Maximum $k$- vs. $\ell$-colourings of graphs

TL;DR

This work studies promise maximum colouring, where a graph that admits a $k$-colouring with high edge-satisfaction is efficiently coloured with $\ell$ colours while preserving a strong fraction of satisfied edges. It extends the Frieze–Jerrum SDP framework to the promise setting, delivering a randomized algorithm with a provable lower bound on the approximation ratio $\alpha_{k\ell}$ that scales as $1 - 1/\ell$ with refinements $\frac{2\ln \ell}{k\ell}$ and lower-order terms, and showing $\alpha_{k\ell} > 1 - 1/\ell$. The paper also provides a derandomised version, and a fixed-$k$ large-$\ell$ algorithm with nearly optimal scaling $1 - o(1/\ell)$, together with a hardness landscape under the Unique Games Conjecture and unconditional results that pin down the problem’s limits. In addition, it develops a unified Fourier-analytic and Markov-operator framework to prove hardness via label-cover reductions and to connect to AGC via 1-approximation, highlighting the problem’s rich interaction between SDP relaxations, probabilistic rounding, and complexity-theoretic barriers.

Abstract

We present polynomial-time SDP-based algorithms for the following problem: For fixed $k \leq \ell$, given a real number $ε>0$ and a graph $G$ that admits a $k$-colouring with a $ρ$-fraction of the edges coloured properly, it returns an $\ell$-colouring of $G$ with an $(αρ- ε)$-fraction of the edges coloured properly in polynomial time in $G$ and $1 / ε$. Our algorithms are based on the algorithms of Frieze and Jerrum [Algorithmica'97] and of Karger, Motwani and Sudan [JACM'98]. When $k$ is fixed and $\ell$ grows large, our algorithm achieves an approximation ratio of $α= 1 - o(1 / \ell)$. When $k, \ell$ are both large, our algorithm achieves an approximation ratio of $α= 1 - 1 / \ell + 2 \ln \ell / k \ell - o(\ln \ell / k \ell) - O(1 / k^2)$; if we fix $d = \ell - k$ and allow $k, \ell$ to grow large, this is $α= 1 - 1 / \ell + 2 \ln \ell / k \ell - o(\ln \ell / k \ell)$. By extending the results of Khot, Kindler, Mossel and O'Donnell [SICOMP'07] to the promise setting, we show that for large $k$ and $\ell$, assuming Khot's Unique Games Conjecture (\UGC), it is \NP-hard to achieve an approximation ratio $α$ greater than $1 - 1 / \ell + 2 \ln \ell / k \ell + o(\ln \ell / k \ell)$, provided that $\ell$ is bounded by a function that is $o(\exp(\sqrt[3]{k}))$. For the case where $d = \ell - k$ is fixed, this bound matches the performance of our algorithm up to $o(\ln \ell / k \ell)$. Furthermore, by extending the results of Guruswami and Sinop [ToC'13] to the promise setting, we prove that it is \NP-hard to achieve an approximation ratio greater than $1 - 1 / \ell + 8 \ln \ell / k \ell + o(\ln \ell / k \ell)$, provided again that $\ell$ is bounded as before (but this time without assuming the \UGC).

Figures (1)

• Figure 1: Plot of $F_T(x), F_T'(x), F_T"(x)$ for $T = 10$.

Theorems & Definitions (63)

• Theorem 1
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• Theorem 4
• Definition 5
• Definition 6
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• Proposition 6