885 research outputs found
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
Constrained Monotone Function Maximization and the Supermodular Degree
The problem of maximizing a constrained monotone set function has many
practical applications and generalizes many combinatorial problems.
Unfortunately, it is generally not possible to maximize a monotone set function
up to an acceptable approximation ratio, even subject to simple constraints.
One highly studied approach to cope with this hardness is to restrict the set
function. An outstanding disadvantage of imposing such a restriction on the set
function is that no result is implied for set functions deviating from the
restriction, even slightly. A more flexible approach, studied by Feige and
Izsak, is to design an approximation algorithm whose approximation ratio
depends on the complexity of the instance, as measured by some complexity
measure. Specifically, they introduced a complexity measure called supermodular
degree, measuring deviation from submodularity, and designed an algorithm for
the welfare maximization problem with an approximation ratio that depends on
this measure.
In this work, we give the first (to the best of our knowledge) algorithm for
maximizing an arbitrary monotone set function, subject to a k-extendible
system. This class of constraints captures, for example, the intersection of
k-matroids (note that a single matroid constraint is sufficient to capture the
welfare maximization problem). Our approximation ratio deteriorates gracefully
with the complexity of the set function and k. Our work can be seen as
generalizing both the classic result of Fisher, Nemhauser and Wolsey, for
maximizing a submodular set function subject to a k-extendible system, and the
result of Feige and Izsak for the welfare maximization problem. Moreover, when
our algorithm is applied to each one of these simpler cases, it obtains the
same approximation ratio as of the respective original work.Comment: 23 page
Randomized Strategies for Robust Combinatorial Optimization
In this paper, we study the following robust optimization problem. Given an
independence system and candidate objective functions, we choose an independent
set, and then an adversary chooses one objective function, knowing our choice.
Our goal is to find a randomized strategy (i.e., a probability distribution
over the independent sets) that maximizes the expected objective value. To
solve the problem, we propose two types of schemes for designing approximation
algorithms. One scheme is for the case when objective functions are linear. It
first finds an approximately optimal aggregated strategy and then retrieves a
desired solution with little loss of the objective value. The approximation
ratio depends on a relaxation of an independence system polytope. As
applications, we provide approximation algorithms for a knapsack constraint or
a matroid intersection by developing appropriate relaxations and retrievals.
The other scheme is based on the multiplicative weights update method. A key
technique is to introduce a new concept called -reductions for
objective functions with parameters . We show that our scheme
outputs a nearly -approximate solution if there exists an
-approximation algorithm for a subproblem defined by
-reductions. This improves approximation ratio in previous
results. Using our result, we provide approximation algorithms when the
objective functions are submodular or correspond to the cardinality robustness
for the knapsack problem
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
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