Tight lower bound for the channel assignment problem

TL;DR

This paper proves that the Channel Assignment problem cannot be solved in significantly faster exponential time than the current algorithms, assuming the Exponential Time Hypothesis, establishing a tight lower bound.

Contribution

It establishes a tight lower bound of $2^{o(n ext{log} n)}$ for solving the Channel Assignment problem, resolving an open complexity question.

Findings

No $2^{o(n ext{log} n)}$-time algorithm exists under ETH.

Current best algorithm runs in $O^*(n!)$ time, matching the lower bound.

The result confirms the optimality of existing algorithms under standard complexity assumptions.

Abstract

We study the complexity of the Channel Assignment problem. A major open problem asks whether Channel Assignment admits an $O(c^n)$-time algorithm, for a constant $c$ independent of the weights on the edges. We answer this question in the negative i.e. we show that there is no $2^{o(n\log n)}$-time algorithm solving Channel Assignment unless the Exponential Time Hypothesis fails. Note that the currently best known algorithm works in time $O^*(n!) = 2^{O(n\log n)}$ so our lower bound is tight.