Special cases of the quadratic shortest path problem

Abstract

The quadratic shortest path problem (QSPP) is the problem of finding a path with prespecified start vertex s and end vertex t in a digraph such that the sum of weights of arcs and the sum of interaction costs over all pairs of arcs on the path is minimized. We first consider a variant of the QSPP known as the adjacent QSPP. It was recently proven that the adjacent QSPP on cyclic digraphs cannot be approximated unless P = NP. Here, we give a simple proof for the same result. We also show that if the quadratic cost matrix is a symmetric weak sum matrix and all s-t paths have the same length, then an optimal solution for the QSPP can be obtained by solving the corresponding instance of the shortest path problem. Similarly, it is shown that the QSPP with a symmetric product cost matrix is solvable in polynomial time. Further, we provide sufficient and necessary conditions for a QSPP instance on a complete symmetric digraph with four vertices to be linearizable. We also characterize linearizable QSPP instances on complete symmetric digraphs with more than four vertices. Finally, we derive an algorithm that examines whether a QSPP instance on the directed grid graph G pq ( p, q ≥ 2) is linearizable. The complexity of this algorithm is O( p 3 q 2 + p 2 q 3 ).

On Solving the Quadratic Shortest Path Problem

“…That is a variant of the QSPP where the interaction costs of all non-adjacent arcs are equal to zero. Hu and Sotirov [12] give an alternative proof for the same result using a simple reduction from the arc-disjoint paths problem.…”

“…The quadratic shortest path problem (QSPP) is the problem of finding a path in a directed graph from the source vertex s to the target vertex t such that the sum of costs of arcs and the sum of interaction costs over all distinct pairs of arcs on the path is minimized. The QSPP is a NP-hard combinatorial optimization problem, see [12,20]. Rostami et al [20] show that the problem remains NP-hard even for the adjacent QSPP.…”

“…In particular, Rostami et al [21] provide a polynomial time algorithm for the adjacent QSPP considered on directed acyclic graphs. Hu and Sotirov [12] show that the QSPP can be efficiently solved if the cost matrix is a nonnegative symmetric product matrix, or if the cost matrix is a sum matrix and every s-t path in the graph has constant length. In [12], it is also shown that the linearizability of the QSPP on grid graphs can be detected in polynomial-time.…”

Geometric and LP-based heuristics for angular travelling salesman problems in the plane

“…Our version of costs only for successive edges is called adjacent QMSTP in these papers. The same terminology applies for the quadratic shortest path problem recently considered in [27] and [17].…”

Representations of quadratic combinatorial optimization problems: A case study using quadratic set covering and quadratic knapsack problems

“…Since ℓ is arbitrary, D must be a zero matrix. Now consider the solution As observed earlier, imposing additional restrictions on the family of feasible solutions, more interesting characterizations for diagonalizability can be obtained [18,30,35,45]. Let us now add a simple restriction that all elements of the underlyingF have the same cardinality.…”