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.     

Ramsey linear families and generalized subdivided graphs

The following results are obtained. (i) Let p, d, and k be fixed positive integers, and let G be a graph whose vertex set can be partitioned into parts V₁, V₂,…, V_a such that for each i at most d vertices in V₁ … V_i have neighbors in V_i+1 and r(K_k, V_i) p | V(G) |, where V_i denotes the subgraph of G induced by V_i. Then there exists a number c depending only on p, d, and k such that r(K_k, G)c | V(G) |. (ii) Let d be a positive integer and let G be a graph in which there is an independent set I V(G) such that each component of G−I has at most d vertices and at most two neighbors in I. Then r(G,G)c | V(G) |, where c is a number depending only on d. As a special case, r(G, G) 6 | V(G) | for a graph G in which all vertices of degree at least three are independent. The constant 6 cannot be replaced by one less than 4.     

On-line algorithms for ordered sets and comparability graphs

We discuss several results concerning on-line algorithms for ordered sets and comparability graphs. The main new result is on the problem of on-line transitive orientation. We view on-line transitive orientation of a comparability graph G as a two-person game. In the ith round of play, 1 i | V(G)|, player A names a graph G_i such that G_i is isomorphic to a subgraph of G, |V(G_i)| = i, and G_i−1 is an induced subgraph of G_i if i> 1. Player B must respond with a transitive orientation of G_i which extends the transitive orientation given to G_i−1 in the previous round of play. Player A wins if and only if player B fails to give a transitive orientation to G_i for some i, 1 i |V(G)|. Our main result shows that player A has at most three winning moves. We also discuss applications to on-line clique covering of comparability graphs, and we mention some open problems.     

A new algorithm to find the shortest paths between all pairs of nodes

A new algorithm to find the shortest paths between all pairs of nodes is presented. This algorithm makes use of a dual cost transformation and of a particular data structure. Its worst case time complexity is of the order of the third power of the number of nodes, and its space complexity is linear with the number of arcs. A comparison with existing algorithms is presented.     

Shortest paths in networks with vector weights

For a directed network in which vector weights are assigned to arcs, the Pareto analog to the shortest path problem is analyzed. An algorithm is presented for obtaining all Pareto shortest paths from a specified node to every other node.     

Partition of a directed bipartite graph into two directed cycles

Let D = (V₁, V₂; A) be a directed bipartite graph with |V₁| = |V₂| = n 2. Suppose that d_D(x) + d_D(y) 3n + 1 for all x ε V₁ and y ε V₂. Then D contains two vertex-disjoint directed cycles of lengths 2n₁ and 2n₂, respectively, for any positive integer partition n = n₁ + n₂. Moreover, the condition is sharp for even n and nearly sharp for odd n.     

A note on formulations for the A-partition problem on hypergraphs

Let H = (V, E) be an undirected hypergraph and AC. We consider the problem of finding a minimum cost partition of V that separates every pair of nodes in A. We consider three formulations of the problem and show that the theoretical lower bounds to the integer optimal objective value provided by the LP-relaxations in all three cases are identical. We describe our empiical findings with each formulation.     

Two linear transformations each tridiagonal with respect to an eigenbasis of the other

Let denote a field, and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A:V→V and A^*:V→V satisfying both conditions below:

1. [(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^* is irreducible tridiagonal.

2. [(ii)] There exists a basis for V with respect to which the matrix representing A^* is diagonal and the matrix representing A is irreducible tridiagonal.

We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A^* on V, there exists a sequence of scalars β,γ,γ^*,,^* taken from such that both

where [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.     

Two observations about free graded hopf algebras

This paper records two results about graded Hopf algebras that do not appear to be stated explicitly in the literature. Let B be a graded set, graded by the positive integers. Let V be the graded vector space with basis B over a field K of characteristic zero and V^'=K⊕V, where K is in grading zero. Let L ne the free graded Lie algebra on B over K and let T be the free graded tensor algebra on B. The first result is the "graded Witt formula" giving the dimension of the subspace of L in each grading. The second result is the observation that any graded coassociative, co-commutative comultiplication Δ:V^'→V^'⊗V^', with co-unit the projection V¹→K. extends to a graded Hopf algebra structure on T that is in fact isomorphic to the natural graded Hopf algebra structure on T. In the ungraded case the statement analogous to the second result is false.     

Quaternionic Numerical Range and Real Subspaces

Let W(A) be the numerical range of an n × n quaternionic matrix A and V a real subspace of the skew field of real quaternions. In this note the authors consider the relation among the shape of W(A), the convexity of V∩W(A): and the validity of the equality V∩W(A) = W_v(A), where W_v (A) is the orthogonal projection of W(A) into V.     

