Algorithms and Adaptivity Gaps for Stochastic k-TSP

Given a metric $(V,d)$ and a $\textsf{root} \in V$, the classic \textsf{k-TSP} problem is to find a tour originating at the $\textsf{root}$ of minimum length that visits at least $k$ nodes in $V$. In this work, motivated by applications where the input to an optimization problem is uncertain, we study two stochastic versions of \textsf{k-TSP}. In Stoch-Reward $k$-TSP, originally defined by Ene-Nagarajan-Saket [ENS17], each vertex $v$ in the given metric $(V,d)$ contains a stochastic reward $R_v$. The goal is to adaptively find a tour of minimum expected length that collects at least reward $k$; here "adaptively" means our next decision may depend on previous outcomes. Ene et al. give an $O(\log k)$-approximation adaptive algorithm for this problem, and left open if there is an $O(1)$-approximation algorithm. We totally resolve their open question and even give an $O(1)$-approximation \emph{non-adaptive} algorithm for this problem. We also introduce and obtain similar results for the Stoch-Cost $k$-TSP problem. In this problem each vertex $v$ has a stochastic cost $C_v$, and the goal is to visit and select at least $k$ vertices to minimize the expected \emph{sum} of tour length and cost of selected vertices. This problem generalizes the Price of Information framework [Singla18] from deterministic probing costs to metric probing costs. Our techniques are based on two crucial ideas: "repetitions" and "critical scaling". We show using Freedman's and Jogdeo-Samuels' inequalities that for our problems, if we truncate the random variables at an ideal threshold and repeat, then their expected values form a good surrogate. Unfortunately, this ideal threshold is adaptive as it depends on how far we are from achieving our target $k$, so we truncate at various different scales and identify a "critical" scale.Comment: ITCS 202

The Vehicle Routing Problem with Service Level Constraints

We consider a vehicle routing problem which seeks to minimize cost subject to service level constraints on several groups of deliveries. This problem captures some essential challenges faced by a logistics provider which operates transportation services for a limited number of partners and should respect contractual obligations on service levels. The problem also generalizes several important classes of vehicle routing problems with profits. To solve it, we propose a compact mathematical formulation, a branch-and-price algorithm, and a hybrid genetic algorithm with population management, which relies on problem-tailored solution representation, crossover and local search operators, as well as an adaptive penalization mechanism establishing a good balance between service levels and costs. Our computational experiments show that the proposed heuristic returns very high-quality solutions for this difficult problem, matches all optimal solutions found for small and medium-scale benchmark instances, and improves upon existing algorithms for two important special cases: the vehicle routing problem with private fleet and common carrier, and the capacitated profitable tour problem. The branch-and-price algorithm also produces new optimal solutions for all three problems

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.

Centralized versus market-based approaches to mobile task allocation problem: State-of-the-art

Centralized approach has been adopted for finding solutions to resource allocation problems (RAPs) in many real-life applications. On the other hand, market-based approach has been proposed as an alternative to solve the problem due to recent advancement in ICT technologies. In spite of the existence of some efforts to review the pros and cons of each approach in RAPs, the studies cannot be directly applied to specific problem domains like mobile task allocation problem which is characterised with high level of uncertainty on the availability of resources (workers). This paper aims to review existing studies on task allocation problems(TAPs) focusing on those two approaches and their comparison and identify major issues that need to be resolved for comparing the two approaches in mobile task allocation problems. Mobile Task Allocation Problem (MTAP) is defined and its problematic structures are explained in relation with task allocation to mobile workers. Solutions produced by each approach to some applications and variations of MTAP are also discussed and compared. Finally, some future research directions are identified in order to compare both approaches in function of uncertainty emerging from the mobile nature of the MTAP