Algorithms for the Shortest Path Improvement Problems under Unit Hamming Distance

Abstract

In a shortest path improvement problem under unit Hamming distance (denoted by SPIUH), an edge weighted graph with a set of source-terminal pairs is given; we need to modify the lengths of edges by a minimum cost under unit Hamming distance such that the modified distances of the shortest paths are upper bounded by given values. The SPIUH problem on arborescent network is formulated as a 0-1 integer programming model. Some strongly polynomial time algorithms are designed for the problems on some special arborescent networks. Firstly, two greedy algorithms are proposed for problems on chain networks and special star-tree networks, respectively. Secondly, a strongly polynomial time algorithm is presented for the problem with a single source and constrained paths. Finally, a heuristic algorithm and its computational experiments are given for the SPIUH problem on general graphs.

Efficient algorithms for the reverse shortest path problem on trees under the hamming distance

“…As a special case of trees, we perform experiments on star-tree networks G where any two origin-destination paths of G have no common arcs [17]. Throughout experiments, we assume that the origin node s is a leaf of G, i.e., one node with degree 1.…”

“…As the RSP problem on trees under the sum-type Hamming distance is NP-hard, this aspect is worthwhile to further study special cases which the problem is polynomially solvable. In [17], the authors proposed polynomial-time algorithms to solve the problem under the unit Hamming distance on chain networks and star-tree networks. Then, they designed an efficient algorithm for solving the TRSP problem with a very special constraint on paths.…”

An Inverse Shortest Path Problem on an Uncertain Graph

“…Consequently, model (15) reduces to a deterministic programming problem with linear constraints as follows,…”

“…In [14], Zhang et al considered the shortest path improvement problem under hamming distance, which is a variant of the inverse shortest path problem, and discussed its computational complexity. Subsequently, Zhang et al [15] designed two greedy algorithms for this problem on chain networks and special star-tree networks, one of which is a strongly polynomial time algorithm for the problem with a single source and constrained paths, and the other is a heuristic algorithm for general graphs.…”