Odd subgraphs and matchings

Let G be a graph and f : V(G)→{1,3,5,…}. Then a subgraph H of G is called a (1,f)-odd subgraph if deg_H(x){1,3,…,f(x)} for all xV(H). If f(x)=1 for all xV(G), then a (1,f)-odd subgraph is nothing but a matching. A (1,f)-odd subgraph H of G is said to be maximum if G has no (1,f)-odd subgraph K such that |K|>|H|. We show that (1,f)-odd subgraphs have some properties similar to those of matchings, in particular, we give a formula for the order of a maximum (1,f)-odd subgraph, which is similar to that for the order of a maximum matching.     

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.     

On determining the congruence of point sets in d dimensions

This paper considers the following problem: given two point sets A and B (|A| = |B| = n) in d dimensional Euclidean space, determine whether or not A is congruent to B. This paper presents an O(n^(d−1)/2 log n) time randomized algorithm. The birthday paradox, which is well-known in combinatorics, is used effectively in this algorithm. Although this algorithm is Monte-Carlo type (i.e., it may give a wrong result), this improves a previous O(n^d−2 log n) time deterministic algorithm considerably. This paper also shows that if d is not bounded, the problem is at least as hard as the graph isomorphism problem in the sense of the polynomiality. Several related results are described too.     

Maximal orders containing local crossed-products

Let Λ = (S/R, ) be the crossed product order in the crossed product algebra A = (L/K, ) with factor set , where L/K is a Galois extension of the local field K, and R (resp. S) the valuation ring of K (resp. L). In this paper the maximal R-orders in A containing Λ and the irreducible Λ-lattices are determined.     

Class Steiner trees and VLSI-design

We consider a generalized version of the Steiner problem in graphs, motivated by the wire routing phase in physical VLSI design: given a connected, undirected distance graph with required classes of vertices and Steiner vertices, find a shortest connected subgraph containing at least one vertex of each required class. We show that this problem is NP-hard, even if there are no Steiner vertices and the graph is a tree. Moreover, the same complexity result holds if the input class Steiner graph additionally is embedded in a unit grid, if each vertex has degree at most three, and each class consists of no more than three vertices. For similar restricted versions, we prove MAX SNP-hardness and we show that there exists no polynomial-time approximation algorithm with a constant bound on the relative error, unless P = NP. We propose two efficient heuristics computing different approximate solutions in time O(¦E¦+¦V¦log¦V¦) and in time O(c(¦E¦+¦V¦log¦V¦)), respectively, where E is the set of edges in the given graph, V is the set of vertices, and c is the number of classes. We present some promising implementation results. kw]Steiner Tree; Heuristic; Approximation complexity; MAX-SNP-hardness     

A heuristic for the Steiner problem in graphs

In this paper, we present a heuristic for the Steiner problem in graphs (SPG) along with some experimental results. The heuristic is based on an approach similar to Prim's algorithm for the minimum spanning tree. However, in this approach, arcs are associated with preference weights which are used to break ties among alternative choices of shortest paths occurring during the course of the algorithm. The preference weights are calculated according to a global view which takes into consideration the effect of all the regular nodes, nodes to be connected, on determining the choice of an arc in the solution tree.     

Long division for Laurent series matrices and the optimal assignment problem

We present a necessary and sufficient condition to represent a Laurent series matrix A(x) as a product where is a Laurent series matrix whose leading scalar matrix is nonsingular and U(x) and V(x) are diagonal matrices whose nonzero entries are powers of x. If A(x) can be written in this form, then the matrix equation A(x)Y(x) = B(x) can be solved by long division. Our result relies on a classical theorem on optimal assignments.     

The energy-constrained quickest path problem

This paper addresses a variant of the quickest path problem in which each arc has an additional parameter associated to it representing the energy consumed during the transmission along the arc while each node is endowed with a limited power to transmit messages. The aim of the energy-constrained quickest path problem is to obtain a quickest path whose nodes are able to support the transmission of a message of a known size. After introducing the problem and proving the main theoretical results, a polynomial algorithm is proposed to solve the problem based on computing shortest paths in a sequence of subnetworks of the original network. In the second part of the paper, the bi-objective variant of this problem is considered in which the objectives are the transmission time and the total energy used. An exact algorithm is proposed to find a complete set of efficient paths. The computational experiments carried out show the performance of both algorithms.     

On the K shortest path trees problem

We address the problem of finding the K best path trees connecting a source node with any other non-source node in a directed network with arbitrary lengths. The main result in this paper is the proof that the kth shortest path tree is adjacent to at least one of the previous (k-1) shortest path trees. Consequently, we design an O(f(n,m,C_max)+Km) time and O(K+m) space algorithm to determine the K shortest path trees, in a directed network with n nodes, m arcs and maximum absolute length C_max, where O(f(n,m,C_max)) is the best time needed to solve the shortest simple paths connecting a source node with any other non-source node.