3,708 research outputs found
Profit-based latency problems on the line.
The latency problem with profits is a generalization of the minimum latency problem. In this generalization it is not necessary to visit all clients, however, visiting a client may bring a certain revenue. More precisely, in the latency problem with profits, a server and a set of n clients, each with corresponding profit p_i (1 ≤ i ≤ n), are given. The single server is positioned at the origin at time t = 0 and travels with unit speed. When visiting a client, the server receives a revenue of p_i - t, with t the time at which the server reaches client i (1 ≤ i ≤ n). The goal is to select clients and find a route for the server such that total collected revenue is maximized. We formulate a dynamic programming algorithm to solve this problem when all clients are located on a line. We also consider the problem on the line with k servers and prove NP-completeness for the latency problem on the line with k non-identical servers and release dates. In this proof we also settle the complexity of an open problem in de Paepe et al. [4].Minimum latency; Traveling repairman; Dynamic programming; Complexity;
Asymmetric Traveling Salesman Path and Directed Latency Problems
We study integrality gaps and approximability of two closely related problems
on directed graphs. Given a set V of n nodes in an underlying asymmetric metric
and two specified nodes s and t, both problems ask to find an s-t path visiting
all other nodes. In the asymmetric traveling salesman path problem (ATSPP), the
objective is to minimize the total cost of this path. In the directed latency
problem, the objective is to minimize the sum of distances on this path from s
to each node. Both of these problems are NP-hard. The best known approximation
algorithms for ATSPP had ratio O(log n) until the very recent result that
improves it to O(log n/ log log n). However, only a bound of O(sqrt(n)) for the
integrality gap of its linear programming relaxation has been known. For
directed latency, the best previously known approximation algorithm has a
guarantee of O(n^(1/2+eps)), for any constant eps > 0. We present a new
algorithm for the ATSPP problem that has an approximation ratio of O(log n),
but whose analysis also bounds the integrality gap of the standard LP
relaxation of ATSPP by the same factor. This solves an open problem posed by
Chekuri and Pal [2007]. We then pursue a deeper study of this linear program
and its variations, which leads to an algorithm for the k-person ATSPP (where k
s-t paths of minimum total length are sought) and an O(log n)-approximation for
the directed latency problem
Approximation and complexity of multi-target graph search and the Canadian traveler problem
In the Canadian traveler problem, we are given an edge weighted graph with two specified vertices s and t and a probability distribution over the edges that tells which edges are present. The goal is to minimize the expected length of a walk from s to t. However, we only get to know whether an edge is active the moment we visit one of its incident vertices. Under the assumption that the edges are active independently, we show NP-hardness on series-parallel graphs and give results on the adaptivity gap. We further show that this problem is NP-hard on disjoint-path graphs and cactus graphs when the distribution is given by a list of scenarios. We also consider a special case called the multi-target graph search problem. In this problem, we are given a probability distribution over subsets of vertices. The distribution specifies which set of vertices has targets. The goal is to minimize the expected length of the walk until finding a target. For the
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
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