arXiv subjects

Marco Molinaro

Publications and source records attributed to Marco Molinaro.

74 records · Page 5Linked to original sources

arXiv2010

Capacitated Vehicle Routing with Non-Uniform Speeds

The capacitated vehicle routing problem (CVRP) 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 non-uniform speeds (that we call Heterogenous CVRP), and present a constant-factor 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 with possibly different speeds, 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 non-uniform speeds introduces difficulties for employing standard tour-splitting techniques. In order to get a better understanding of this technique in our context, we appeal to ideas from the 2-approximation for scheduling in parallel machine of Lenstra et al.. This motivates the introduction of a new approximate MST construction called Level-Prim, which is related to Light Approximate Shortest-path Trees. The last component of our algorithm involves partitioning the Level-Prim 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.

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arXiv2009

On the Complexity of Searching in Trees: Average-case Minimization

We focus on the average-case analysis: A function w : V -> Z+ is given which defines the likelihood for a node to be the one marked, and we want the strategy that minimizes the expected number of queries. Prior to this paper, very little was known about this natural question and the complexity of the problem had remained so far an open question. We close this question and prove that the above tree search problem is NP-complete even for the class of trees with diameter at most 4. This results in a complete characterization of the complexity of the problem with respect to the diameter size. In fact, for diameter not larger than 3 the problem can be shown to be polynomially solvable using a dynamic programming approach. In addition we prove that the problem is NP-complete even for the class of trees of maximum degree at most 16. To the best of our knowledge, the only known result in this direction is that the tree search problem is solvable in O(|V| log|V|) time for trees with degree at most 2 (paths). We match the above complexity results with a tight algorithmic analysis. We first show that a natural greedy algorithm attains a 2-approximation. Furthermore, for the bounded degree instances, we show that any optimal strategy (i.e., one that minimizes the expected number of queries) performs at most O(Δ(T) (log |V| + log w(T))) queries in the worst case, where w(T) is the sum of the likelihoods of the nodes of T and Δ(T) is the maximum degree of T. We combine this result with a non-trivial exponential time algorithm to provide an FPTAS for trees with bounded degree.

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