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A Polynomial Time Algorithm for Steiner Tree when Terminals Avoid a $K_4$-Minor

Carla Groenland, Jesper Nederlof, Tomohiro Koana

Abstract

We study a special case of the Steiner Tree problem in which the input graph does not have a minor model of a complete graph on 4 vertices for which all branch sets contain a terminal. We show that this problem can be solved in $O(n^4)$ time, where $n$ denotes the number of vertices in the input graph. This generalizes a seminal paper by Erickson et al. [Math. Oper. Res., 1987] that solves Steiner tree on planar graphs with all terminals on one face in polynomial time.

A Polynomial Time Algorithm for Steiner Tree when Terminals Avoid a $K_4$-Minor

Abstract

We study a special case of the Steiner Tree problem in which the input graph does not have a minor model of a complete graph on 4 vertices for which all branch sets contain a terminal. We show that this problem can be solved in time, where denotes the number of vertices in the input graph. This generalizes a seminal paper by Erickson et al. [Math. Oper. Res., 1987] that solves Steiner tree on planar graphs with all terminals on one face in polynomial time.

Paper Structure

This paper contains 8 sections, 4 theorems, 3 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finding a cycle passing through terminals and virtual edges
  4. Description of algorithm
  5. Virtual edges
  6. Dynamic programming when roots lie on a cycle
  7. Preprocessing
  8. Our algorithm

Key Result

Theorem 1

Steiner Tree without $K_4$ minor rooted at terminals can be solved in $O(n^4)$ time.

Figures (1)

  • Figure 1: Terminals are depicted in red. The left figure has $3$ terminals and hence no rooted $K_4$ minor, but it is a $K_5$ and hence not planar. The right figure has no rooted $K_4$ minor (since the middle vertex and two non-adjacent terminals separate the remaining terminals), and has many $K_5$ subgraphs and hence is not planar.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 3: Lemma 7 in Fabila-MonroyAndWood
  • Lemma 4
  • Lemma 6