1 Steiner Minimal Trees

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Steiner Minimal Trees^∗

Bang Ye Wu
Kun-Mao Chao

1 Steiner Minimal Trees

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1.1 Approximation by MST

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Problem Instance Goal

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^∗An excerpt from the book “Spanning Trees and Optimization Problems,” by Bang Ye Wu and Kun-Mao Chao
(2004), Chapman & Hall/CRC Press, USA.

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Bibliographic Notes and Further Reading

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References

Recommended publications

‣ Dijkstra's Algorithm ‣ Minimum Spanning Trees ‣ Prim, Kruskal, Boruvka ‣ Single-Link Clustering ‣ Min-Cost Arborescences

4. GREEDY ALGORITHMS II ‣ Dijkstra's algorithm ‣ minimum spanning trees ‣ Prim, Kruskal, Boruvka ‣ single-link clustering ‣ min-cost arborescences Lecture slides by Kevin Wayne Copyright © 2005 Pearson-Addison Wesley http://www.cs.princeton.edu/~wayne/kleinberg-tardos Last updated on Feb 18, 2013 6:08 AM 4. GREEDY ALGORITHMS II ‣ Dijkstra's algorithm ‣ minimum spanning trees ‣ Prim, Kruskal, Boruvka ‣ single-link clustering ‣ min-cost arborescences SECTION 4.4 Shortest-paths problem Problem. Given a digraph G = (V, E), edge weights ℓe ≥ 0, source s ∈ V, and destination t ∈ V, find the shortest directed path from s to t. 1 15 3 5 4 12 source s 0 3 8 7 7 2 9 9 6 1 11 5 5 4 13 4 20 6 destination t length of path = 9 + 4 + 1 + 11 = 25 3 Car navigation 4 Shortest path applications ・PERT/CPM. ・Map routing. ・Seam carving. ・Robot navigation. ・Texture mapping. ・Typesetting in LaTeX. ・Urban traffic planning. ・Telemarketer operator scheduling. ・Routing of telecommunications messages. ・Network routing protocols (OSPF, BGP, RIP). ・Optimal truck routing through given traffic congestion pattern. Reference: Network Flows: Theory, Algorithms, and Applications, R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Prentice Hall, 1993. 5 Dijkstra's algorithm Greedy approach. Maintain a set of explored nodes S for which algorithm has determined the shortest path distance d(u) from s to u. ・Initialize S = { s }, d(s) = 0. ・Repeatedly choose unexplored node v which minimizes shortest path to some node u in explored part, followed by a single edge (u, v) ℓe v d(u) u S s 6 Dijkstra's algorithm Greedy approach.
Converting MST to TSP Path by Branch Elimination

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View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Applied Mathematics 39 (1992) 147- 153 147 North-Holland Minimum perfect bipartite matchings and spanning trees under categorization Michael B. Richey Operations Research & Applied Statistics Department, George Mason University, Fairfax, VA 22030, USA Abraham P. Punnen Mathematics Department, Indian Institute of Technology. Kanpur-208 016, India Received 2 February 1989 Revised 1 May 1990 Abstract Richey, M.B. and A.P. Punnen, Minimum perfect bipartite mats-Yngs and spanning trees under catego- rization, Discrete Applied Mathematics 39 (1992) 147-153. Network optimization problems under categorization arise when the edge set is partitioned into several sets and the objective function is optimized over each set in one sense, then among the sets in some other sense. Here we find that unlike some other problems under categorization, the minimum perfect bipar- tite matching and minimum spanning tree problems (under categorization) are NP-complete, even on bipartite outerplanar graphs. However for one objective function both problems can be solved in polynomial time on arbitrary graphs if the number of sets is fixed. 1. Introduction In operations research and in graph-theoretic optimization, two types of objective functions have dominated the attention of researchers, namely optimization of the weighted sum of the variables and optimization of the most extreme value of the variables. Recently the following type of (minimization) objective function has appeared in the literature [1,6,7,14]: divide the variables into categories, find the Correspondence to: Professor M.B. Richey, Operations Research & Applied Statistics Department, George Mason University, Fairfax, VA 22030, USA.
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‣ Dijkstra′S Algorithm ‣ Minimum Spanning Trees ‣ Prim, Kruskal, Boruvka ‣ Single-Link Clustering ‣ Min-Cost Arborescences

