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Given a set of points in the Euclidean plane, the Euclidean textit{$delta$-minimum spanning tree} ($delta$-MST) problem is the problem of finding a spanning tree with maximum degree no more than $delta$ for the set of points such the sum of the total length of its edges is minimum. Similarly, the Euclidean textit{$delta$-minimum bottleneck spanning tree} ($delta$-MBST) problem, is the problem of finding a degree-bounded spanning tree for a set of points in the plane such that the length of the longest edge is minimum. When $delta leq 4$, these two problems may yield disjoint sets of optimal solutions for the same set of points. In this paper, we perform computational experiments to compare the accuracies of a variety of heuristic and approximation algorithms for both these problems. We develop heuristics for these problems and compare them with existing algorithms. We also describe a new type of edge swap algorithm for these problems that outperforms all the algorithms we tested.
The minimum degree spanning tree (MDST) problem requires the construction of a spanning tree $T$ for graph $G=(V,E)$ with $n$ vertices, such that the maximum degree $d$ of $T$ is the smallest among all spanning trees of $G$. In this paper, we present
The geometric $delta$-minimum spanning tree problem ($delta$-MST) is the problem of finding a minimum spanning tree for a set of points in a normed vector space, such that no vertex in the tree has a degree which exceeds $delta$, and the sum of the l
We consider the task of approximating the ground state energy of two-local quantum Hamiltonians on bounded-degree graphs. Most existing algorithms optimize the energy over the set of product states. Here we describe a family of shallow quantum circui
In the Priority Steiner Tree (PST) problem, we are given an undirected graph $G=(V,E)$ with a source $s in V$ and terminals $T subseteq V setminus {s}$, where each terminal $v in T$ requires a nonnegative priority $P(v)$. The goal is to compute a min
This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all $k