# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

__Mittagsseminar Talk Information__ | |

**Date and Time**: Tuesday, March 01, 2016, 12:15 pm

**Duration**: 30 minutes

**Location**: OAT S15/S16/S17

**Speaker**: Martin Nägele

## Refuting a Conjecture of Goemans on Minimum Bounded Degree Spanning Trees

In the bounded degree spanning tree problem, we are given an undirected graph with edge costs and a degree bound for every vertex. The task is to find a spanning tree T whose degree at each vertex does not exceed the degree bound and T is of minimum cost among all such spanning trees. Even checking whether a feasible spanning tree exists is well known to be NP-hard. Thus, interest surged in understanding whether a small violation of the degree constraints may make it possible to efficiently obtain a spanning tree whose cost is not larger than the optimum spanning tree, which does not violate the degree constraints. In 2006, Goemans presented a nearly optimal algorithm, based on matroid intersection, which leads to a degree violation of at most 2 units. In 2007, Singh and Lau closed the gap by showing that iterative relaxation allows for obtaining the same result with degree violation of at most 1. Besides iterative relaxation, no other technique is known to lead to the same result. Interestingly, Goemans stated a conjecture which, if true, would imply that his matroid intersection approach would as well lead to a violation of at most 1 unit. In this work, we refute Goemans' conjecture, by refuting an even weaker version of it. (Joint work with Stephen Chestnut and Rico Zenklusen.)

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