29 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.
Top-k Route Search through Submodularity Modeling of Recurrent POI Features
We consider a practical top-k route search problem: given a collection of
points of interest (POIs) with rated features and traveling costs between POIs,
a user wants to find k routes from a source to a destination and limited in a
cost budget, that maximally match her needs on feature preferences. One
challenge is dealing with the personalized diversity requirement where users
have various trade-off between quantity (the number of POIs with a specified
feature) and variety (the coverage of specified features). Another challenge is
the large scale of the POI map and the great many alternative routes to search.
We model the personalized diversity requirement by the whole class of
submodular functions, and present an optimal solution to the top-k route search
problem through indices for retrieving relevant POIs in both feature and route
spaces and various strategies for pruning the search space using user
preferences and constraints. We also present promising heuristic solutions and
evaluate all the solutions on real life data.Comment: 11 pages, 7 figures, 2 table
Constant-Factor Approximation to Deadline TSP and Related Problems in (Almost) Quasi-Polytime
We investigate a genre of vehicle-routing problems (VRPs), that we call max-reward VRPs, wherein nodes located in a metric space have associated rewards that depend on their visiting times, and we seek a path that earns maximum reward. A prominent problem in this genre is deadline TSP, where nodes have deadlines and we seek a path that visits all nodes by their deadlines and earns maximum reward. Our main result is a constant-factor approximation for deadline TSP running in time O(n^O(log(n?))) in metric spaces with integer distances at most ?. This is the first improvement over the approximation factor of O(log n) due to Bansal et al. [N. Bansal et al., 2004] in over 15 years (but is achieved in super-polynomial time). Our result provides the first concrete indication that log n is unlikely to be a real inapproximability barrier for deadline TSP, and raises the exciting possibility that deadline TSP might admit a polytime constant-factor approximation.
At a high level, we obtain our result by carefully guessing an appropriate sequence of O(log (n?)) nodes appearing on the optimal path, and finding suitable paths between any two consecutive guessed nodes. We argue that the problem of finding a path between two consecutive guessed nodes can be relaxed to an instance of a special case of deadline TSP called point-to-point (P2P) orienteering. Any approximation algorithm for P2P orienteering can then be utilized in conjunction with either a greedy approach, or an LP-rounding approach, to find a good set of paths overall between every pair of guessed nodes. While concatenating these paths does not immediately yield a feasible solution, we argue that it can be covered by a constant number of feasible solutions. Overall our result therefore provides a novel reduction showing that any ?-approximation for P2P orienteering can be leveraged to obtain an O(?)-approximation for deadline TSP in O(n^O(log n?)) time.
Our results extend to yield the same guarantees (in approximation ratio and running time) for a substantial generalization of deadline TSP, where the reward obtained by a client is given by an arbitrary non-increasing function (specified by a value oracle) of its visiting time. Finally, we discuss applications of our results to variants of deadline TSP, including settings where both end-nodes are specified, nodes have release dates, and orienteering with time windows
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
On rooted -connectivity problems in quasi-bipartite digraphs
We consider the directed Rooted Subset -Edge-Connectivity problem: given a
set of terminals in a digraph with edge costs and
an integer , find a min-cost subgraph of that contains edge disjoint
-paths for all . The case when every edge of positive cost has
head in admits a polynomial time algorithm due to Frank, and the case when
all positive cost edges are incident to is equivalent to the -Multicover
problem. Recently, [Chan et al. APPROX20] obtained ratio for
quasi-bipartite instances, when every edge in has an end in . We give
a simple proof for the same ratio for a more general problem of covering an
arbitrary -intersecting supermodular set function by a minimum cost edge
set, and for the case when only every positive cost edge has an end in