4,121 research outputs found
Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets
Consider the following problem: given a set system (U,I) and an edge-weighted
graph G = (U, E) on the same universe U, find the set A in I such that the
Steiner tree cost with terminals A is as large as possible: "which set in I is
the most difficult to connect up?" This is an example of a max-min problem:
find the set A in I such that the value of some minimization (covering) problem
is as large as possible.
In this paper, we show that for certain covering problems which admit good
deterministic online algorithms, we can give good algorithms for max-min
optimization when the set system I is given by a p-system or q-knapsacks or
both. This result is similar to results for constrained maximization of
submodular functions. Although many natural covering problems are not even
approximately submodular, we show that one can use properties of the online
algorithm as a surrogate for submodularity.
Moreover, we give stronger connections between max-min optimization and
two-stage robust optimization, and hence give improved algorithms for robust
versions of various covering problems, for cases where the uncertainty sets are
given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and
http://arxiv.org/abs/0912.1045 appeared in ICALP 201
Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms
Constrained submodular maximization problems have long been studied, with
near-optimal results known under a variety of constraints when the submodular
function is monotone. The case of non-monotone submodular maximization is less
understood: the first approximation algorithms even for the unconstrainted
setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC
'09, APPROX '09) show how to approximately maximize non-monotone submodular
functions when the constraints are given by the intersection of p matroid
constraints; their algorithm is based on local-search procedures that consider
p-swaps, and hence the running time may be n^Omega(p), implying their algorithm
is polynomial-time only for constantly many matroids. In this paper, we give
algorithms that work for p-independence systems (which generalize constraints
given by the intersection of p matroids), where the running time is poly(n,p).
Our algorithm essentially reduces the non-monotone maximization problem to
multiple runs of the greedy algorithm previously used in the monotone case.
Our idea of using existing algorithms for monotone functions to solve the
non-monotone case also works for maximizing a submodular function with respect
to a knapsack constraint: we get a simple greedy-based constant-factor
approximation for this problem.
With these simpler algorithms, we are able to adapt our approach to
constrained non-monotone submodular maximization to the (online) secretary
setting, where elements arrive one at a time in random order, and the algorithm
must make irrevocable decisions about whether or not to select each element as
it arrives. We give constant approximations in this secretary setting when the
algorithm is constrained subject to a uniform matroid or a partition matroid,
and give an O(log k) approximation when it is constrained by a general matroid
of rank k.Comment: In the Proceedings of WINE 201
Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints
We investigate two new optimization problems -- minimizing a submodular
function subject to a submodular lower bound constraint (submodular cover) and
maximizing a submodular function subject to a submodular upper bound constraint
(submodular knapsack). We are motivated by a number of real-world applications
in machine learning including sensor placement and data subset selection, which
require maximizing a certain submodular function (like coverage or diversity)
while simultaneously minimizing another (like cooperative cost). These problems
are often posed as minimizing the difference between submodular functions [14,
35] which is in the worst case inapproximable. We show, however, that by
phrasing these problems as constrained optimization, which is more natural for
many applications, we achieve a number of bounded approximation guarantees. We
also show that both these problems are closely related and an approximation
algorithm solving one can be used to obtain an approximation guarantee for the
other. We provide hardness results for both problems thus showing that our
approximation factors are tight up to log-factors. Finally, we empirically
demonstrate the performance and good scalability properties of our algorithms.Comment: 23 pages. A short version of this appeared in Advances of NIPS-201
The Price of Information in Combinatorial Optimization
Consider a network design application where we wish to lay down a
minimum-cost spanning tree in a given graph; however, we only have stochastic
information about the edge costs. To learn the precise cost of any edge, we
have to conduct a study that incurs a price. Our goal is to find a spanning
tree while minimizing the disutility, which is the sum of the tree cost and the
total price that we spend on the studies. In a different application, each edge
gives a stochastic reward value. Our goal is to find a spanning tree while
maximizing the utility, which is the tree reward minus the prices that we pay.
Situations such as the above two often arise in practice where we wish to
find a good solution to an optimization problem, but we start with only some
partial knowledge about the parameters of the problem. The missing information
can be found only after paying a probing price, which we call the price of
information. What strategy should we adopt to optimize our expected
utility/disutility?
