Minimum-weight Spanning Tree Problem Research Articles

Sorting can exponentially speed up pure dynamic programming

Many discrete minimization problems, including various versions of the shortest path problem, can be efficiently solved by dynamic programming (DP) algorithms that are “pure” in that they only perform basic operations, as min, max, +, but no conditional branchings via if-then-else in their recursion equations. It is known that any pure (min⁡,+) DP algorithm solving the minimum weight spanning tree problem on undirected n-vertex graphs must perform at least 2Ω(n) operations. We show that this problem can be solved by a pure (min⁡,max⁡,+) DP algorithm performing only O(n3) operations. The algorithm is essentially a (min⁡,max⁡) algorithm: addition operations are only used to output the final values. The presence of both min and max operations means that now DP algorithms can sort: this explains the title of the paper.

Greedy can beat pure dynamic programming

Many dynamic programming algorithms for discrete 0–1 optimization problems are “pure” in that their recursion equations only use min/max and addition operations, and do not depend on actual input weights. The well-known greedy algorithm of Kruskal solves the minimum weight spanning tree problem on n-vertex graphs using only O(n2log⁡n) operations. We prove that any pure DP algorithm for this problem must perform 2Ω(n) operations. Since the greedy algorithm can also badly fail on some optimization problems, easily solvable by pure DP algorithms, our result shows that the computational powers of these two types of algorithms are incomparable.

An algebraic framework for minimum spanning tree problems

We formally prove that Prim's minimum spanning tree algorithm is correct for various optimisation problems with different aggregation functions. The original minimum weight spanning tree problem and the minimum bottleneck spanning tree problem are special cases, but the framework covers many other problems. To this end we work in new generalisations of relation algebras and Kleene algebras and in new algebraic structures that capture key operations used in Prim's algorithm and its specification. Weighted graphs form instances of these algebraic structures, where edge weights are taken from a linear order with a binary aggregation operation. Many existing results from relation algebras and Kleene algebras generalise from the relation model to the weighted-graph model with no or small changes. The overall structure of the proof uses Hoare logic. All results are formally verified in Isabelle/HOL heavily using its integrated automated theorem provers. This paper is an extended version of [36].

Optimality conditions in preference-based spanning tree problems

Abstract Spanning tree problems defined in a preference-based environment are addressed. In this approach, optimality conditions for the minimum-weight spanning tree problem (MST) are generalized for use with other, more general preference orders. The main goal of this paper is to determine which properties of the preference relations are sufficient to assure that the set of ‘most-preferred’ trees is the set of spanning trees verifying the optimality conditions. Finally, algorithms for the construction of the set of spanning trees fulfilling the optimality conditions are designed, improving the methods in previous papers.

New metaheuristic approaches for the edge-weighted k-cardinality tree problem

In this paper we propose three metaheuristic approaches, namely a Tabu Search, an Evolutionary Computation and an Ant Colony Optimization approach, for the edge-weighted k-cardinality tree (KCT) problem. This problem is an NP-hard combinatorial optimization problem that generalizes the well-known minimum weight spanning tree problem. Given an edge-weighted graph G=( V, E), it consists of finding a tree in G with exactly k⩽| V|−1 edges, such that the sum of the weights is minimal. First, we show that our new metaheuristic approaches are competitive by applying them to a set of existing benchmark instances and comparing the results to two different Tabu Search methods from the literature. The results show that these benchmark instances are not challenging enough for our metaheuristics. Therefore, we propose a diverse set of benchmark instances that are characterized by different features such as density and variance in vertex degree. We show that the performance of our metaheuristics depends on the characteristics of the tackled instance, as well as on the cardinality. For example, for low cardinalities the Ant Colony Optimization approach is best, whereas for high cardinalities the Tabu Search approach has advantages.

Minimum-weight spanning tree algorithms A survey and empirical study

The minimum-weight spanning tree problem is one of the most typical and well-known problems of combinatorial optimisation. Efficient solution techniques had been known for many years. However, in the last two decades asymptotically faster algorithms have been invented. Each new algorithm brought the time bound one step closer to linearity and finally Karger, Klein and Tarjan proposed the only known expected linear-time method. Modern algorithms make use of more advanced data structures and appear to be more complicated to implement. Most authors and practitioners refer to these but still use the classical ones, which are considerably simpler but asymptotically slower. The paper first presents a survey of the classical methods and the more recent algorithmic developments. Modern algorithms are then compared with the classical ones and their relative performance is evaluated through extensive empirical tests, using reasonably large-size problem instances. Randomly generated problem instances used in the tests range from small networks having 512 nodes and 1024 edges to quite large ones with 16 384 nodes and 524 288 edges. The purpose of the comparative study is to investigate the conjecture that modern algorithms are also easy to apply and have constants of proportionality small enough to make them competitive in practice with the older ones. Scope and purpose The minimum-weight spanning tree (MST) problem is a well-known combinatorial optimisation problem concerned with finding a spanning tree of an undirected, connected graph, such that the sum of the weights of the selected edges is minimum. The importance of this problem derives from its direct applications in the design of computer, communication, transportation, power and piping networks; from its appearance as part of solution methods to other problems to which it applies less directly such as network reliability, clustering and classification problems and from its occurrence as a subproblem in the solution of other problems like the travelling salesman problem, the multi-terminal flow problem, the matching problem and the capacitated MST problem. Although efficient solution techniques capable of solving large instances had existed, there has been sustained effort over the last two decades to invent asymptotically faster algorithms. With each new algorithm found the time bound approached linearity. Finally, an expected linear-time method has been proposed. The purpose of this work is to survey the classical and modern solution techniques and empirically compare the performance of the existing methods.

Optimal vertex ordering of graphs

In this paper we consider an optimal vertex ordering problem of graphs. The vertex ordering and an optimality measure are defined. It is proved that the optimal ordering problem under consideration can be transformed into the well-known minimum-weight spanning tree problem and is therefore solvable in low-polynomial time. We also investigate some properties of optimal vertex orderings.