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.     

On the analysis of optimization problems in arc-dependent networks

This paper is concerned with the design and analysis of algorithms for optimization problems in arc-dependent networks. A network is said to be arc-dependent if the cost of an arc $a$ depends upon the arc taken to enter $a$. These networks are fundamentally different from traditional networks in which the cost associated with an arc is a fixed constant and part of the input. We first study the arc-dependent shortest path (ADSP) problem, which is also known as the suffix-1 path-dependent shortest path problem in the literature. This problem has a polynomial time solution if the shortest paths are not required to be simple. The ADSP problem finds applications in a number of domains, including highway engineering, turn penalties and prohibitions, and fare rebates. In this paper, we are interested in the ADSP problem when restricted to simple paths. We call this restricted version the simple arc-dependent shortest path (SADSP) problem. We show that the SADSP problem is NP-complete. We present inapproximability results and an exact exponential algorithm for this problem. We also extend our results for the longest path problem in arc-dependent networks. Additionally, we explore the problem of detecting negative cycles in arc-dependent networks and discuss its computational complexity. Our results include variants of the negative cycle detection problem such as longest, shortest, heaviest, and lightest negative simple cycles.²     

AN INVERSE MAXIMUM CAPACITY PATH PROBLEM WITH LOWER BOUND CONSTRAINTS

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Finding K shortest looping paths with waiting time in a time–window network

A time-constrained shortest path problem is a shortest path problem including time constraints that are commonly modeled by the form of time windows. Finding K shortest paths are suitable for the problem associated with constraints that are difficult to define or optimize simultaneously. Depending on the types of constraints, these K paths are generally classified into either simple paths or looping paths. In the presence of time–window constraints, waiting time occurs but is largely ignored. Given a network with such constraints, the contribution of this paper is to develop a polynomial time algorithm that finds the first K shortest looping paths including waiting time. The time complexity of the algorithm is O(rK²|V₁|³), where r is the number of different windows of a node and |V₁| is the number of nodes in the original network.     

Search for shortest path around semialgebraic obstacles in the plane

An algorithm is described which reduces in polynomial time the problem of constructing a shortest path (between two points) around semialgebraic obstacles in the plane to the problem of constructing a path of smallest weight in a graph the weights of whose vertices are integrals of positive algebraic functions (without singularities). As consequences of this construction one can get algorithms of polynomial complexity for constructing approximations to shortest paths. One such algorithm is given in the paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 192, pp. 163–173, 1991.     

The shortest path problem with forbidden paths

We consider a variant of the constrained shortest path problem, where the constraints come from a set of forbidden paths (arc sequences) that cannot be part of any feasible solution. Two solution approaches are proposed for this variant. The first uses Aho and Corasick's keyword matching algorithm to filter paths produced by a k-shortest paths algorithm. The second generalizes Martins' deviation path approach for the k-shortest paths problem by merging the original graph with a state graph derived from Aho and Corasick's algorithm. Like Martins' approach, the second method amounts to a polynomial reduction of the shortest path problem with forbidden paths to a classic shortest path problem. Its significant advantage over the first approach is that it allows considering forbidden paths in more general shortest path problems such as the shortest path problem with resource constraints.     

A lexicographic path order with slow growing derivation bounds

This paper is concerned with implicit computational complexity of the exptime computable functions. Modifying the lexicographic path order, we introduce a path order EPO. It is shown that a termination proof for a term rewriting system via EPO implies an exponential bound on the lengths of derivations. The path order EPO is designed so that every exptime function is representable as a term rewrite system compatible with EPO (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bottleneck shortest paths on a partially ordered scale

In bottleneck combinatorial problems, admissible solutions are compared with respect to their maximal elements. In such problems, one may work with an ordinal evaluation scale instead of a numerical scale. We consider here a generalization of this problem in which one only has a partially ordered scale (instead of a completely ordered scale). After the introduction of a mappimax comparison operator between sets of evaluations (which boils down to the classical operator when the order is complete), we establish computational complexity results for this variation of the shortest path problem. Finally, we formulate our problem as an algebraic shortest path problem and suggest adequate algorithms to solve it in the subsequent semiring.Received: June 2002, Revised: March 2003, AMS classification: 05C50, 16Y60, 90B06Olivier Spanjaard: Corresponding author.     

