Complexity, approximation, and extension of proper $\{a,b\}$-edge-weightings

Abstract

For distinct integers $a$ and $b$, an $\{a,b\}$-edge-weighting assigns $a$ or $b$ to each edge and labels each vertex by the sum of its incident weights. Such a weighting is proper if adjacent vertices receive distinct labels. We prove that, for every fixed pair of distinct integers, deciding whether a proper weighting exists is NP-complete even for simple cubic planar graphs. On planar multigraphs with $m$ edges, we give an exact $2^{O(\sqrt m)}$-time algorithm and, assuming the Exponential Time Hypothesis (ETH), exclude $2^{o(\sqrt m)}$-time algorithms even for simple cubic planar graphs. As a consequence, locally irregular $2$-edge-coloring is NP-complete on simple cubic planar graphs, admits a deterministic $2^{O(\sqrt n)}$-time algorithm on $n$-vertex graphs in this class, and admits no $2^{o(\sqrt n)}$-time algorithm under ETH. For maximizing the number of edges joining vertices with distinct labels, we give a deterministic efficient polynomial-time approximation scheme (EPTAS) on planar multigraphs, a polynomial-time $1/2$-approximation on multigraphs, and APX-completeness even on simple cubic graphs. Extending a partial $\{a,b\}$-edge-weighting to a proper one is NP-complete for every fixed pair even on simple cubic planar bipartite graphs, while it is polynomial-time solvable on trees. The hardness persists even when the prescribed edges form disjoint paths of length $6$ and all edges of each path have the same prescribed weight.