Probabilistische analyse von heuristiken der kombinatorischen optimierung – ein überbllck

To various problems of combinatorial optimization we consider the question how the value of the optimal solution resp. the values of some approximative solutions are predetermined with high probability to a given distribution. We present results to probabilistic analysis of heuristics. We consider the problems Traveling Salesman, Minimum Perfect Matching. Minimum Spanning Tree, Linear Optimization, Bin Packing, Multi-processor-Scheduling, Subset Sum and some problems to random graphs.     

Minimum Sum Coloring of P4-sparse graphs

In this paper, we study the Minimum Sum Coloring (MSC) problem on P₄-sparse graphs. In the MSC problem, we aim to assign natural numbers to vertices of a graph such that adjacent vertices get different numbers, and the sum of the numbers assigned to the vertices is minimum. First, we introduce the concept of maximal sequence associated with an optimal solution of the MSC problem of any graph. Next, based in such maximal sequences, we show that there is a large sub-family of P₄-sparse graphs for which the MSC problem can be solved in polynomial-time.     

Optimization problems in multiple subtree graphs

We study various optimization problems in t-subtree graphs, the intersection graphs of t-subtrees, where a t-subtree is the union of t disjoint subtrees of some tree. This graph class generalizes both the class of chordal graphs and the class of t-interval graphs, a generalization of interval graphs that has recently been studied from a combinatorial optimization point of view. We present approximation algorithms for the Maximum Independent Set, Minimum Coloring, Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique problems.     

Coloring Fuzzy Circular Interval Graphs

Computing the weighted coloring number of graphs is a classical topic in combinatorics and graph theory. Recently these problems have again attracted a lot of attention for the class of quasi-line graphs and more specifically fuzzy circular interval graphs.The problem is NP-complete for quasi-line graphs. For the subclass of fuzzy circular interval graphs however, one can compute the weighted coloring number in polynomial time using recent results of Chudnovsky and Ovetsky and of King and Reed. Whether one could actually compute an optimal weighted coloring of a fuzzy circular interval graph in polynomial time however was still open.We provide a combinatorial algorithm that computes weighted colorings and the weighted coloring number for fuzzy circular interval graphs efficiently. The algorithm reduces the problem to the case of circular interval graphs, then making use of an algorithm by Gijswijt to compute integer decompositions.     

The Complexity of Minimum Ratio Spanning Tree Problems

We examine the complexity of two minimum spanning tree problems with rational objective functions. We show that the Minimum Ratio Spanning Tree problem is NP-hard when the denominator is unrestricted in sign, thereby sharpening a previous complexity result. We then consider an extension of this problem where the objective function is the sum of two linear ratios whose numerators and denominators are strictly positive. This problem is shown to be NP-hard as well. We conclude with some results characterizing sufficient conditions for a globally optimal solution.     

(Some of) the many uses of Eulerian graphs in graph theory (plus some applications)

In this survey type article, various connections between eulerian graphs and other graph properties such as being hamiltonian, nowhere-zero flows, the cycle-plus-triangles problem and problems derived from it, are demonstrated. It is also shown how compatible cycle decompositions can be used to construct loopless 4-regular graphs having precisely one hamiltonian cycle, or to prove the equivalence between the Chinese Postman Problem and the Minimum Cycle Covering Problem in the planar bridgeless case.     

Node-searching problem on block graphs

The node-searching problem, introduced by Kirousis and Papadimitriou, is equivalent to several important problems, such as the interval thickness problem, the path-width problem, the vertex separation problem, and so on. In this paper, we generalize the avenue concept, originally proposed for trees, to block graphs whereby we design an efficient algorithm for computing both the search numbers and optimal search strategies for block graphs. It answers the question proposed by Peng et al. of whether the node-searching problem on block graphs can be solved in polynomial time.     

New Models of the Generalized Minimum Spanning Tree Problem

We consider a generalization of the Minimum Spanning Tree Problem, called the Generalized Minimum Spanning Tree Problem, denoted by GMST. It is known that the GMST problem is NP-hard. We present a stronger result regarding its complexity, namely, the GMST problem is NP-hard even on trees as well an exact exponential time algorithm for the problem based on dynamic programming. We describe new mixed integer programming models of the GMST problem, mainly containing a polynomial number of constraints. We establish relationships between the polytopes corresponding to their linear relaxations. Based on a new model of the GMST we present a solution procedure that solves the problem to optimality for graphs with nodes up to 240. We discuss the advantages of our method in comparison with earlier methods.     

