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Approximation Algorithms for VRP with Stochastic Demands
"... We consider the vehicle routing problem with stochastic demands (VRPSD). We give randomized approximation algorithms achieving approximation guarantees of 1 + α for splitdelivery VRPSD, and 2 + α for unsplitdelivery VRPSD; here α is the best approximation guarantee for the traveling salesman probl ..."
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We consider the vehicle routing problem with stochastic demands (VRPSD). We give randomized approximation algorithms achieving approximation guarantees of 1 + α for splitdelivery VRPSD, and 2 + α for unsplitdelivery VRPSD; here α is the best approximation guarantee for the traveling salesman problem. These bounds match the best known for even the respective deterministic problems [1, 2]. We also show that the ‘cyclic heuristic ’ for splitdelivery VRPSD achieves a constant approximation ratio, as conjectured in [4]. Subject classifications: Analysis of algorithms: suboptimal algorithms. Transportation: vehicle routing.
Capacitated Vehicle Routing with NonUniform Speeds
"... Abstract. The capacitated vehicle routing problem (CVRP) [17] involves distributing (identical) items from a depot to a set of demand locations, using a single capacitated vehicle. We study a generalization of this problem to the setting of multiple vehicles having nonuniform speeds (that we call H ..."
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Abstract. The capacitated vehicle routing problem (CVRP) [17] involves distributing (identical) items from a depot to a set of demand locations, using a single capacitated vehicle. We study a generalization of this problem to the setting of multiple vehicles having nonuniform speeds (that we call Heterogenous CVRP), and present a constantfactor approximation algorithm. The technical heart of our result lies in achieving a constant approximation to the following TSP variant (called Heterogenous TSP). Given a metric denoting distances between vertices, a depot r containing k vehicles having respective speeds {λi} k i=1, the goal is to find a tour for each vehicle (starting and ending at r), so that every vertex is covered in some tour and the maximum completion time is minimized. This problem is precisely Heterogenous CVRP when vehicles are uncapacitated. The presence of nonuniform speeds introduces difficulties for employing standard toursplitting techniques. In order to get a better understanding of this technique in our context, we appeal to ideas from the 2approximation for scheduling in parallel machine of Lenstra et al. [15]. This motivates the introduction of a new approximate MST construction called LevelPrim, which is related to Light Approximate Shortestpath Trees [14]. The last component of our algorithm involves partitioning the LevelPrim tree and matching the resulting parts to vehicles. This decomposition is more subtle than usual since now we need to enforce correlation between the size of the parts and their distances to the depot. 1
An InclusionExclusion Algorithm for the ktour Problem
"... Abstract — Consider an undirected graph G with n vertices, among them a distinguished vertex s called the origin, and nonnegativeinteger edge weights in {1,..., M}. The ktour problem for G is to cover all vertices of G with cycles such that: each cycle passes through s and includes at most k other ..."
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Abstract — Consider an undirected graph G with n vertices, among them a distinguished vertex s called the origin, and nonnegativeinteger edge weights in {1,..., M}. The ktour problem for G is to cover all vertices of G with cycles such that: each cycle passes through s and includes at most k other vertices, each vertex different from s is visited exactly once by the cycles, and the total weight of the cycles is minimal. This problem is a special case of the general vehicle routing problem and it is known to be NPhard for k ≥ 3. We show that the ktour problem for G can be solved in time O(2 n n 7 k 2 M 2 (n log n + log M)) and space O(n 4 kM(n logn + log M)).
Capacitated Vehicle Routing with NonUniform Speeds
"... The capacitated vehicle routing problem (CVRP) [TV02] involves distributing (identical) items from a depot to a set of demand locations, using a single capacitated vehicle. We study a generalization of this problem to the setting of multiple vehicles having nonuniform speeds (that we call Heteroge ..."
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The capacitated vehicle routing problem (CVRP) [TV02] involves distributing (identical) items from a depot to a set of demand locations, using a single capacitated vehicle. We study a generalization of this problem to the setting of multiple vehicles having nonuniform speeds (that we call Heterogenous CVRP), and present a constantfactor approximation algorithm. The technical heart of our result lies in achieving a constant approximation to the following TSP variant (called Heterogenous TSP). Given a metric denoting distances between vertices, a depot r containing k vehicles having respective speeds {λi}ki=1, the goal is to find a tour for each vehicle (starting and ending at r), so that every vertex is covered in some tour and the maximum completion time is minimized. This problem is precisely Heterogenous CVRP when vehicles are uncapacitated. The presence of nonuniform speeds introduces difficulties for employing standard toursplitting techniques. In order to get a better understanding of this technique in our context, we appeal to ideas from the 2approximation for scheduling in parallel machine of Lenstra et al. [LST90]. This motivates the introduction of a new approximate MST construction called LevelPrim, which is related to Light Approximate Shortestpath Trees [KRY95]. The last component of our algorithm involves partitioning the LevelPrim tree and matching the resulting parts to vehicles. This decomposition is more subtle than usual since now we need to enforce correlation between the size of the parts and their distances to the depot.