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Bond Polytope under Vertex- and Edge-sums

Petr Kolman, Hans Raj Tiwary

TL;DR

The paper studies the Max-Bond problem, where a bond is a cut with both sides connected, and bonds correspond to circuits of the co-graphic matroid; while NP-hard in general, Max-Bond is solvable in linear time on $K_5\setminus e$-minor-free graphs and more broadly for bounded-treewidth graphs via clique-sum decompositions. It introduces a subdirect-sum framework for combining bond polytopes under $1$- and $2$-sums, and develops extended formulations that combine additively under these sum operations, yielding linear extension complexity for $(K_5\setminus e)$-minor-free graphs. A key contribution is an elementary linear-time algorithm for Max-Bond on wheel graphs, which replaces heavier treewidth machinery used in prior work, and together with the decomposition results, gives a linear-time Max-Bond algorithm for all $(K_5\setminus e)$-minor-free graphs. The work also clarifies when linear-size descriptions of the bond polytope are possible (via extension formulations) and shows that, in general, a linear description is not attainable without such formulations, while providing practical, linear-time procedures for the targeted graph classes.

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.

Bond Polytope under Vertex- and Edge-sums

TL;DR

The paper studies the Max-Bond problem, where a bond is a cut with both sides connected, and bonds correspond to circuits of the co-graphic matroid; while NP-hard in general, Max-Bond is solvable in linear time on -minor-free graphs and more broadly for bounded-treewidth graphs via clique-sum decompositions. It introduces a subdirect-sum framework for combining bond polytopes under - and -sums, and develops extended formulations that combine additively under these sum operations, yielding linear extension complexity for -minor-free graphs. A key contribution is an elementary linear-time algorithm for Max-Bond on wheel graphs, which replaces heavier treewidth machinery used in prior work, and together with the decomposition results, gives a linear-time Max-Bond algorithm for all -minor-free graphs. The work also clarifies when linear-size descriptions of the bond polytope are possible (via extension formulations) and shows that, in general, a linear description is not attainable without such formulations, while providing practical, linear-time procedures for the targeted graph classes.

Abstract

A cut in a graph is called a {\em bond} if both parts of the cut induce connected subgraphs in , 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 -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 - or -sum of graphs and from the bond polytopes of . Using this we show that the extension complexity of the bond polytope of -minor-free graphs is linear. Prior to this work, a linear size description of the bond polytope was known only for -connected planar -minor-free graphs, essentially only for wheel graphs. We also describe an elementary linear time algorithm for the \MaxBond problem on -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.
Paper Structure (10 sections, 14 theorems, 11 equations, 2 figures, 1 algorithm)

This paper contains 10 sections, 14 theorems, 11 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Other Related Results
  3. Preliminaries
  4. The Subdirect Sum of Polytopes
  5. The Bond Polytope
  6. Description of the Bond Polytope under $k$-sums $\oplus_{1}$ and $\oplus_{2}$
  7. Extension Complexity of Bond Polytope under $k$-sums $\oplus_{1}$, $\oplus_{2}$ and $\oplus_{2}^-$
  8. Extension Complexity of the Bond Polytope for $(K_5\setminus e)$-minor-free Graphs
  9. Linear Time Algorithm for $(K_5\setminus e)$-minor-free Graphs
  10. Finding the optimal bond from $\mathcal{F} _1$

Key Result

theorem 1

Each maximal $(K_5\setminus e)$-minor-free graph $G$ can be decomposed as $G=G_1 \oplus^1 \dots \oplus^{l-1} G_\ell$ where each $G_i$ is isomorphic to a wheel graph, $\textit{Prism}$, $K_2$, $K_3$, or $K_{3,3}$, and each operation $\oplus^i$ is $\oplus_1$ or $\oplus_2$.

Figures (2)

  • Figure 1: $\{1,2\}$-sum of graphs $G_1$ and $G_2$
  • Figure 2: The wheel graph $W_n$

Theorems & Definitions (23)

  • theorem 1: Satz 7, Wagner Wagner:60
  • theorem 2
  • theorem 3: Balas Balas:98, Theorem 2.1
  • lemma thmcounterlemma: Gluing lemma Margot_thesisKolmanKT:20
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 13 more