1,672 research outputs found
Packing a Knapsack of Unknown Capacity
We study the problem of packing a knapsack without knowing its capacity.
Whenever we attempt to pack an item that does not fit, the item is discarded;
if the item fits, we have to include it in the packing. We show that there is
always a policy that packs a value within factor 2 of the optimum packing,
irrespective of the actual capacity. If all items have unit density, we achieve
a factor equal to the golden ratio. Both factors are shown to be best possible.
In fact, we obtain the above factors using packing policies that are universal
in the sense that they fix a particular order of the items and try to pack the
items in this order, independent of the observations made while packing. We
give efficient algorithms computing these policies. On the other hand, we show
that, for any alpha>1, the problem of deciding whether a given universal policy
achieves a factor of alpha is coNP-complete. If alpha is part of the input, the
same problem is shown to be coNP-complete for items with unit densities.
Finally, we show that it is coNP-hard to decide, for given alpha, whether a set
of items admits a universal policy with factor alpha, even if all items have
unit densities
Maximizing a Submodular Function with Bounded Curvature Under an Unknown Knapsack Constraint
This paper studies the problem of maximizing a monotone submodular function under an unknown knapsack constraint. A solution to this problem is a policy that decides which item to pack next based on the past packing history. The robustness factor of a policy is the worst case ratio of the solution obtained by following the policy and an optimal solution that knows the knapsack capacity. We develop an algorithm with a robustness factor that is decreasing in the curvature c of the submodular function. For the extreme cases c = 0 corresponding to a modular objective, it matches a previously known and best possible robustness factor of 1/2. For the other extreme case of c = 1 it yields a robustness factor of ? 0.35 improving over the best previously known robustness factor of ? 0.06
Maximizing a Submodular Function with Bounded Curvature under an Unknown Knapsack Constraint
This paper studies the problem of maximizing a monotone submodular function
under an unknown knapsack constraint. A solution to this problem is a policy
that decides which item to pack next based on the past packing history. The
robustness factor of a policy is the worst case ratio of the solution obtained
by following the policy and an optimal solution that knows the knapsack
capacity. We develop an algorithm with a robustness factor that is decreasing
in the curvature of the submodular function. For the extreme cases
corresponding to a modular objective, it matches a previously known and best
possible robustness factor of . For the other extreme case of it
yields a robustness factor of improving over the best previously
known robustness factor of
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
Stochastic Combinatorial Optimization via Poisson Approximation
We study several stochastic combinatorial problems, including the expected
utility maximization problem, the stochastic knapsack problem and the
stochastic bin packing problem. A common technical challenge in these problems
is to optimize some function of the sum of a set of random variables. The
difficulty is mainly due to the fact that the probability distribution of the
sum is the convolution of a set of distributions, which is not an easy
objective function to work with. To tackle this difficulty, we introduce the
Poisson approximation technique. The technique is based on the Poisson
approximation theorem discovered by Le Cam, which enables us to approximate the
distribution of the sum of a set of random variables using a compound Poisson
distribution.
We first study the expected utility maximization problem introduced recently
[Li and Despande, FOCS11]. For monotone and Lipschitz utility functions, we
obtain an additive PTAS if there is a multidimensional PTAS for the
multi-objective version of the problem, strictly generalizing the previous
result.
For the stochastic bin packing problem (introduced in [Kleinberg, Rabani and
Tardos, STOC97]), we show there is a polynomial time algorithm which uses at
most the optimal number of bins, if we relax the size of each bin and the
overflow probability by eps.
For stochastic knapsack, we show a 1+eps-approximation using eps extra
capacity, even when the size and reward of each item may be correlated and
cancelations of items are allowed. This generalizes the previous work [Balghat,
Goel and Khanna, SODA11] for the case without correlation and cancelation. Our
algorithm is also simpler. We also present a factor 2+eps approximation
algorithm for stochastic knapsack with cancelations. the current known
approximation factor of 8 [Gupta, Krishnaswamy, Molinaro and Ravi, FOCS11].Comment: 42 pages, 1 figure, Preliminary version appears in the Proceeding of
the 45th ACM Symposium on the Theory of Computing (STOC13
Online Knapsack Problem under Expected Capacity Constraint
Online knapsack problem is considered, where items arrive in a sequential
fashion that have two attributes; value and weight. Each arriving item has to
be accepted or rejected on its arrival irrevocably. The objective is to
maximize the sum of the value of the accepted items such that the sum of their
weights is below a budget/capacity. Conventionally a hard budget/capacity
constraint is considered, for which variety of results are available. In modern
applications, e.g., in wireless networks, data centres, cloud computing, etc.,
enforcing the capacity constraint in expectation is sufficient. With this
motivation, we consider the knapsack problem with an expected capacity
constraint. For the special case of knapsack problem, called the secretary
problem, where the weight of each item is unity, we propose an algorithm whose
probability of selecting any one of the optimal items is equal to and
provide a matching lower bound. For the general knapsack problem, we propose an
algorithm whose competitive ratio is shown to be that is significantly
better than the best known competitive ratio of for the knapsack
problem with the hard capacity constraint.Comment: To appear in IEEE INFOCOM 2018, April 2018, Honolulu H
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