Shortest paths with ordinal weights

We investigate the single-source-single-destination “shortest” path problem in directed, acyclic graphs with ordinal weighted arc costs. We define the concepts of ordinal dominance and efficiency for paths and their associated ordinal levels, respectively. Further, we show that the number of ordinally non-dominated path vectors from the source node to every other node in the graph is polynomially bounded and we propose a polynomial time labeling algorithm for solving the problem of finding the set of ordinally non-dominated path vectors from source to sink.     

An Algorithm for Solving the Shortest Path Improvement Problem on Rooted Trees Under Unit Hamming Distance

Shortest path problems play important roles in computer science, communication networks, and transportation networks. In a shortest path improvement problem under unit Hamming distance, an edge-weighted graph with a set of source–terminal pairs is given. The objective is to modify the weights of the edges at a minimum cost under unit Hamming distance such that the modified distances of the shortest paths between some given sources and terminals are upper bounded by the given values. As the shortest path improvement problem is NP-hard, it is meaningful to analyze the complexity of the shortest path improvement problem defined on rooted trees with one common source. We first present a preprocessing algorithm to normalize the problem. We then present the proofs of some properties of the optimal solutions to the problem. A dynamic programming algorithm is proposed for the problem, and its time complexity is analyzed. A comparison of the computational experiments of the dynamic programming algorithm and MATLAB functions shows that the algorithm is efficient although its worst-case complexity is exponential time.     

The first <Emphasis Type="Italic">K</Emphasis> minimum cost paths in a time-schedule network

The time-constrained shortest path problem is an important generalisation of the classical shortest path problem and in recent years has attracted much research interest. We consider a time-schedule network, where every node in the network has a list of pre-specified departure times and departure from a node may take place only at one of these departure times. The objective of this paper is to find the first K minimum cost simple paths subject to a total time constraint. An efficient polynomial time algorithm is developed. It is also demonstrated that the algorithm can be modified for finding the first K paths for all possible values of total time.     

Polynomial auction algorithms for shortest paths

In this paper we consider strongly polynomial variations of the auction algorithm for the single origin/many destinations shortest path problem. These variations are based on the idea of graph reduction, that is, deleting unnecessary arcs of the graph by using certain bounds naturally obtained in the course of the algorithm. We study the structure of the reduced graph and we exploit this structure to obtain algorithms withO (n min{m, n logn}) andO(n ²) running time. Our computational experiments show that these algorithms outperform their closest competitors on randomly generated dense all destinations problems, and on a broad variety of few destination problems.Research supported by NSF under Grant No. DDM-8903385, by the ARO under Grant DAAL03-86-K-0171, by a CNR-GNIM grant, and by a Fullbright grant     

Preconditioning trace coupled 3d‐1d systems using fractional Laplacian

Multiscale or multiphysics problems often involve coupling of partial differential equations posed on domains of different dimensionality. In this work, we consider a simplified model problem of a 3d‐1d coupling and the main objective is to construct algorithms that may utilize standard multilevel algorithms for the 3d domain, which has the dominating computational complexity. Preconditioning for a system of two elliptic problems posed, respectively, in a three‐dimensional domain and an embedded one dimensional curve and coupled by the trace constraint is discussed. Investigating numerically the properties of the well‐defined discrete trace operator, it is found that negative fractional Sobolev norms are suitable preconditioners for the Schur complement of the system. The norms are employed to construct a robust block diagonal preconditioner for the coupled problem.     

