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Submodular functions and perfect graphs

Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Kristina Vušković

TL;DR

The paper addresses Maximum Weight Independent Set in perfect graphs of bounded degree that are free of prisms and 4-holes, producing a combinatorial polynomial-time algorithm. It introduces two main tools—even set separators and iterated decompositions with central bags—to structure the graph and reduce MWIS to submodular function minimization, avoiding reliance on ellipsoid-based SDP methods. A key development is showing that the core bag $\mathcal{R}$ is a $(\delta^2+2)$-iterated even set, built from a star-separator framework in paw-friendly graphs, which enables a recursive, combinatorial MWIS algorithm. The work culminates in establishing tameness for paw-friendly graphs with bounded degree, thereby providing a concrete, purely combinatorial polynomial-time method for MWIS in a substantial restricted class of perfect graphs and expanding the toolkit for graph decomposition-based optimization. This advances the understanding of how structural graph properties—via central bags, star separations, and even sets—can interface with submodular minimization to solve classic combinatorial problems efficiently.

Abstract

We give a combinatorial polynomial-time algorithm to find a maximum weight independent set in perfect graphs of bounded degree that do not contain a prism or a hole of length four as an induced subgraph. An even pair in a graph is a pair of vertices all induced paths between which are even. An even set is a set of vertices every two of which are an even pair. We show that every perfect graph that does not contain a prism or a hole of length four as an induced subgraph has a balanced separator which is the union of a bounded number of even sets, where the bound depends only on the maximum degree of the graph. This allows us to solve the maximum weight independent set problem using the well-known submodular function minimization algorithm.

Submodular functions and perfect graphs

TL;DR

The paper addresses Maximum Weight Independent Set in perfect graphs of bounded degree that are free of prisms and 4-holes, producing a combinatorial polynomial-time algorithm. It introduces two main tools—even set separators and iterated decompositions with central bags—to structure the graph and reduce MWIS to submodular function minimization, avoiding reliance on ellipsoid-based SDP methods. A key development is showing that the core bag is a -iterated even set, built from a star-separator framework in paw-friendly graphs, which enables a recursive, combinatorial MWIS algorithm. The work culminates in establishing tameness for paw-friendly graphs with bounded degree, thereby providing a concrete, purely combinatorial polynomial-time method for MWIS in a substantial restricted class of perfect graphs and expanding the toolkit for graph decomposition-based optimization. This advances the understanding of how structural graph properties—via central bags, star separations, and even sets—can interface with submodular minimization to solve classic combinatorial problems efficiently.

Abstract

We give a combinatorial polynomial-time algorithm to find a maximum weight independent set in perfect graphs of bounded degree that do not contain a prism or a hole of length four as an induced subgraph. An even pair in a graph is a pair of vertices all induced paths between which are even. An even set is a set of vertices every two of which are an even pair. We show that every perfect graph that does not contain a prism or a hole of length four as an induced subgraph has a balanced separator which is the union of a bounded number of even sets, where the bound depends only on the maximum degree of the graph. This allows us to solve the maximum weight independent set problem using the well-known submodular function minimization algorithm.

Paper Structure

This paper contains 12 sections, 44 theorems, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions and notation.
  3. Submodular functions and even set separators
  4. Iterated decompositions and their central bags
  5. Star separations
  6. Forcers for $(C_4$, prism)-free perfect graphs
  7. The star-free bag of paw-friendly graphs
  8. The core bag $\mathcal{R}$ is a $(\delta^2+2)$-iterated even set
  9. Even sets in $\beta$
  10. Even sets in $\mathcal{R} \setminus \beta$
  11. Even set separators and $(k, c, d, m)$-tame graphs
  12. Acknowledgments

Key Result

Lemma 1.1

There is a function $f: \mathbb{N} \times \mathbb{R} \rightarrow \mathbb{N}$ with the following property. Let $G$ be a graph with $|V(G)| = n$. Let $c \in [\frac{1}{2}, 1)$, let $k$ be a nonnegative integer, and suppose $G$ has a $(w, c)$-balanced separator of size at most $k$ for every uniform weig

Figures (4)

  • Figure 1: Illustrations of two separations $S_1 = (A_1, C_1, B_1)$ and $S_2 = (A_2, C_2, B_2)$ being (a) non-crossing and (b) loosely non-crossing.
  • Figure 2: Star twins $u$ and $v$, where $u$ and $v$ may or may not be adjacent
  • Figure 3: The paw graph
  • Figure 4: A drawing showing the central, core, and intermediate bags. Recall that $G \setminus \gamma = \bigcup_{x \in X_2} A_{x}$, $\gamma \setminus \beta \subseteq \bigcup_{x \in X_1} A_x$, $\mathcal{R} = \beta \cup \bigcup_{x \in X_2} C_{x}$, and $\beta = X_1 \cup X_2$.

Theorems & Definitions (78)

  • Lemma 1.1: HW
  • Theorem 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: O-Sbm
  • Lemma 2.2
  • proof
  • Theorem 2.2
  • ...and 68 more