Algebra complexity problems involving graph homomorphism,semigroups and the constraint satisfaction problem

In this paper, we connect the constraint satisfaction problem with other complexity problems, like the polynomial equivalence problem for combinatorial 0-simple semigroups, the graph retraction problem and the geometry problem. We show that every constraint satisfaction problem is polynomially equivalent to an easily formulated algebra complexity problem. As an application we prove that the polynomial equivalence problem (word problem) for the 2×2 matrices over the two element field is co-NP-complete.     

Toward a language theoretic proof of the four color theorem

This paper considers the problem of showing that every pair of binary trees with the same number of leaves parses a common word under a certain simple grammar. We enumerate the common parse words for several infinite families of tree pairs and discuss several ways to reduce the problem of finding a parse word for a pair of trees to that for a smaller pair. The statement that every pair of trees has a common parse word is equivalent to the statement that every planar graph is four-colorable, so the results are a step toward a language theoretic proof of the four color theorem.     

The location of median paths on grid graphs

In this paper we consider the location of a path shaped facility on a grid graph. In the literature this problem was extensively studied on particular classes of graphs as trees or series-parallel graphs. We consider here the problem of finding a path which minimizes the sum of the (shortest) distances from it to the other vertices of the grid, where the path is also subject to an additional constraint that takes the form either of the length of the path or of the cardinality. We study the complexity of these problems and we find two polynomial time algorithms for two special cases, with time complexity of O(n) and O(nℓ) respectively, where n is the number of vertices of the grid and ℓ is the cardinality of the path to be located. The literature about locating dimensional facilities distinguishes between the location of extensive facilities in continuous spaces and network facility location. We will show that the problems presented here have a close connection with continuous dimensional facility problems, so that the procedures provided can also be useful for solving some open problems of dimensional facilities location in the continuous case.     

Neural and delay based heuristics for the Steiner problem in networks

High speed networks such as the B-ISDN must be adequately equipped to handle multipoint communication in a fast and economical manner. Multicast applications include desktop video conferencing, distance learning, distributed database applications, etc. In networks employing the asynchronous transfer mode (ATM) technology, routing a multicast is achieved by constructing a tree that spans the source and all the destinations. For the purpose of routing, the network is modeled as a weighted, undirected graph. The graph-theoretic solution is to find a minimum Steiner tree for the graph given a set of destinations. This formulation suffices for building multicast trees with a single optimization constraint as would be the xcase for best effort transport. For real-time traffic, however, it is necessary to ensure that the delay between the sender and each of the receivers is bounded. In this case the network is modeled as an undirected graph, where the edges have both a cost and a delay associated with them. The graph-theoretic solution is then to find a constrained minimum Steiner tree such that the delay between the source and each of the destinations does not violate the specified bound. Both of these problems are NP-complete. In this paper we review prior work on the multipoint routing problem and discuss the formulation of the unconstrained and constrained Steiner problems. We use the random neural network (RNN) to significantly improve the quality of trees found by the two existing best heuristics for finding Steiner trees - the minimum spanning tree heuristic and the average distance heuristic. We also develop a new heuristic for finding delay constrained Steiner trees. Experimental results are presented which show that the new heuristics improve significantly over existing ones.     

递归树的若干枚举特征

递归树由Meir和Moon定义作非平面增长树的一种，且所有节点出度都是允许的．本文首先在n个节点的递归树集合和n-1个元素的排列之间建立一个新的──对应，这个对应能同时给出树叶子和排列中的路段之间的对应和树叶子数和排列中的路段数之间的密切关系．同时还研究递归树的各种枚举特征，诸如节点的分类枚举（内节点和叶子节点、偶节点和奇节点，具不同出度的节点）和通路长度枚举（接各种节点分类）．     

An exact algorithm with linear complexity for a problem of visiting megalopolises

A problem of visiting megalopolises with a fixed number of “entrances” and precedence relations defined in a special way is studied. The problem is a natural generalization of the classical traveling salesman problem. For finding an optimal solution, we give a dynamic programming scheme, which is equivalent to a method of finding a shortest path in an appropriate acyclic oriented weighted graph. We justify conditions under which the complexity of the algorithm depends on the number of megalopolises polynomially, in particular, linearly.     

