47 research outputs found
Approximation Algorithms for the A Priori TravelingRepairman
We consider the a priori traveling repairman problem, which is a stochastic
version of the classic traveling repairman problem (also called the traveling
deliveryman or minimum latency problem). Given a metric with a root
, the traveling repairman problem (TRP) involves finding a tour
originating from that minimizes the sum of arrival-times at all vertices.
In its a priori version, we are also given independent probabilities of each
vertex being active. We want to find a master tour originating from
and visiting all vertices. The objective is to minimize the expected sum of
arrival-times at all active vertices, when is shortcut over the inactive
vertices. We obtain the first constant-factor approximation algorithm for a
priori TRP under non-uniform probabilities. Previously, such a result was only
known for uniform probabilities
Improved Approximation Algorithms for the Expanding Search Problem
A searcher faces a graph with edge lengths and vertex weights, initially having explored only a given starting vertex. In each step, the searcher adds an edge to the solution that connects an unexplored vertex to an explored vertex. This requires an amount of time equal to the edge length. The goal is to minimize the weighted sum of the exploration times over all vertices. We show that this problem is hard to approximate and provide algorithms with improved approximation guarantees. For the general case, we give a (2e+?)-approximation for any ? > 0. For the case that all vertices have unit weight, we provide a 2e-approximation. Finally, we provide a PTAS for the case of a Euclidean graph. Previously, for all cases only an 8-approximation was known
Degree-constrained Minimum Latency Trees are APX-Hard
When transmitting data from a single source to many recipients, it is often desirable to use some recipients of the stream to re-broadcast the stream to other users. In such multicast systems each client may be used as a server, serving up to B other clients. Formally, we will take a set X of clients, along with a distance function d that specifies the latency (in the host network) between each pair of clients in X. Our goal will be to produce a directed spanning tree of X, rooted at some specified root 2 X, with out-degree bounded by B, and minimizing the sum of the latencies from root to every point in X. In addition to being motivated by current experimental algorithms work, the problem also interpolates naturally between the traveling repairman problem (when B = 1) and single source shortest paths (when B = n − 1). The former problem is APX-complete (in metric spaces) and the latter is in P. We explore the hardness of the problem for other values of B. In particular, we show that the problem remains APX-Hard at least up to B = Cpn for some universal constant C when the host space is a general semi-metric
Efficient GRASP+VND and GRASP+VNS metaheuristics for the traveling repairman problem
The traveling repairman problem is a customer-centric routing problem, in which the total waiting time of the customers is minimized, rather than the total travel time of a vehicle. To date, research on this problem has focused on exact algorithms and approximation methods. This paper presents the first metaheuristic approach for the traveling repairman problem
On the minimum latency problem
We are given a set of points and a symmetric distance
matrix giving the distance between and . We wish to
construct a tour that minimizes , where is the
{\em latency} of , defined to be the distance traveled before first
visiting . This problem is also known in the literature as the {\em
deliveryman problem} or the {\em traveling repairman problem}. It arises in a
number of applications including disk-head scheduling, and turns out to be
surprisingly different from the traveling salesman problem in character. We
give exact and approximate solutions to a number of cases, including a
constant-factor approximation algorithm whenever the distance matrix satisfies
the triangle inequality.Comment: 9 page
Probabilistic bounds on the Traveling Salesman Problem and the Traveling Repairman Problem
The traveling salesman problem (-TSP) seeks a tour of minimal length
that visits a subset of points. The traveling repairman problem (TRP)
seeks a complete tour with minimal latency. This paper provides constant-factor
probabilistic approximations of both problems. We first show that the optimal
length of the -TSP path grows at a rate of
. The proof
provides a constant-factor approximation scheme, which solves a TSP in a
high-concentration zone -- leveraging large deviations of local concentrations.
Then, we show that the optimal TRP latency grows at a rate of . This result extends the classical Beardwood-Halton-Hammersley theorem to
the TRP. Again, the proof provides a constant-factor approximation scheme,
which visits zones by decreasing order of probability density. We discuss
practical implications of this result in the design of transportation and
logistics systems. Finally, we propose dedicated notions of fairness --
randomized population-based fairness for the -TSP and geographical fairness
for the TRP -- and give algorithms to balance efficiency and fairness
Approximation Algorithms for Capacitated k-Travelling Repairmen Problems
We study variants of the capacitated vehicle routing problem. In the multiple depot capacitated k-travelling repairmen problem (MD-CkTRP), we have a collection of clients to be served by one vehicle in a fleet of k identical vehicles based at given depots. Each client has a given demand that must be satisfied, and each vehicle can carry a total of at most Q demand before it must resupply at its original depot. We wish to route the vehicles in a way that obeys the constraints while minimizing the average time (latency) required to serve a client. This generalizes the Multi-depot k-Travelling Repairman Problem (MD-kTRP) [Chekuri and Kumar, IEEE-FOCS, 2003; Post and Swamy, ACM-SIAM SODA, 2015] to the capacitated vehicle setting, and while it has been previously studied [Lysgaard and Wohlk, EJOR, 2014; Rivera et al, Comput Optim Appl, 2015], no approximation algorithm with a proven ratio is known.
We give a 42.49-approximation to this general problem, and refine this constant to 25.49 when clients have unit demands. As far as we are aware, these are the first constant-factor approximations for capacitated vehicle routing problems with a latency objective. We achieve these results by developing a framework allowing us to solve a wider range of latency problems, and crafting various orienteering-style oracles for use in this framework. We also show a simple LP rounding algorithm has a better approximation ratio for the maximum coverage problem with groups (MCG), first studied by Chekuri and Kumar [APPROX, 2004], and use it as a subroutine in our framework. Our approximation ratio for MD-CkTRP when restricted to uncapacitated setting matches the best known bound for it [Post and Swamy, ACM-SIAM SODA, 2015]. With our framework, any improvements to our oracles or our MCG approximation will result in improved approximations to the corresponding k-TRP problem