4. GREEDY ALGORITHMS II ‣ Dijkstra′s algorithm ‣ minimum spanning trees ‣ Prim, Kruskal, Boruvka ‣ single-link clustering ‣ min-cost arborescences Lecture slides by Kevin Wayne Copyright © 2005 Pearson-Addison Wesley http://www.cs.princeton.edu/~wayne/kleinberg-tardos Last updated on 2/25/20 4:16 PM 4. GREEDY ALGORITHMS II ‣ Dijkstra′s algorithm ‣ minimum spanning trees ‣ Prim, Kruskal, Boruvka ‣ single-link clustering ‣ min-cost arborescences SECTION 4.4 Single-pair shortest path problem Problem. Given a digraph G = (V, E), edge lengths ℓe ≥ 0, source s ∈ V, and destination t ∈ V, find a shortest directed path from s to t. 1 15 3 5 4 12 source s 0 3 8 7 7 2 9 9 6 1 11 5 5 4 13 4 20 6 destination t length of path = 9 + 4 + 1 + 11 = 25 3 Single-source shortest paths problem Problem. Given a digraph G = (V, E), edge lengths ℓe ≥ 0, source s ∈ V, find a shortest directed path from s to every node. 1 15 3 5 4 12 source s 0 3 8 7 7 2 9 9 6 1 11 5 5 4 13 4 20 6 shortest-paths tree 4 Shortest paths: quiz 1 Suppose that you change the length of every edge of G as follows. For which is every shortest path in G a shortest path in G′? A. Add 17. B. Multiply by 17. C. Either A or B. D. Neither A nor B. 24 —7 s —1 —2 3— t 18 19 20 5 Shortest paths: quiz 2 Which variant in car GPS? A.
$Arxiv:1710.09672V1 [Math.CO] 26 Oct 2017 Minimum Spanning Tree (MST)$

Arxiv:1710.09672V1 [Math.CO] 26 Oct 2017 Minimum Spanning Tree (MST)

1-SKELETONS OF THE SPANNING TREE PROBLEMS WITH ADDITIONAL CONSTRAINTS VLADIMIR BONDARENKO, ANDREI NIKOLAEV, DZHAMBOLET SHOVGENOV Abstract. We consider the polyhedral properties of two spanning tree problems with addi- tional constraints. In the first problem, it is required to find a tree with a minimum sum of edge weights among all spanning trees with the number of leaves less or equal a given value. In the second problem, an additional constraint is the assumption that the degree of all vertices of the spanning tree does not exceed a given value. The decision versions of both problems are NP-complete. We consider the polytopes of these problems and their 1-skeletons. We prove that in both cases it is a NP-complete problem to determine whether the vertices of 1-skeleton are adjacent. Although it is possible to obtain a superpolynomial lower bounds on the clique numbers of these graphs. These values characterize the time complexity in a broad class of algorithms based on linear comparisons. The results indicate a fundamental difference in combinatorial and geometric properties between the considered problems and the classical minimum spanning tree problem. 1. Introduction A significant number of works related to the computational complexity of combinatorial prob- lems is aimed at the study of geometric objects associated with problems. Usually such objects are the polytopes of the problems and their 1-skeletons. In particular, the clique number of the 1-skeleton (the size of the maximum clique) of the problem serves as a lower bound on compu- tational complexity in a broad class of algorithms based on linear comparisons.
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Combinatorial Characterisations of accessible Graphical Computational Search Problems: P= 6 NP. Kayibi Koko-Kalambay School of Mathematics. Department of Sciences University of Bristol Abstract Let G be a connected graph with edge-set E and vertex-set V . Given an input X, where X is the edge-set or the vertex-set of the graph G, a graphical computational search problem associated with a predicate γ con- sists of finding a solution Y , where Y ⊆ X, and Y satisfies the condition γ. We denote such a problem as Π(X; γ), and we let Π^ to denote the decision problem associated with Π(X; γ). A sub-solution of Π(X; γ) is a subset Y 0 that is a solution of the problem Π(X0; γ), where X0 ⊂ X, and X0 is the edge-set or the vertex-set of the contraction-minor G=A, where A ⊆ E. To each search problem Π(X; γ), we associate the set system (X; I), where I denotes the set of all the solutions and sub-solutions of Π(X; γ). The predicate γ is an accessible predicate if, given Y 6= ;, Y satisfies γ in X implies that there is an element y 2 Y such that Y n y satisfies γ in X0. If γ is an accessible property, we then show in Theorem 1 that a decision problem Π^ is in P if and only if, for all input X, the set system (X; I) satisfies Axioms M2', where M2', called the Augmentability Axiom, is an extension of both the Exchange Axiom and the Accessibility Axiom of a greedoid.
Minimum Spanning Trees

Minimum Spanning Trees • G = (V; E) is an undirected graph with non-negative edge weights w : E ! Z+ • We assume wlog that edge weights are distinct • A spanning tree is a tree with V − 1 edges, i.e. a tree that connects all the vertices. P • The total cost (weight) of a spanning tree T is deﬁned as e2T w(e) • A minimum spanning tree is a tree of minimum total weight. 6 4 5 9 14 10 2 3 15 8 Minimum Spanning Trees • G = (V; E) is an undirected graph with non-negative edge weights w : E ! Z+ • We assume wlog that edge weights are distinct • A spanning tree is a tree with V − 1 edges, i.e. a tree that connects all the vertices. P • The total cost (weight) of a spanning tree T is deﬁned as e2T w(e) • A minimum spanning tree is a tree of minimum total weight. 6 4 5 9 14 10 2 3 15 8 Cuts • A cut in a graph is a partition of the vertices into two sets S and T . • An edge (u; v) with u 2 S and v 2 T is said to cross the cut. 6 4 5 9 14 10 2 3 15 8 Cuts • A cut in a graph is a partition of the vertices into two sets S and T . • An edge (u; v) with u 2 S and v 2 T is said to cross the cut. 6 4 5 9 14 10 2 3 15 8 Cuts • A cut in a graph is a partition of the vertices into two sets S and T .