Stochastic Vehicle Routing with Recourse

We study the classic Vehicle Routing Problem in the setting of stochastic optimization with recourse. StochVRP is a two-stage optimization problem, where demand is satisfied using two routes: fixed and recourse. The fixed route is computed using only a demand distribution. Then after observing the demand instantiations, a recourse route is computed -- but costs here become more expensive by a factor lambda. We present an O(log^2 n log(n lambda))-approximation algorithm for this stochastic routing problem, under arbitrary distributions. The main idea in this result is relating StochVRP to a special case of submodular orienteering, called knapsack rank-function orienteering. We also give a better approximation ratio for knapsack rank-function orienteering than what follows from prior work. Finally, we provide a Unique Games Conjecture based omega(1) hardness of approximation for StochVRP, even on star-like metrics on which our algorithm achieves a logarithmic approximation.Comment: 20 Pages, 1 figure Revision corrects the statement and proof of Theorem 1.

Dial a Ride from k-forest

The k-forest problem is a common generalization of both the k-MST and the dense-$k$-subgraph problems. Formally, given a metric space on $n$ vertices $V$, with $m$ demand pairs $\subseteq V \times V$ and a ``target'' $k\le m$, the goal is to find a minimum cost subgraph that connects at least $k$ demand pairs. In this paper, we give an $O(\min\{\sqrt{n},\sqrt{k}\})$-approximation algorithm for $k$-forest, improving on the previous best ratio of $O(n^{2/3}\log n)$ by Segev & Segev. We then apply our algorithm for k-forest to obtain approximation algorithms for several Dial-a-Ride problems. The basic Dial-a-Ride problem is the following: given an $n$ point metric space with $m$ objects each with its own source and destination, and a vehicle capable of carrying at most $k$ objects at any time, find the minimum length tour that uses this vehicle to move each object from its source to destination. We prove that an $\alpha$-approximation algorithm for the $k$-forest problem implies an $O(\alpha\cdot\log^2n)$-approximation algorithm for Dial-a-Ride. Using our results for $k$-forest, we get an $O(\min\{\sqrt{n},\sqrt{k}\}\cdot\log^2 n)$- approximation algorithm for Dial-a-Ride. The only previous result known for Dial-a-Ride was an $O(\sqrt{k}\log n)$-approximation by Charikar & Raghavachari; our results give a different proof of a similar approximation guarantee--in fact, when the vehicle capacity $k$ is large, we give a slight improvement on their results.Comment: Preliminary version in Proc. European Symposium on Algorithms, 200

Tractable Pathfinding for the Stochastic On-Time Arrival Problem

We present a new and more efficient technique for computing the route that maximizes the probability of on-time arrival in stochastic networks, also known as the path-based stochastic on-time arrival (SOTA) problem. Our primary contribution is a pathfinding algorithm that uses the solution to the policy-based SOTA problem---which is of pseudo-polynomial-time complexity in the time budget of the journey---as a search heuristic for the optimal path. In particular, we show that this heuristic can be exceptionally efficient in practice, effectively making it possible to solve the path-based SOTA problem as quickly as the policy-based SOTA problem. Our secondary contribution is the extension of policy-based preprocessing to path-based preprocessing for the SOTA problem. In the process, we also introduce Arc-Potentials, a more efficient generalization of Stochastic Arc-Flags that can be used for both policy- and path-based SOTA. After developing the pathfinding and preprocessing algorithms, we evaluate their performance on two different real-world networks. To the best of our knowledge, these techniques provide the most efficient computation strategy for the path-based SOTA problem for general probability distributions, both with and without preprocessing.Comment: Submission accepted by the International Symposium on Experimental Algorithms 2016 and published by Springer in the Lecture Notes in Computer Science series on June 1, 2016. Includes typographical corrections and modifications to pre-processing made after the initial submission to SODA'15 (July 7, 2014

Towards the solution of variants of Vehicle Routing Problem

Some of the problems that are used extensively in -real life are NP complete problems. There is no any algorithm which can give the optimal solution to NP complete problems in the polynomial time in the worst case. So researchers are applying their best efforts to design the approximation algorithms for these NP complete problems. Approximation algorithm gives the solution of a particular problem, which is close to the optimal solution of that problem. In this paper, a study on variants of vehicle routing problem is being done along with the difference in the approximation ratios of different approximation algorithms as being given by researchers and it is found that Researchers are continuously applying their best efforts to design new approximation algorithms which have better approximation ratio as compared to the previously existing algorithms

Revenue Maximization in Transportation Networks

We study the joint optimization problem of pricing trips in a transportation network and serving the induced demands by routing a fleet of available service vehicles to maximize revenue. Our framework encompasses applications that include traditional transportation networks (e.g., airplanes, buses) and their more modern counterparts (e.g., ride-sharing systems). We describe a simple combinatorial model, in which each edge in the network is endowed with a curve that gives the demand for traveling between its endpoints at any given price. We are supplied with a number of vehicles and a time budget to serve the demands induced by the prices that we set, seeking to maximize revenue. We first focus on a (preliminary) special case of our model with unit distances and unit time horizon. We show that this version of the problem can be solved optimally in polynomial time. Switching to the general case of our model, we first present a two-stage approach that separately optimizes for prices and routes, achieving a logarithmic approximation to revenue in the process. Next, using the insights gathered in the first two results, we present a constant factor approximation algorithm that jointly optimizes for prices and routes for the supply vehicles. Finally, we discuss how our algorithms can handle capacitated vehicles, impatient demands, and selfish (wage-maximizing) drivers

Pruning 2-Connected Graphs

Given an edge-weighted undirected graph $G$ with a specified set of terminals, let the emph{density} of any subgraph be the ratio of its weight/cost to the number of terminals it contains. If $G$ is 2-connected, does it contain smaller 2-connected subgraphs of density comparable to that of $G$? We answer this question in the affirmative by giving an algorithm to emph{prune} $G$ and find such subgraphs of any desired size, at the cost of only a logarithmic increase in density (plus a small additive factor). We apply the pruning techniques to give algorithms for two NP-Hard problems on finding large 2-vertex-connected subgraphs of low cost; no previous approximation algorithm was known for either problem. In the kv problem, we are given an undirected graph $G$ with edge costs and an integer $k$; the goal is to find a minimum-cost 2-vertex-connected subgraph of $G$ containing at least $k$ vertices. In the bv problem, we are given the graph $G$ with edge costs, and a budget $B$; the goal is to find a 2-vertex-connected subgraph $H$ of $G$ with total edge cost at most $B$ that maximizes the number of vertices in $H$. We describe an $O(log n log k)$ approximation for the kv problem, and a bicriteria approximation for the bv problem that gives an $O(frac{1}{eps}log^2 n)$ approximation, while violating the budget by a factor of at most $3+eps$