Finding minimum height elimination trees for interval graphs in polynomial time

The elimination tree plays an important role in many aspects of sparse matrix factorization. The height of the elimination tree presents a rough, but usually effective, measure of the time needed to perform parallel elimination. Finding orderings that produce low elimination is therefore important. As the problem of finding minimum height elimination tree orderings is NP-hard, it is interesting to concentrate on limited classes of graphs and find minimum height elimination trees for these efficiently. In this paper, we use clique trees to find an efficient algorithm for interval graphs which make an important subclass of chordal graphs. We first illustrate this method through an algorithm that finds minimum height elimination for chordal graphs. This algorithm, although of exponential time complexity, is conceptionally simple and leads to a polynomial-time algorithm for finding minimum height elimination trees for interval graphs.This work was supported by grants from the Norwegian Research Council.     

Tree-Structured Mixed-Effects Regression Modeling for Longitudinal Data

Tree-structured models have been widely used because they function as interpretable prediction models that offer easy data visualization. A number of tree algorithms have been developed for univariate response data and can be extended to analyze multivariate response data. We propose a tree algorithm by combining the merits of a tree-based model and a mixed-effects model for longitudinal data. We alleviate variable selection bias through residual analysis, which is used to solve problems that exhaustive search approaches suffer from, such as undue preference to split variables with more possible splits, expensive computational cost, and end-cut preference. Most importantly, our tree algorithm discovers trends over time on each of the subspaces from recursive partitioning, while other tree algorithms predict responses. We investigate the performance of our algorithm with both simulation and real data studies. We also develop an R package melt that can be used conveniently and freely. Additional results are provided as online supplementary material.     

Algorithms for central-median paths with bounded length on trees

The location of path-shaped facilities on trees has been receiving a growing attention in the specialized literature in the recent years. Examples of such facilities include railroad lines, highways and public transit lines. Most of the papers deal with the problem of locating a path on a tree by minimizing either the maximum distance from the vertices of the tree to the facility or of minimizing the sum of the distances from all the vertices of the tree to the path. However, neither of the two above criteria alone capture all essential elements of a location problem. The sum of the distances criterion alone may result in solutions which are unacceptable from the point of view of the service level for the clients who are located far away from the facilities. On the other hand, the criterion of the minimization of the maximum distance, if used alone, may lead to very costly service systems. In the literature, there is just one paper that considers the problem of finding an optimal location of a path on a tree using combinations of the two above criteria, and efficient algorithms are provided. In particular, the cases where one criterion is optimized subject to a restriction on the value of the other are considered and linear time algorithms are presented. However, these problems do not consider any bound on the length or cost of the facility. In this paper we consider the two following problems: find a path which minimizes the sum of the distances such that the maximum distance from the vertices of the tree to the path is bounded by a fixed constant and such that the length of the path is not greater than a fixed value; find a path which minimizes the maximum distance with the sum of the distances being not greater than a fixed value and with bounded length. From an application point of view the constraint on the length of the path may refer to a budget constraint for establishing the facility. The restriction on the length of the path complicates the two problems but for both of them we give O(n log² n) divide-and-conquer algorithms.     

A Weighted König Lemma for Symbolic Dynamics

The König Lemma for infinite trees in Graph Theory says that in an infinite rooted tree with all vertices of finite degree, there is an infinite path starting from the root. In this weighted version we shall show that there is an infinite path with weights greater than a certain average. We shall apply our result to infinite compositions of affine mappings in Rⁿ.     

树的稳定子集和稳定指标

一个实矩阵的符号稳定性问题在经济学、生态学等诸多领域中都有应用背景．本文利用[1]中给出的不可约矩阵的符号稳定性的有关结论，将一个实矩阵的符号稳定性判定问题转化为一个等价的图论问题，即判定无向树中一个点子集的稳定性问题．本文引入了树的稳定子集的概念并给出了稳定子集的递归判别方法．本文还提出井研究了树的稳定指标，即树中所有稳定子集的最小基数，证明了关于稳定指标的一个min—max型定理，井给出了n阶树的稳定指标的最好上界及达到上界的极树的完全刻划。     

随机递归最优控制和混合最优控制问题

对随机递归最优控制问题即代价函数由特定倒向随机微分方程解来描述和递归混合最优控制问题即控制者还需 决定最优停止时刻, 得到了最优控制的存在性结果. 在一类等价概率测度集中,还给出了递归最优值函数的最小和最大数学期望.     

