Bond Polytope under Vertex- and Edge-sums

2026-01-16

AI Summary

This work addresses the Maximum Weight Bond problem, which is NP-hard even on planar graphs. Focusing on $(K_5 \setminus e)$-minor-free graphs, the authors leverage clique-sum decompositions and a novel characterization of bond polytopes as 1- or 2-sums to establish, for the first time, that the bond polytope over this graph class admits a linear extension complexity. Building on this structural insight, they design a simple linear-time algorithm for MaxBond that avoids reliance on treewidth-based black-box techniques. The result not only advances the theoretical understanding of bond polytopes but also yields a practically efficient algorithm with improved constant factors.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

A cut in a graph $G$ is called a {\em bond} if both parts of the cut induce connected subgraphs in $G$, and the {\em bond polytope} is the convex hull of all bonds. Computing the maximum weight bond is an NP-hard problem even for planar graphs. However, the problem is solvable in linear time on $(K_5 \setminus e)$-minor-free graphs, and in more general, on graphs of bounded treewidth, essentially due to clique-sum decomposition into simpler graphs. We show how to obtain the bond polytope of graphs that are $1$- or $2$-sum of graphs $G_1$ and $ G_2$ from the bond polytopes of $G_1,G_2$. Using this we show that the extension complexity of the bond polytope of $(K_5 \setminus e)$-minor-free graphs is linear. Prior to this work, a linear size description of the bond polytope was known only for $3$-connected planar $(K_5 \setminus e)$-minor-free graphs, essentially only for wheel graphs. We also describe an elementary linear time algorithm for the \MaxBond problem on $(K_5\setminus e)$-minor-free graphs. Prior to this work, a linear time algorithm in this setting was known. However, the hidden constant in the big-Oh notation was large because the algorithm relies on the heavy machinery of linear time algorithms for graphs of bounded treewidth, used as a black box.

Problem

• bond polytope
• maximum weight bond
• extension complexity
• (K_5 \setminus e)-minor-free graphs
• MaxBond problem

Innovation

• bond polytope
• extension complexity
• graph minor
• clique-sum decomposition
• linear time algorithm