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Updated: Sep 11, 2025

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Linear-Time MaxCut in Multigraphs Parameterized Above the Poljak-Turzík Bound.

Jonas Lill1, Kalina Petrova1,2, Simon Weber1

  • 1Department of Computer Science, ETH Zürich, Rämistrasse 101, 8092 Zurich, Switzerland.

Algorithmica
|August 11, 2025
PubMed
Summary

This study enhances the MaxCut algorithm for multigraphs and weighted graphs. It improves efficiency by using the Poljak-Turzík bound, achieving parameterized linear time complexity for finding maximum cuts.

Keywords:
Edwards-Erdös boundFixed-parameter tractabilityInteger-weighted graphsMaximum cutMultigraphsPoljak-Turzík bound

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Area of Science:

  • Computer Science
  • Discrete Mathematics
  • Graph Theory
  • Algorithm Analysis

Background:

  • The MaxCut problem is a classical NP-complete problem with significant applications in combinatorial optimization.
  • Existing algorithms for MaxCut on simple graphs have parameterized linear time complexity based on the Edwards-Erdös bound.
  • The Poljak-Turzík bound offers a potentially tighter lower bound for MaxCut, especially in weighted graphs.

Purpose of the Study:

  • To extend existing parameterized algorithms for the MaxCut problem to handle multigraphs and weighted graphs.
  • To improve the parameterization of MaxCut algorithms by utilizing the Poljak-Turzík bound instead of the Edwards-Erdös bound.
  • To achieve a parameterized linear time complexity for MaxCut on multigraphs and weighted graphs.

Main Methods:

  • Adaptation of existing fixed-parameter tractable (FPT) algorithms for MaxCut.
  • Extension of the algorithm's applicability to graphs with parallel edges (multigraphs) or positive integer weights.
  • Re-parameterization of the algorithm using the difference between the desired cut size and the Poljak-Turzík bound.

Main Results:

  • The developed algorithm successfully extends the parameterized MaxCut solution to multigraphs and weighted graphs.
  • The new parameterization based on the Poljak-Turzík bound can lead to smaller parameter values (k) for multigraphs.
  • The algorithm achieves a parameterized linear time complexity of f(k)·O(m+n), an improvement over previous O(m) bounds.

Conclusions:

  • This work provides a more versatile and efficient parameterized algorithm for the MaxCut problem applicable to a broader class of graphs.
  • The use of the Poljak-Turzík bound offers a refined approach to parameterizing MaxCut, enhancing algorithmic performance.
  • The achieved parameterized linear time complexity represents a significant theoretical advancement in solving the MaxCut problem.