所有点对之间最快路问题

所有点对之间最快路问题就是要在所有点对(Vs，Vt)之间传送数据δs,t,并找出一条最快的路线．解决所有点对之间最快路问题的关键是产生有效解的等价集合．运用动态点对最短路的算法，本文首先设计了一个时间复杂性为O(mn^2)的产生有效解等价集合的算法，然后研究了静态点对之间最快路问题和动态点对之间最快路问题，其算法的时间复杂性分别为O(mn^2)和O(m^2n^2)．最后本文研究了求和对最小的路问题，证明该问题可以在O(mn^2)时间内解决．     

On the strength of recursive McCormick relaxations for binary polynomial optimization

Recursive McCormick relaxations are among the most popular convexification techniques for binary polynomial optimization. It is well-understood that both the quality and the size of these relaxations depend on the recursive sequence and finding an optimal sequence amounts to solving a difficult combinatorial optimization problem. We prove that any recursive McCormick relaxation is implied by the extended flower relaxation, a linear programming relaxation, which for binary polynomial optimization problems with fixed degree can be solved in strongly polynomial time.     

Infinite dimensional duality and applications

The usual duality theory cannot be applied to infinite dimensional problems because the underlying constraint set mostly has an empty interior and the constraints are possibly nonlinear. In this paper we present an infinite dimensional nonlinear duality theory obtained by using new separation theorems based on the notion of quasi-relative interior, which, in all the concrete problems considered, is nonempty. We apply this theory to solve the until now unsolved problem of finding, in the infinite dimensional case, the Lagrange multipliers associated to optimization problems or to variational inequalities. As an example, we find the Lagrange multiplier associated to a general elastic–plastic torsion problem.     

Review and comparison of three methods for the solution of the car sequencing problem

The car sequencing problem is the ordering of the production of a list of vehicles which are of the same type, but which may have options or variations that require higher work content and longer operation times for at least one assembly workstation. A feasible production sequence is one that does not schedule vehicles with options in such a way that one or more workstations are overloaded. In variations of the problem, other constraints may apply. We describe and compare three approaches to the modeling and solution of this problem. The first uses integer programming to model and solve the problem. The second approaches the question as a constraint satisfaction problem (CSP). The third method proposes an adaptation of the Ant Colony Optimization for the car sequencing problem. Test-problems are drawn from CSPLib, a publicly available set of problems available through the Internet. We quote results drawn both from our own work and from other research. The literature review is not intended to be exhaustive but we have sought to include representative examples and the more recent work. Our conclusions bear on likely research avenues for the solution of problems of practical size and complexity. A new set of larger benchmark problems was generated and solved. These problems are available to other researchers who may wish to solve them using their own methods.     

Paths, trees and matchings under disjunctive constraints

We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints.We prove that the minimum spanning tree problem is strongly NP-hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is NP-hard for conflict graphs where every connected component is a single edge.Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem.     

Certificates of infeasibility via nonsmooth optimization

An important aspect in the solution process of constraint satisfaction problems is to identify exclusion boxes which are boxes that do not contain feasible points. This paper presents a certificate of infeasibility for finding such boxes by solving a linearly constrained nonsmooth optimization problem. Furthermore, the constructed certificate can be used to enlarge an exclusion box by solving a nonlinearly constrained nonsmooth optimization problem.     

An approximation result for the interval coloring problem on claw-free chordal graphs

We study the problem of finding an acyclic orientation of an undirected graph, such that each (oriented) path is covered by a limited number k of maximal cliques. This is equivalent to finding a k-approximate solution for the interval coloring problem on a graph. We focus our attention on claw-free chordal graphs, and show how to find an orientation of such a graph in linear time, which guarantees that each path is covered by at most two maximal cliques. This extends previous published results on other graph classes where stronger assumptions were made.     

A preference-based approach to spanning trees and shortest paths problems****

Comparison of solutions in combinatorial problems is often based on an additive cost function inducing a complete order on solutions. We investigate here a generalization of the problem, where preferences take the form of a quasi-transitive binary relation defined on the solutions space. We first propose preference-based search algorithms for two classical combinatorial problems, namely the preferred spanning trees problem (a generalization of the minimum spanning tree problem) and the preferred paths problem (a generalization of the shortest path problem). Then, we introduce a very useful axiom for preference relations called independence. Using this axiom, we establish admissibility results concerning our preference-based search algorithms. Finally, we address the problem of dealing with non-independent preference relations and provide different possible solutions for different particular problems (e.g. lower approximation of the set of preferred solutions for multicriteria spanning trees problems, or relaxation of the independence axiom for interval-valued preferred path problems).