coloring Reconfiguration is one of the most well-studied reconfiguration problems, which asks to transform a given proper $k$-coloring of a graph to another by repeatedly recoloring a single vertex. Its approximate version, Maxmin $k$-Cut Reconfiguration, is defined as an optimization problem of maximizing the minimum fraction of bichromatic edges during the transformation between (not necessarily proper) $k$-colorings. In this paper, we prove that the optimal approximation factor of this problem is $1 - Θ\left(\frac{1}{k}\right)$ for every $k \ge 2$. Specifically, we show the $\mathsf{PSPACE}$-hardness of approximating the objective value within a factor of $1 - \frac{\varepsilon}{k}$ for some universal constant $\varepsilon > 0$, whereas we present a deterministic polynomial-time algorithm that achieves the approximation factor of $1 - \frac{2}{k}$. To prove the hardness result, we develop a new probabilistic verifier that tests a ``striped'' pattern. Our polynomial-time algorithm is based on ``a random reconfiguration via a random solution,'' i.e., the transformation that goes through one random $k$-coloring.

Figures (10)

• Figure 1: A Yes instance of 3-Coloring Reconfiguration. There is a reconfiguration sequence $(f_\mathsf{start} = f^{(1)}, f^{(2)}, f^{(3)}, f^{(4)} = f_\mathsf{end})$ such that each $3$-coloring is proper and is obtained by the previous one by recoloring a single vertex.
• Figure 2: A No instance of 3-Coloring Reconfiguration. There is no reconfiguration sequence from $g_\mathsf{start}$ to $g_\mathsf{end}$, because $g_\mathsf{end}$ is "frozen" in that any vertex cannot recolored. Considering this input as an instance of Maxmin 3-Cut Reconfiguration, we can transform $g_\mathsf{start}$ into $g_\mathsf{end}$ via $g^{(2)}$, which contains a single monochromatic edge.
• Figure 3: A failed attempt to reduce Maxmin 2-Cut Reconfiguration to Maxmin $k$-Cut Reconfiguration ($k=8$) using kann1997hardness. Given a graph $G$ and a pair of its $2$-colorings $f_\mathsf{start},f_\mathsf{end}$, we construct a new graph $H$ and a pair of its $k$-colorings $f'_\mathsf{start},f'_\mathsf{end}$. Consider a reconfiguration sequence $\mathscr{F}'$ from $f'_\mathsf{start}$ to $f'_\mathsf{end}$ obtained by recoloring vertices of $V_1, V_2, \ldots, V_{\frac{k}{2}}$ in this order. For any intermediate $k$-coloring of $\mathscr{F}'$, all but one induced subgraph $H[V_i]$ do not contain any monochromatic edges.
• Figure 4: Our proposed encoding and the stripe test are motivated by the graph structure formed by two different proper $k$-colorings.
• Figure 5: A $k$-coloring $f$ of $[k]^2$ that is far from being striped. Obviously, $f$ is closest to an $8 \times 8$ horizontally striped pattern but differs in $16$ entries; thus, $f$ is $0.25$-far from being striped.

Theorems & Definitions (85)

• Theorem 1.1: informal; see \ref{['thm:Cut-hard']}
• Theorem 1.2: informal; see \ref{['thm:Cut-alg']}
• Proposition 2.1: informal; see \ref{['prp:Cut-hard:2Cut']}
• Lemma 2.2: informal; see \ref{['lem:Cut-hard:crazy']}
• Lemma 2.3: informal; see \ref{['lem:Cut-hard:quadratic']}
• Lemma 2.4: informal; see \ref{['lem:Cut-hard:stripe:striped', 'lem:Cut-hard:stripe:far']}
• Lemma 2.5: informal; see \ref{['lem:Cut-hard:cons:striped', 'lem:Cut-hard:cons:far']}
• Lemma 2.6: informal; see \ref{['lem:Cut-hard:edge:striped', 'lem:Cut-hard:edge:mismatch', 'lem:Cut-hard:edge:any']}
• Lemma 2.7: informal; see \ref{['lem:Cut-hard:complete', 'lem:Cut-hard:sound']}
• Remark 2.8