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

Avinandan Das

TL;DR

While the semi-streaming complexity of 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.

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 $\min\{k, °(v)\}$ 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 $(Δ+1)$-proper coloring and $Δ$-proper coloring, respectively. Prior works of Assadi, Chen, and Khanna [SODA~2019] and Assadi, Kumar, and Mittal [TheoretiCS~2023] show that both $(Δ+1)$-proper coloring and $Δ$-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.

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

TL;DR

While the semi-streaming complexity of k-partial k-partial $k-coloring remains semi-streaming intractable, effectively demonstrating a ``dichotomy of one color'' in the streaming model.

Abstract

This paper investigates the semi-streaming complexity of \textit{-partial coloring}, a generalization of proper graph coloring. For , a -partial coloring requires that each vertex in an -node graph is assigned a color such that at least of its neighbors are assigned colors different from its own. This framework naturally extends classical coloring problems: specifically, -partial -coloring and -partial -coloring generalize -proper coloring and -proper coloring, respectively. Prior works of Assadi, Chen, and Khanna [SODA~2019] and Assadi, Kumar, and Mittal [TheoretiCS~2023] show that both -proper coloring and -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 -partial -coloring admits a one-pass randomized semi-streaming algorithm, the -partial -coloring remains semi-streaming intractable, effectively demonstrating a ``dichotomy of one color'' in the streaming model.
Paper Structure (22 sections, 18 theorems, 19 equations, 1 figure, 1 algorithm)

This paper contains 22 sections, 18 theorems, 19 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Partial coloring.
  3. A simple Streaming Algorithm.
  4. Challenges in getting down to $k+1$ colors.
  5. $k$-partial $k$-coloring.
  6. Related Work
  7. Our Results
  8. Technical Overview
  9. Preliminaries
  10. Semi-Streaming Algorithm for $k$-partial $(k+1)$-coloring
  11. Palette Sparsification for Partial Coloring
  12. Phase 1: Coloring the Independent Set $S$.
  13. Phase 2: Extending the coloring to $R = V \setminus S$.
  14. Algorithm
  15. Description of the Algorithm
  16. ...and 7 more sections

Key Result

Theorem 1

There exists a one-pass randomized semi-streaming algorithm that, with high probability, produces a $k$-partial coloring of an input graph $G$ using $k+1$ colors, where $k \in \mathbb{N}$ is given as input prior to the stream and $G$ is presented as an insertion-only stream.

Figures (1)

  • Figure 1: Example of a graph where $3$-partial coloring implies a proper coloring. Observe that the vertices $v$ and $u_2$ have degree greater than $3$ but are surrounded by degree $3$ vertices which enforce a proper coloring on them.

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Theorem 3
  • Definition 1: Witness for Partial Coloring
  • Lemma 2
  • proof
  • Theorem 4: Palette sparsification for $(\deg+1)$-list coloring
  • Theorem 5
  • ...and 24 more