One Color Makes All the Difference in the Tractability of Partial Coloring in Semi-Streaming

TL;DR

This paper explores the semi-streaming complexity of k-partial coloring in graphs, revealing a sharp threshold where certain coloring problems are tractable with one-pass algorithms, but others are inherently intractable.

Contribution

It establishes a clear computational dichotomy in semi-streaming graph coloring, showing that adding just one color can drastically change problem tractability.

Findings

k-partial (k+1)-coloring admits a one-pass semi-streaming algorithm

k-partial k-coloring remains semi-streaming intractable

Demonstrates a sharp threshold in streaming complexity based on the number of colors

Abstract

This paper investigates the semi-streaming complexity of \textit{$k$-partial coloring}, a generalization of proper graph coloring. For $k \geq 1$, a $k$-partial coloring requires that each vertex $v$ in an $n$-node graph is assigned a color such that at least $\min\{k, \deg(v)\}$ of its neighbors are assigned colors different from its own. This framework naturally extends classical coloring problems: specifically, $k$-partial $(k+1)$-coloring and $k$-partial $k$-coloring generalize $(\Delta+1)$-proper coloring and $\Delta$-proper coloring, respectively. Prior works of Assadi, Chen, and Khanna [SODA~2019] and Assadi, Kumar, and Mittal [TheoretiCS~2023] show that both $(\Delta+1)$-proper coloring and $\Delta$-proper coloring admit one-pass randomized semi-streaming algorithms. We explore whether these efficiency gains extend to their partial coloring generalizations and reveal a sharp computational threshold : while $k$-partial $(k+1)$-coloring admits a one-pass randomized semi-streaming algorithm, the $k$-partial $k$-coloring remains semi-streaming intractable, effectively demonstrating a ``dichotomy of one color'' in the streaming model.