- Algorithms for the power-$p$ Steiner tree problem in the Euclidean plane Christina Burt(christina.naomi.burtgmail.com) Alysson Costa(alysson.costaunimelb.edu.au) Charl Ras(cjrasunimelb.edu.au) Abstract: We study the problem of constructing minimum power-$p$ Euclidean $k$-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most $k$ additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of $p$ (where $p\geq 1$), and the cost of a network is the sum of all edge costs. We propose two heuristics: a beaded" minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio $\kappa$ of the beaded-MST heuristic satisfies $\sqrt{3}^{p-1}(1+2^{1-p})\leq \kappa\leq 3(2^{p-1})$. We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the $p=2$ case. Keywords: Euclidean Steiner Tree Problem, Mixed-integer Quadratically Constrained Program, Heuristics Category 1: Combinatorial Optimization (Other ) Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization ) Citation: Download: [PDF]Entry Submitted: 09/17/2015Entry Accepted: 09/17/2015Entry Last Modified: 09/17/2015Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.