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Maximum Biclique for Star 1,2,3 -free and Bounded Bimodularwidth Twin-free Bipartite Graphs $\star$

Fabien de Montgolfier, Renaud Torfs

TL;DR

The paper tackles the problem of finding maximum bicliques in bipartite graphs under three definitions by leveraging a bimodular decomposition framework. It introduces the Maximum Bisize Set ($MBS_G$) as a unifying dynamic programming primitive and derives explicit recurrence relations for how $MBS_G$ behaves under Parallel, Series, K+S, and Prime decompositions, including prime-quotient handling via a quotient graph $H_G$. The authors provide $O(n^2)$-time algorithms for two targeted graph classes: twin-free $Star_{1,2,3}$-free graphs and twin-free graphs of bounded bimodularwidth, given a decomposition. This yields efficient, exact solutions for maximum vertex, maximum edge, and maximum balanced bicliques in structured bipartite graphs, with practical implications for applications where the graphs admit bimodular decompositions.

Abstract

There are three usual definitions of a maximum bipartite clique (biclique) in a bipartite graph\,: either maximizing the number of vertices, or of edges, or finding a maximum balanced biclique. The first problem can be solved in polynomial time, the last ones are NP-complete. Here we show how these three problems may be efficiently solved for two classes of bipartite graphs: Star123-free twin-free graphs, and bounded bimodularwidth twin-free graphs, a class that may be defined using bimodular decomposition. Our computation requires O(n^2) time and requires a decomposition is provided, which takes respectively O(n + m) and O(mn^3) time.

Maximum Biclique for Star 1,2,3 -free and Bounded Bimodularwidth Twin-free Bipartite Graphs $\star$

TL;DR

The paper tackles the problem of finding maximum bicliques in bipartite graphs under three definitions by leveraging a bimodular decomposition framework. It introduces the Maximum Bisize Set () as a unifying dynamic programming primitive and derives explicit recurrence relations for how behaves under Parallel, Series, K+S, and Prime decompositions, including prime-quotient handling via a quotient graph . The authors provide -time algorithms for two targeted graph classes: twin-free -free graphs and twin-free graphs of bounded bimodularwidth, given a decomposition. This yields efficient, exact solutions for maximum vertex, maximum edge, and maximum balanced bicliques in structured bipartite graphs, with practical implications for applications where the graphs admit bimodular decompositions.

Abstract

There are three usual definitions of a maximum bipartite clique (biclique) in a bipartite graph\,: either maximizing the number of vertices, or of edges, or finding a maximum balanced biclique. The first problem can be solved in polynomial time, the last ones are NP-complete. Here we show how these three problems may be efficiently solved for two classes of bipartite graphs: Star123-free twin-free graphs, and bounded bimodularwidth twin-free graphs, a class that may be defined using bimodular decomposition. Our computation requires O(n^2) time and requires a decomposition is provided, which takes respectively O(n + m) and O(mn^3) time.

Paper Structure

This paper contains 12 sections, 19 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Bimodular decomposition
  3. Maximum Biclique Size set
  4. Bicliques with respect to bimodular decomposition
  5. Parallel case
  6. Series case
  7. K+S case
  8. Prime case
  9. Base cases: paths and cycles and their bipartite complements
  10. Algorithms
  11. $Star_{1,2,3}$-free twin-free graphs
  12. Bounded Bimodularwidth twin-free graphs

Key Result

theorem 1

Let $G$ be a twin-free bipartite graph without induced $Star_{1,2,3}$. Then

Figures (2)

  • Figure 1: A bipartite graph and its bimodular decomposition tree. Bimodularwidth is 7
  • Figure 2: An example of $G$ with non-trivial maximal bimodules highlighted in green, yellow and blue; $H_G$ drawn below with the edges corresponding to non-trivial maximal bimodules colored the same color. A biclique of $G$ whose size is a maxbisize is drawn with thicker edges; the corresponding biclique $C$ in $H_G$$C$ (bottom), and $Rroc(C)$ (top), are highlighted in red.

Theorems & Definitions (42)

  • theorem 1: Lozin lozin2002bipartite
  • theorem 2: Fouquet et. al WG04
  • definition 1: bimodularwidth
  • definition 2: Domination
  • definition 3: Bisize and maxbisize
  • remark 1
  • definition 4: maxbisize set and operators
  • theorem 3
  • proof
  • proposition 1
  • ...and 32 more