We consider a variant of the vehicle routing problem (VRP) where each customer has a unit demand and the goal is to minimize the total cost of routing a fleet of capacitated vehicles from one or multiple depots to visit all customers. We propose two parallel algorithms to efficiently solve the columngenerationbased linearprogramming relaxation for this VRP. Specifically, we focus on algorithms for the “pricing problem,” which corresponds to the resourceconstrained elementary shortest path problem. The first algorithm extends the pulse algorithm for which we derive a new bounding scheme on the maximum load of any route. The second algorithm is based on random coloring from parameterized complexity which can be also combined with other techniques in the literature for improving VRPs, including cutting planes and column enumeration. We conduct numerical studies using VRP benchmarks (with 50–957 nodes) and instances of a medical home care delivery problem using census data in Wayne County, Michigan. Using parallel computing, both pulse and random coloring can significantly improve column generation for solving the linear programming relaxations and we can obtain heuristic integer solutions with small optimality gaps. Combining random coloring with column enumeration, we can obtain improved integer solutions having less than 2% optimality gaps for most VRP benchmark instances and less than 1% optimality gaps for the medical home care delivery instances, both under a 30minute computational time limit. The use of cutting planes (e.g., robust cuts) can further reduce optimality gaps on some hard instances, without much increase in the run time. Summary of Contribution: The vehicle routing problem (VRP) is a fundamental combinatorial problem, and its variants have been studied extensively in the literature of operations research and computer science. In this paper, we consider generalpurpose algorithms for solving VRPs, including the columngeneration approach for the linear programming relaxations of the integer programs of VRPs and the columnenumeration approach for seeking improved integer solutions. We revise the pulse algorithm and also propose a randomcoloring algorithm that can be used for solving the elementary shortest path problem that formulates the pricing problem in the columngeneration approach. We show that the parallel implementation of both algorithms can significantly improve the performance of column generation and the random coloring algorithm can improve the solution time and quality of the VRP integer solutions produced by the columnenumeration approach. We focus on algorithmic design for VRPs and conduct extensive computational tests to demonstrate the performance of various approaches.
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Brief Announcement: A Parallel (Δ, Γ)Stepping Algorithm for the Constrained Shortest Path Problem
We design a parallel algorithm for the Constrained Shortest Path (CSP) problem. The CSP problem is known to be NPhard and there exists a pseudopolynomial time sequential algorithm that solves it. To design the parallel algorithm, we extend the techniques used in the design of the Δstepping algorithm for the singlesource shortest paths problem.
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 Award ID(s):
 1724227
 NSFPAR ID:
 10349904
 Date Published:
 Journal Name:
 SPAA '22: Proceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures
 Page Range / eLocation ID:
 287 to 289
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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