On the Complexity of Bipartite Degree Realizability

TL;DR

This paper investigates the computational complexity of the Bipartite Degree Realization problem, identifying parameter regimes where it is polynomial-time solvable and others where it remains NP-complete, thus clarifying its algorithmic landscape.

Contribution

It provides a detailed parameterized analysis of BDR, establishing new polynomial-time solvability conditions and NP-completeness boundaries based on degree sequence parameters.

Findings

BDR is polynomial-time solvable for certain degree interval parameters.

BDR remains NP-complete outside these tractable parameter regimes.

The paper delineates a sharp boundary between tractable and intractable cases.

Abstract

We study the \emph{Bipartite Degree Realization} (BDR) problem: given a graphic degree sequence $D$, decide whether it admits a realization as a bipartite graph. While bipartite realizability for a fixed vertex partition can be decided in polynomial time via the Gale--Ryser theorem, the computational complexity of BDR without a prescribed partition remains unresolved. We address this question through a parameterized analysis. For constants $0 \le c_1 \le c_2 \le 1$, we define $\mathrm{BDR}_{c_1,c_2}$ as the restriction of BDR to degree sequences of length $n$ whose degrees lie in the interval $[c_1 n, c_2 n]$. Our main result shows that $\mathrm{BDR}_{c_1,c_2}$ is solvable in polynomial time whenever $0 \le c_1 \le c_2 \le \frac{\sqrt{c_1(c_1+4)}-c_1}{2}$, as well as for all $c_1 > \tfrac12$. The proof relies on a reduction to extremal \emph{least balanced degree sequences} and a detailed verification of the critical Gale--Ryser inequalities, combined with a bounded subset-sum formulation. We further show that, assuming the NP-completeness of unrestricted BDR, the problem $\mathrm{BDR}_{c_1,c_2}$ remains NP-complete for all $0 < c_2 < \frac{1}{2}$ and $c_1 < 1 - c_2 - \sqrt{1-2c_2}$. % This establishes a sharp conditional boundary between tractable and intractable parameter regimes. Our results clarify the algorithmic landscape of bipartite degree realization and contribute to the broader study of potentially bipartite graphic degree sequences.