Characterization of the Imbalance Problem on Complete Bipartite Graphs

TL;DR

This paper characterizes the optimal solutions for the imbalance problem on complete bipartite graphs, providing efficient algorithms for solving and verifying solutions, and extends results to a special class of interval bipartite graphs.

Contribution

It offers a characterization of optimal solutions for the imbalance problem on complete bipartite graphs and develops efficient algorithms for solving and verification.

Findings

Optimal solutions can be characterized for complete bipartite graphs.

The imbalance problem can be solved in logarithmic time given graph part sizes.

Verification of optimality is achievable in linear time.

Abstract

We study the imbalance problem on complete bipartite graphs. The imbalance problem is a graph layout problem and is known to be NP-complete. Graph layout problems find their applications in the optimization of networks for parallel computer architectures, VLSI circuit design, information retrieval, numerical analysis, computational biology, graph theory, scheduling and archaeology. In this paper, we give characterizations for the optimal solutions of the imbalance problem on complete bipartite graphs. Using the characterizations, we can solve the imbalance problem in $\mathcal{O}(\log(|V|) \cdot \log(\log(|V|)))$ time, when given the cardinalities of the parts of the graph, and verify whether a given solution is optimal in $O(|V|)$ time on complete bipartite graphs. We also introduce a restricted form of proper interval bipartite graphs on which the imbalance problem is solvable in $\mathcal{O}(c \cdot \log(|V|) \cdot \log(\log(|V|)))$ time, where $c = \mathcal{O}(|V|)$, by using the aforementioned characterizations.