Multivariate Polynomial Minimization and Its Application in Signal Processing

We make a conjecture that the number of isolated local minimum points of a 2n-degree or (2n+1)-degree r-variable polynomial is not greater than n ^r when n 2. We show that this conjecture is the minimal estimate, and is true in several cases. In particular, we show that a cubic polynomial of r variables may have at most one local minimum point though it may have 2^r critical points. We then study the global minimization problem of an even-degree multivariate polynomial whose leading order coefficient tensor is positive definite. We call such a multivariate polynomial a normal multivariate polynomial. By giving a one-variable polynomial majored below a normal multivariate polynomial, we show the existence of a global minimum of a normal multivariate polynomial, and give an upper bound of the norm of the global minimum and a lower bound of the global minimization value. We show that the quartic multivariate polynomial arising from broad-band antenna array signal processing, is a normal polynomial, and give a computable upper bound of the norm of the global minimum and a computable lower bound of the global minimization value of this normal quartic multivariate polynomial. We give some sufficient and necessary conditions for an even order tensor to be positive definite. Several challenging questions remain open.     

Reconstructing shortest paths

This paper develops a polynomial-time algorithm that reconstructs a shortest path between two vertices using only the all pairs shortest path distance matrix. For graphs with positive edge weights, the algorithm requiresO(⦹V|log|V|) time. When the graph contains both positive and negative, but not zero, edge weights, and all cycles have positive length, the algorithm runs inO(|V|²) time. The remarkable feature about the algorithm is that it does not require explicit information about edges in the original graph.     

Covering hypercubes by isometric paths

An isometric path is merely any shortest path between two vertices. If the vertices of the hypercube Q_n are represented by the set of 0–1 vectors of length n, an isometric path is obtained by changing the coordinates of a vector one at a time, never changing the same coordinate more than once. The minimum number of isometric paths required to cover the vertices of Q_n is at least 2ⁿ/(n+1). We show that when n+1 is a power of 2, the lower bound is in fact the minimum. In doing so, we construct a family of disjoint isometric paths which can be used to find an upper bound for additional classes of hypercubes.     

Efficient Solution of the Single-item,Capacitated Lot-sizing Problem with Start-up and Reservation Costs

A capacitated dynamic lot-sizing model, where the costs incurred are a start-up cost for switching the production facility on and another reservation cost for keeping the facility on, whether or not it is producing, is considered. The resulting scheduling problem is NP-hard. An efficient shortest path model of the uncapacitated version of the problem is developed. This model is then included, via a redefinition of variables, into a tight capacitated model; tight in the sense that sharp lower bounds can be produced from it. The lower bound problems are solved efficiently by recovering the shortest path structure through column generation, and effective upper bounds are generated by solving a small capacitated trans-shipment problem. The results of computational tests to verify the computational efficiency of the resulting solution scheme are presented.     

The Complexity of Factors of Multivariate Polynomials

The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of polynomial functions of polynomially bounded degree over fields of characteristic zero. The proof relies on a polynomial upper bound on the approximative complexity of a factor g of a polynomial f in terms of the (approximative) complexity of f and the degree of the factor g. This extends a result by Kaltofen. The concept of approximative complexity allows us to cope with the case that a factor has an exponential multiplicity, by using a perturbation argument. Our result extends to randomized (two-sided error) decision complexity.     

Computational bounds on polynomial differential equations

In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations.We consider elementary (in the sense of Analysis) discrete-time dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines.We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial-value problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56.Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.     

The multiperiod assignment problem: A multicommodity network flow model and specialized branch and bound algorithm

The multiperiod assignment problem, a specialization of the three dimensional assignment problem, is an optimization model that describes the situation of assigning people to activities, or jobs over time. We consider the most general case which has both a cost of assigning a person to an activity in each time period, and a cost of transferring the person from one activity in each period to another activity in the next period. In general, the number of time periods need not equal the number of persons and activities. We present a new formulation of the multiperiod assignment problem, that of an integer, multicommodity network flow model. The special structure of the model allows us to find a good feasible solution relatively quickly by a shortest path heuristic algorithm. We discuss a new branch and bound algorithm for solving this problem to optimality. The subproblems of the branch and bound tree are evaluated by solving a set of special-structured, shortest path problems all of which can be solved in order n²T time, where n is the number of persons and activities, and T is the number of time periods. We present computational tests of the algorithm on moderately sized problems.