A classical example of the above setting is Weitzman's "Pandora's box"
problem where we are given probability distributions on values of
independent random variables. The goal is to choose a single variable with a
large value, but we can find the actual outcomes only after paying a price. Our
work is a generalization of this model to other combinatorial optimization
problems such as matching, set cover, facility location, and prize-collecting
Steiner tree. We give a technique that reduces such problems to their non-price
counterparts, and use it to design exact/approximation algorithms to optimize
our utility/disutility. Our techniques extend to situations where there are
additional constraints on what parameters can be probed or when we can
simultaneously probe a subset of the parameters.Comment: SODA 201
General Bounds for Incremental Maximization
We propose a theoretical framework to capture incremental solutions to
cardinality constrained maximization problems. The defining characteristic of
our framework is that the cardinality/support of the solution is bounded by a
value that grows over time, and we allow the solution to be
extended one element at a time. We investigate the best-possible competitive
ratio of such an incremental solution, i.e., the worst ratio over all
between the incremental solution after steps and an optimum solution of
cardinality . We define a large class of problems that contains many
important cardinality constrained maximization problems like maximum matching,
knapsack, and packing/covering problems. We provide a general
-competitive incremental algorithm for this class of problems, and show
that no algorithm can have competitive ratio below in general.
In the second part of the paper, we focus on the inherently incremental
greedy algorithm that increases the objective value as much as possible in each
step. This algorithm is known to be -competitive for submodular objective
functions, but it has unbounded competitive ratio for the class of incremental
problems mentioned above. We define a relaxed submodularity condition for the
objective function, capturing problems like maximum (weighted) (-)matching
and a variant of the maximum flow problem. We show that the greedy algorithm
has competitive ratio (exactly) for the class of problems that satisfy
this relaxed submodularity condition.
Note that our upper bounds on the competitive ratios translate to
approximation ratios for the underlying cardinality constrained problems.Comment: fixed typo
Budget-Feasible Mechanism Design for Non-Monotone Submodular Objectives: Offline and Online
The framework of budget-feasible mechanism design studies procurement
auctions where the auctioneer (buyer) aims to maximize his valuation function
subject to a hard budget constraint. We study the problem of designing truthful
mechanisms that have good approximation guarantees and never pay the
participating agents (sellers) more than the budget. We focus on the case of
general (non-monotone) submodular valuation functions and derive the first
truthful, budget-feasible and -approximate mechanisms that run in
polynomial time in the value query model, for both offline and online auctions.
Prior to our work, the only -approximation mechanism known for
non-monotone submodular objectives required an exponential number of value
queries.
At the heart of our approach lies a novel greedy algorithm for non-monotone
submodular maximization under a knapsack constraint. Our algorithm builds two
candidate solutions simultaneously (to achieve a good approximation), yet
ensures that agents cannot jump from one solution to the other (to implicitly
enforce truthfulness). Ours is the first mechanism for the problem
where---crucially---the agents are not ordered with respect to their marginal
value per cost. This allows us to appropriately adapt these ideas to the online
setting as well.
To further illustrate the applicability of our approach, we also consider the
case where additional feasibility constraints are present. We obtain
-approximation mechanisms for both monotone and non-monotone submodular
objectives, when the feasible solutions are independent sets of a -system.
With the exception of additive valuation functions, no mechanisms were known
for this setting prior to our work. Finally, we provide lower bounds suggesting
that, when one cares about non-trivial approximation guarantees in polynomial
time, our results are asymptotically best possible.Comment: Accepted to EC 201
Algorithm Engineering in Robust Optimization
Robust optimization is a young and emerging field of research having received
a considerable increase of interest over the last decade. In this paper, we
argue that the the algorithm engineering methodology fits very well to the
field of robust optimization and yields a rewarding new perspective on both the
current state of research and open research directions.
To this end we go through the algorithm engineering cycle of design and
analysis of concepts, development and implementation of algorithms, and
theoretical and experimental evaluation. We show that many ideas of algorithm
engineering have already been applied in publications on robust optimization.
Most work on robust optimization is devoted to analysis of the concepts and the
development of algorithms, some papers deal with the evaluation of a particular
concept in case studies, and work on comparison of concepts just starts. What
is still a drawback in many papers on robustness is the missing link to include
the results of the experiments again in the design
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