5 research outputs found
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.
Minimum Latency Submodular Cover
We study the Minimum Latency Submodular Cover problem (MLSC), which consists
of a metric with source and monotone submodular functions
. The goal is to find a path
originating at that minimizes the total cover time of all functions. This
generalizes well-studied problems, such as Submodular Ranking [AzarG11] and
Group Steiner Tree [GKR00]. We give a polynomial time O(\log \frac{1}{\eps}
\cdot \log^{2+\delta} |V|)-approximation algorithm for MLSC, where
is the smallest non-zero marginal increase of any
and is any constant.
We also consider the Latency Covering Steiner Tree problem (LCST), which is
the special case of \mlsc where the s are multi-coverage functions. This
is a common generalization of the Latency Group Steiner Tree
[GuptaNR10a,ChakrabartyS11] and Generalized Min-sum Set Cover [AzarGY09,
BansalGK10] problems. We obtain an -approximation algorithm for
LCST.
Finally we study a natural stochastic extension of the Submodular Ranking
problem, and obtain an adaptive algorithm with an O(\log 1/ \eps)
approximation ratio, which is best possible. This result also generalizes some
previously studied stochastic optimization problems, such as Stochastic Set
Cover [GoemansV06] and Shared Filter Evaluation [MunagalaSW07, LiuPRY08].Comment: 23 pages, 1 figur
An Improved Approximation Ratio for the Covering Steiner Problem
In the Covering Steiner problem, we are given an undirected graph with edge-costs, and some subsets of vertices called groups, with each group being equipped with a non-negative integer value (called its requirement); the problem is to find a minimum-cost tree which spans at least the required number of vertices from every group. The Covering Steiner problem is a common generalization of the k-MST and the Group Steiner problems; indeed, when all the vertices of the graph lie in one group with a requirement of k, we get the k-MST problem, and when there are multiple groups with unit requirements, we obtain the Group Steiner problem. While many covering problems (e.g., the covering integer programs such as set cover) become easier to approximate as the requirements increase, the Covering Steiner problem remains at least as hard to approximate as the Group Steiner problem; in fact, the best guarantees previously known for the Covering Steiner problem were worse than those for Group Steiner as the requirements became large. In this work, we present an improved approximation algorithm whose guarantee equals the best known guarantee for the Group Steiner problem