Assigning channels via the meet-in-the-middle approach

TL;DR

This paper introduces a novel meet-in-the-middle algorithm for the bounded Channel Assignment problem that surpasses previous complexity barriers, and explores related complexity questions for Generalized T-Coloring.

Contribution

It presents the first sub-$O( ext{ell})^n$ algorithm for $ ext{ell}$-bounded Channel Assignment and extends it to counting variants, also analyzing complexity limits for Generalized T-Coloring.

Findings

New algorithm runs in $O^*((2\sqrt{ ext{ell}+1})^n)$ time

Breaks the $(O( ext{ell}))^n$ barrier for bounded Channel Assignment

Shows no $2^{2^{o(\sqrt{n} ight)}}$-time algorithm exists for Generalized T-Coloring

Abstract

We study the complexity of the Channel Assignment problem. By applying the meet-in-the-middle approach we get an algorithm for the $\ell$-bounded Channel Assignment (when the edge weights are bounded by $\ell$) running in time $O^*((2\sqrt{\ell+1})^n)$. This is the first algorithm which breaks the $(O(\ell))^n$ barrier. We extend this algorithm to the counting variant, at the cost of slightly higher polynomial factor. A major open problem asks whether Channel Assignment admits a $O(c^n)$-time algorithm, for a constant $c$ independent of $\ell$. We consider a similar question for Generalized T-Coloring, a CSP problem that generalizes \CA. We show that Generalized T-Coloring does not admit a $2^{2^{o\left(\sqrt{n}\right)}} {\rm poly}(r)$-time algorithm, where $r$ is the size of the instance.