单体型装配问题及其算法

单核苷酸多态性(SNP)单体型装配问题就是从给定的来自某人染色体的SNP片段中去除错误,重构出尽可能与原来片段一致的单体型.这个问题有几个不同的模型最少片段去除(MFR)问题,最少SNP去除(MSR)问题以及最少错误纠正(MEC)问题.前两个问题的复杂性与算法已有一些学者研究过.第三个问题已被证明是NP完全问题,但这个问题的实际算法还没有.该文对MEC问题给出了一个分支定界算法,这个算法能得到问题的全局最优解.通过这个算法对实际数据的计算说明了MEC模型的合理性,即在一定条件下,通过修正最少的错误重构出的单体型确实是真实的单体型.由于分支定界算法对这样一个NP完全问题不能在可接受的时间内解规模较大的问题,文中又给出了求解MEC问题的两个基于动态聚类的算法,以便对规模较大的问题在可接受的时间内得到近似最优解.数值实际表明这两个算法很快,很有效.这两个算法总能得到与分支定界找到的全局最优解很接近的近似最优解.鉴于MEC问题是NP完全的,这两个算法是有效的、实际的算法.     

Complexity results for minimum sum edge coloring

In the Minimum Sum Edge Coloring problem we have to assign positive integers to the edges of a graph such that adjacent edges receive different integers and the sum of the assigned numbers is minimal. We show that the problem is (a) NP-hard for planar bipartite graphs with maximum degree 3, (b) NP-hard for 3-regular planar graphs, (c) NP-hard for partial 2-trees, and (d) APX-hard for bipartite graphs.     

Block-iterative methods for consistent and inconsistent linear equations

Summary We shall in this paper consider the problem of computing a generalized solution of a given linear system of equations. The matrix will be partitioned by blocks of rows or blocks of columns. The generalized inverses of the blocks are then used as data to Jacobi- and SOR-types of iterative schemes. It is shown that the methods based on partitioning by rows converge towards the minimum norm solution of a consistent linear system. The column methods converge towards a least squares solution of a given system. For the case with two blocks explicit expressions for the optimal values of the iteration parameters are obtained. Finally an application is given to the linear system that arises from reconstruction of a two-dimensional object by its one-dimensional projections.     

基于模拟植物生长算法构造Steiner最优树问题研究

Steiner最优树问题是指对于给定区域内的点集,通过引入Steiner点集将区域中的点连接并保证连通的网络达到最小.该问题已成为经典的优化组合问题之一.提出一种基于模拟植物生长算法生成Steiner最优树的连通算法来实现网络连通.通过对实例的实验及结果分析,结果表明本算法不仅可获得最优解,精度和性能也有提高,明显优于其它方法.     

Crown reductions for the Minimum Weighted Vertex Cover problem

The paper studies crown reductions for the Minimum Weighted Vertex Cover problem introduced recently in the unweighted case by Fellows et al. [Blow-Ups, Win/Win's and crown rules: some new directions in FPT, in: Proceedings of the 29th International Workshop on Graph Theoretic Concepts in Computer Science (WG’03), Lecture notes in computer science, vol. 2880, 2003, pp. 1-12, Kernelization algorithms for the vertex cover problem: theory and experiments, in: Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX), New Orleans, Louisiana, January 2004, pp. 62-69]. We describe in detail a close relation of crown reductions to Nemhauser and Trotter reductions that are based on the linear programming relaxation of the problem. We introduce and study the so-called strong crown reductions, suitable for finding (or counting) all minimum vertex covers, or finding a minimum vertex cover under some additional constraints. It is described how crown decompositions and strong crown decompositions suitable for such problems can be computed in polynomial time. For weighted König-Egerváry graphs (G,w) we observe that the set of vertices belonging to all minimum vertex covers, and the set of vertices belonging to no minimum vertex covers, can be efficiently computed.Further, for some specific classes of graphs, simple algorithms for the MIN-VC problem with a constant approximation factor r<2 are provided. On the other hand, we conclude that for the regular graphs, or for the Hamiltonian connected graphs, the problem is as hard to approximate as for general graphs.It is demonstrated how the results about strong crown reductions can be used to achieve a linear size problem kernel for some related vertex cover problems.     

Minimum Dominating Trail Set for Two-Terminal Series Parallel Graphs

Given a graph G, the Minimum Dominating Trail Set (MDTS) problem consists in finding a minimum cardinality collection of pairwise edge-disjoint trails such that each edge of G has at least one endvertex on some trail. The MDTS problem is NP–hard for general graphs. In this paper an algorithmic approach for the MDTS problem on (two-terminal) series parallel graphs is presented.     

