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NP-Completeness of Neighborhood Balanced Colorings

Saeed Asaeedi

TL;DR

The paper addresses the problem of determining whether a graph admits a Neighborhood Balanced Coloring (NBC), proving its NP-completeness via a partition-based reduction that uses $n$-pack gadgets attached to a $K_{2,2}$ base so NBC existence corresponds to a two-way equal-sum partition of an integer set. It contributes an exact algorithm, a single-crossover genetic algorithm, and a random-coloring baseline, all evaluated with a penalty metric $\text{penalty} = \sum_{v\in V} | r_v - b_v |$ to measure deviation from an NBC, on both small and large graphs. The results establish a practical toolkit for NBC search: the exact method is feasible only for small instances, while the genetic algorithm offers scalable performance with competitive accuracy against random coloring. The work advances understanding of NBC's computational hardness and provides heuristic methods for discovering NBCs in larger graphs or when an exact solution is intractable.

Abstract

A Neighborhood Balanced Coloring (NBC) of a graph is a red-blue coloring where each vertex has the same number of red and blue neighbors. This work proves that determining if a graph admits an NBC is NP-complete. We present a genetic algorithm to solve this problem, which we implemented and compared against exact and randomized algorithms.

NP-Completeness of Neighborhood Balanced Colorings

TL;DR

The paper addresses the problem of determining whether a graph admits a Neighborhood Balanced Coloring (NBC), proving its NP-completeness via a partition-based reduction that uses -pack gadgets attached to a base so NBC existence corresponds to a two-way equal-sum partition of an integer set. It contributes an exact algorithm, a single-crossover genetic algorithm, and a random-coloring baseline, all evaluated with a penalty metric to measure deviation from an NBC, on both small and large graphs. The results establish a practical toolkit for NBC search: the exact method is feasible only for small instances, while the genetic algorithm offers scalable performance with competitive accuracy against random coloring. The work advances understanding of NBC's computational hardness and provides heuristic methods for discovering NBCs in larger graphs or when an exact solution is intractable.

Abstract

A Neighborhood Balanced Coloring (NBC) of a graph is a red-blue coloring where each vertex has the same number of red and blue neighbors. This work proves that determining if a graph admits an NBC is NP-complete. We present a genetic algorithm to solve this problem, which we implemented and compared against exact and randomized algorithms.
Paper Structure (4 sections, 1 theorem, 1 equation, 6 figures, 3 algorithms)

This paper contains 4 sections, 1 theorem, 1 equation, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. NP-Completeness of the Problem
  3. Implementation and results
  4. Conclusion

Key Result

Theorem 1

Given a set $S=\{a_1, a_2, \dots, a_n\}$ of positive integers, and let $G$ be the graph constructed using the presented reduction. Then, $G$ admits an NBC if and only if $S$ can be partitioned into subsets $S_1$ and $S_2$ such that $\sum_{a\in S_1} a =\sum_{a\in S_2} a$.

Figures (6)

  • Figure 1: In the $4$-pack graph, the top blue vertex represents the base vertex, the horizontal vertices represent support vertices, and the vertical red vertices represent numeric vertices.
  • Figure 2: The converted graph $G$, resulting from the reduction steps, includes the bipartite graph $K_{2,2}$, a $4$-pack, a $3$-pack, and a $1$-pack.
  • Figure 3: The converted graph $G$, resulting from the reduction steps, includes the bipartite graph $K_{2,2}$, a $4$-pack, a $3$-pack, and a $1$-pack.
  • Figure 4: The results of the genetic algorithm on a 20-vertex random graph.
  • Figure 5: Comparing the performance of exact algorithm (Exact), genetic algorithm (GA), and random coloring algorithm (Random) on graphs with varying sizes from $n=4$ to $n=25$.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 1
  • Remark 1
  • Theorem 1
  • proof