Odd-Cycle-Packing-treewidth: On the Maximum Independent Set problem in odd-minor-free graph classes

2025-11-13

AI Summary

This paper investigates the Maximum Independent Set problem on odd-minor-free graph classes. To characterize their structural properties, we introduce a novel treewidth-like parameter—Odd-Cycle Packing Treewidth (OCP-tw)—based on tree decompositions, and prove its monotonicity under the odd-minor relation. We establish a grid-like theorem: graphs with large OCP-tw necessarily contain prescribed odd-minor structures. Combining structural graph theory and combinatorial optimization, we design polynomial-time algorithms: (i) an approximation algorithm for computing OCP-tw, and (ii) an exact polynomial-time algorithm for Maximum Independent Set when OCP-tw is bounded. Furthermore, we extend these results to totally Δ-modular integer programming, thereby broadening the interface between parameterized algorithms and structural graph theory.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We introduce the tree-decomposition-based graph parameter Odd-Cycle-Packing-treewidth (OCP-tw) as a width parameter that asks to decompose a given graph into pieces of bounded odd cycle packing number. The parameter OCP-tw is monotone under the odd-minor-relation and we provide an analogue to the celebrated Grid Theorem of Robertson and Seymour for OCP-tw. That is, we identify two infinite families of grid-like graphs whose presence as odd-minors implies large OCP-tw and prove that their absence implies bounded OCP-tw. This structural result is constructive and implies a 2^(poly(k))poly(n)-time parameterized poly(k)-approximation algorithm for OCP-tw. Moreover, we show that the (weighted) Maximum Independent Set problem (MIS) can be solved in polynomial time on graphs of bounded OCP-tw. Finally, we lift the concept of OCP-tw to a parameter for matrices of integer programs. To this end, we show that our strategy can be applied to efficiently solve integer programs whose matrices can be"tree-decomposed"into totally delta-modular matrices with at most two non-zero entries per row.

Problem

• Introducing Odd-Cycle-Packing-treewidth as a new graph width parameter
• Solving Maximum Independent Set problem in odd-minor-free graph classes
• Developing efficient algorithms for integer programs with structured matrices

Innovation

• Introduces Odd-Cycle-Packing-treewidth as decomposition parameter
• Uses tree-decomposition to bound odd cycle packing number
• Solves Maximum Independent Set via bounded OCP-tw graphs