1,134 research outputs found
Robust optimization with incremental recourse
In this paper, we consider an adaptive approach to address optimization
problems with uncertain cost parameters. Here, the decision maker selects an
initial decision, observes the realization of the uncertain cost parameters,
and then is permitted to modify the initial decision. We treat the uncertainty
using the framework of robust optimization in which uncertain parameters lie
within a given set. The decision maker optimizes so as to develop the best cost
guarantee in terms of the worst-case analysis. The recourse decision is
``incremental"; that is, the decision maker is permitted to change the initial
solution by a small fixed amount. We refer to the resulting problem as the
robust incremental problem. We study robust incremental variants of several
optimization problems. We show that the robust incremental counterpart of a
linear program is itself a linear program if the uncertainty set is polyhedral.
Hence, it is solvable in polynomial time. We establish the NP-hardness for
robust incremental linear programming for the case of a discrete uncertainty
set. We show that the robust incremental shortest path problem is NP-complete
when costs are chosen from a polyhedral uncertainty set, even in the case that
only one new arc may be added to the initial path. We also address the
complexity of several special cases of the robust incremental shortest path
problem and the robust incremental minimum spanning tree problem
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.
Stochastic shortest path problems with recourse
Caption title.Includes bibliographical references (p. 22-23).Supported by the C.S. Draper Laboratory. DL-H-441625 Supported by a grant from Siemens A.G.George H. Polychronopoulos, John N. Tsitsiklis
A Dynamic Shortest Paths Toolbox: Low-Congestion Vertex Sparsifiers and their Applications
We present a general toolbox, based on new vertex sparsifiers, for designing
data structures to maintain shortest paths in dynamic graphs.
In an -edge graph undergoing edge insertions and deletions, our data
structures give the first algorithms for maintaining (a) -approximate
all-pairs shortest paths (APSP) with \emph{worst-case} update time
and query time , and (b) a tree that has diameter no larger
than a subpolynomial factor times the diameter of the underlying graph, where
each update is handled in amortized subpolynomial time.
In graphs undergoing only edge deletions, we develop a simpler and more
efficient data structure to maintain a -approximate single-source
shortest paths (SSSP) tree in a graph undergoing edge deletions in
amortized time per update.
Our data structures are deterministic. The trees we can maintain are not
subgraphs of , but embed with small edge congestion into . This is in
stark contrast to previous approaches and is useful for algorithms that
internally use trees to route flow.
To illustrate the power of our new toolbox, we show that our SSSP data
structure gives simple deterministic implementations of flow-routing MWU
methods in several contexts, where previously only randomized methods had been
known.
To obtain our toolbox, we give the first algorithm that, given a graph
undergoing edge insertions and deletions and a dynamic terminal set ,
maintains a vertex sparsifier that approximately preserves distances
between terminals in , consists of at most vertices and edges,
and can be updated in worst-case time .
Crucially, our vertex sparsifier construction allows us to maintain a low
edge-congestion embedding of into , which is needed for our
applications
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