[Paper Review] Tight Lower Bounds for List Edge Coloring

Łukasz Kowalik, Arkadiusz Socała|arXiv (Cornell University)|Apr 7, 2018

Advanced Graph Theory Research12 references3 citations

TL;DR

This paper establishes a tight lower bound for list edge coloring in simple graphs, proving that no algorithm can solve it in time $2^{o(n^2)}$ unless the Exponential Time Hypothesis (ETH) fails. The authors reduce a sparse 3-CNF-SAT instance to a list edge coloring instance with $O(n)$ vertices, showing that the standard $2^m n^{O(1)}$ algorithm for edge coloring generalizes to list edge coloring without slowdown, making the bound essentially optimal.

ABSTRACT

The fastest algorithms for edge coloring run in time $2^m n^{O(1)}$, where $m$ and $n$ are the number of edges and vertices of the input graph, respectively. For dense graphs, this bound becomes $2^{Θ(n^2)}$. This is a somewhat unique situation, since most of the studied graph problems admit algorithms running in time $2^{O(n\log n)}$. It is a notorious open problem to either show an algorithm for edge coloring running in time $2^{o(n^2)}$ or to refute it, assuming Exponential Time Hypothesis (ETH) or other well established assumption. We notice that the same question can be asked for list edge coloring, a well-studied generalization of edge coloring where every edge comes with a set (often called a list) of allowed colors. Our main result states that list edge coloring for simple graphs does not admit an algorithm running in time $2^{o(n^2)}$, unless ETH fails. Interestingly, the algorithm for edge coloring running in time $2^m n^{O(1)}$ generalizes to the list version without any asymptotic slow-down. Thus, our lower bound is essentially tight. This also means that in order to design an algorithm running in time $2^{o(n^2)}$ for edge coloring, one has to exploit its special features compared to the list version.

Motivation & Objective

• To close the gap between known upper bounds and the conjectured lower bounds for edge coloring algorithms.
• To investigate whether list edge coloring, a generalization of edge coloring, admits faster algorithms than the standard $2^m n^{O(1)}$ approach.
• To determine whether the $2^{O(n^2)}$ barrier for edge coloring can be broken using methods that do not apply to list edge coloring.
• To establish a tight lower bound for list edge coloring in simple graphs under the Exponential Time Hypothesis (ETH).

Proposed method

• Reduction from a sparse 3-CNF-SAT instance to a list edge coloring instance with $O( olimits\sqrt{n})$ vertices.
• Construction of a graph $G_r$ with variable and clause gadgets to encode truth assignments via alternating edge colorings.
• Use of literal-based color sequences ($x_i, \neg x_i$) on paths to simulate variable assignments and ensure proper coloring.
• Encoding of clause satisfaction via specific edge colorings that depend on the truth value of variables.
• Proof that all edges can be properly colored using only colors from their assigned lists, preserving proper list edge coloring.
• Establishing that a $2^{o(n^2)}$ algorithm for list edge coloring would contradict ETH by solving $ (3,4) $-SAT in $2^{o(n)}$ time.

Experimental results

Research questions

• RQ1 Can list edge coloring be solved in time $2^{o(n^2)}$ for simple graphs?
• RQ2 Is the $2^m n^{O(1)}$ algorithm for edge coloring optimal, or can it be improved?
• RQ3 Does the list edge coloring problem exhibit a tighter lower bound than standard edge coloring?
• RQ4 Can a $2^{O(n)}$-time algorithm for edge coloring be ruled out under ETH?
• RQ5 What structural differences between edge coloring and list edge coloring prevent the use of general techniques for breaking the $2^{O(n^2)}$ barrier?

Key findings

• List edge coloring in simple graphs does not admit an algorithm running in time $2^{o(n^2)}$ unless the Exponential Time Hypothesis (ETH) fails.
• The reduction from $ (3,4) $-SAT to list edge coloring preserves the number of variables and produces an instance with $O(\sqrt{n})$ vertices.
• The standard $2^m n^{O(1)}$ algorithm for edge coloring generalizes to list edge coloring without asymptotic slowdown, making the lower bound tight.
• Any algorithm breaking the $2^{O(n^2)}$ barrier for edge coloring must exploit color symmetry and cannot be generalized from list edge coloring.
• The result implies that list edge coloring is computationally as hard as the densest edge coloring instances, and no faster algorithm exists under ETH.
• The construction ensures that all edges are properly colored using only colors from their lists, and no vertex has incident edges of the same color.