A dual algorithm for the constrained shortest path problem

Abstract

In this paper we develop a Lagrangian relaxation algorithm for the problem of finding a shortest path between two nodes in a network, subject to a knapsack-type constraint. For example, we may wish to find a minimum cost route subject to a total time constraint in a multimode transportation network. Furthermore, the problem, which is shown to be at least as hard as NP-complete problems, is generic to a class of problems that arise in the solution of integer linear programs and discrete state/stage deterministic dynamic programs. One approach to solving the problem is to utilize a kth shortest path algorithm, terminating with the first path that satisfies the constraint.This approach is impractical when the terminal value of k is large. Using Lagrangian relaxation we propose a method that is designed to reduce this value of k. Computational results indicate orders of magnitude savings when the approach is applied to large networks.

On the Computational Complexity of Policy Routing

“…Note also that the NP-hard constrained shortest path problem (CS), which is also in C ∩ SND, [6] can also be posed as a path selection problem over a special algebra.…”

“…Even our very capability to compute the preferred path is under scrutiny now, as it is no longer obvious whether what we are looking for is in fact a path or a walk. Or, in a similarly troubling scenario, a network operator might accidentally pose a service chaining rule that, even though extremely desirable from an operational perspective, happens to induce an intractable path selection problem, making that policy essentially unrealizable in practice 6 . Regrettably, conventional routing theory does not provide sufficient guidelines to identify such pathologic cases.…”

BDD-Constrained A^* Search: A Fast Method for Solving Constrained Shortest-Path Problems

“…However, most studies of CSP handle a special type of constraint, which limits the total consumption of edge-associate non-negative values, called resources in [9], [10]. This type of CSP is specifi-cally called the resource-constrained shortest path problem (RCSP) [11], [12].…”

“…for all BDD node α which is key to Cost [v][α] do 5: β ← followBDD(e i j , α) 6: if β = ⊥ then continue 7: if label(β) = e i j then 8: β ← hi(β) 9: Back [v][α] stores the state just before reaching state (v, α). Subroutine followBDD(e, α) visits BDD nodes by following lo-edges from node α until the label of the visited node is not less than e and returns the last visited node β. Subroutines hi(α) and lo(α) return the node pointed to by the hi-edge and lo-edge of α, respectively.…”

Lagrangean/Surrogate Heuristics for p-Median Problems

“…The best Lagrangean/surrogate relaxation value gives an improved bound to the usual Lagrangean relaxation. An exact solution to ( λ t D ) may be obtained by a search over different values of t (see Minoux (1975) and Handler and Zang (1980)). However, in general, we have an interval of values t 0 ≤ t ≤ t 1 (with t 0 =1 or t 1 =1) which also produces improved bounds (see Figure 1, for the case t 1 =1).…”