A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems

We study a class of time-domain decomposition-based methods for the numerical solution of large-scale linear quadratic optimal control problems. Our methods are based on a multiple shooting reformulation of the linear quadratic optimal control problem as a discrete-time optimal control (DTOC) problem. The optimality conditions for this DTOC problem lead to a linear block tridiagonal system. The diagonal blocks are invertible and are related to the original linear quadratic optimal control problem restricted to smaller time-subintervals. This motivates the application of block Gauss–Seidel (GS)-type methods for the solution of the block tridiagonal systems. Numerical experiments show that the spectral radii of the block GS iteration matrices are larger than one for typical applications, but that the eigenvalues of the iteration matrices decay to zero fast. Hence, while the GS method is not expected to convergence for typical applications, it can be effective as a preconditioner for Krylov-subspace methods. This is confirmed by our numerical tests.A byproduct of this research is the insight that certain instantaneous control techniques can be viewed as the application of one step of the forward block GS method applied to the DTOC optimality system.     

Finding Postal Carrier Walk Paths in Mixed Graphs

The postman problem requires finding a lowest cost tour in a connected graph that traverses each edge at least once. In this paper we first give a brief survey of the literature on postman problems including, the original Chinese postman problem on undirected graphs, the windy Chinese postman problem on graphs where the cost of an arc depends on the direction the arc is transversed, the directed postman problem on graphs with directed edges, and the mixed postman problem on graphs in which there are some directed and some undirected arcs.We show how the mixed postman problem can be solved as an integer program, using the formulation of Gendreau, Laporte and Zhao, by a new row addition branch and bound algorithm, which is a modification of the column subtraction algorithm for set partitioning problems of Harche and Thompson. Computational experience shows that a slack variable heuristic is very effective in finding good solutions that are frequently optimal for these problems.     

Approximating the minimum clique cover and other hard problems in subtree filament graphs

Subtree filament graphs are the intersection graphs of subtree filaments in a tree. This class of graphs contains subtree overlap graphs, interval filament graphs, chordal graphs, circle graphs, circular-arc graphs, cocomparability graphs, and polygon-circle graphs. In this paper we show that, for circle graphs, the clique cover problem is NP-complete and the h-clique cover problem for fixed h is solvable in polynomial time. We then present a general scheme for developing approximation algorithms for subtree filament graphs, and give approximation algorithms developed from the scheme for the following problems which are NP-complete on circle graphs and therefore on subtree filament graphs: clique cover, vertex colouring, maximum k-colourable subgraph, and maximum h-coverable subgraph.     

Minimum cycle bases of graphs on surfaces

In this paper we study the cycle base structures of embedded graphs on surfaces. We first give a sufficient and necessary condition for a set of facial cycles to be contained in a minimum cycle base (or MCB in short) and then set up a 1-1 correspondence between the set of MCBs and the set of collections of nonseparating cycles which are in general positions on surfaces and are of shortest total length. This provides a way to enumerate MCBs in a graph via nonseparating cycles. In particular, some known results such as P.F. Stadler's work on Halin graphs [Minimum cycle bases of Halin graphs, J. Graph Theory 43 (2003) 150-155] and Leydold and Stadler's results on outer-planar graphs [Minimum cycle bases of outerplanar graphs, Electronic J. Combin. 5(16) (1998) 14] are concluded. As applications, the number of MCBs in some types of graphs embedded in lower surfaces (with arbitrarily high genera) is found. Finally, we present an interpolation theorem for the number of one-sided cycles contained in MCB of an embedded graph.     

关于图的运算与图的Pfaffian性

关于一般的图的完美匹配计数的问题已证实是NP-hard问题.但Pfaffian图的完美匹配计数问题(以及其它相关问题)却能够在多项式时间内解决.由此可见图的Pfaffian性的重要性.在这篇文章中,我们研究了若干种影响图的Pfaffian性的运算.     

Telecommunication Link Restoration Planning with Multiple Facility Types

To ensure uninterrupted service, telecommunication networks contain excess (spare) capacity for rerouting (restoring) traffic in the event of a link failure. We study the NP-hard capacity planning problem of economically installing spare capacity on a network to permit link restoration of steady-state traffic. We present a planning model that incorporates multiple facility types, and develop optimization-based heuristic solution methods based on solving a linear programming relaxation and minimum cost network flow subproblems. We establish bounds on the performance of the algorithms, and discuss problem instances that nearly achieve these worst-case bounds. In tests on three real-world problems and numerous randomly-generated problems containing up to 50 nodes and 150 edges, the heuristics provide good solutions (often within 0.5% of optimality) to problems with single facility type, in equivalent or less time than methods from the literature. For multi-facility problems, the gap between our heuristic solution values and the linear programming bounds are larger. However, for small graphs, we show that the optimal linear programming value does not provide a tight bound on the optimal integer value, and our heuristic solutions are closer to optimality than implied by the gaps.