2,009 research outputs found
Improved Online Algorithms for Knapsack and GAP in the Random Order Model
The knapsack problem is one of the classical problems in combinatorial optimization: Given a set of items, each specified by its size and profit, the goal is to find a maximum profit packing into a knapsack of bounded capacity. In the online setting, items are revealed one by one and the decision, if the current item is packed or discarded forever, must be done immediately and irrevocably upon arrival. We study the online variant in the random order model where the input sequence is a uniform random permutation of the item set.
We develop a randomized (1/6.65)-competitive algorithm for this problem, outperforming the current best algorithm of competitive ratio 1/8.06 [Kesselheim et al. SIAM J. Comp. 47(5)]. Our algorithm is based on two new insights: We introduce a novel algorithmic approach that employs two given algorithms, optimized for restricted item classes, sequentially on the input sequence. In addition, we study and exploit the relationship of the knapsack problem to the 2-secretary problem.
The generalized assignment problem (GAP) includes, besides the knapsack problem, several important problems related to scheduling and matching. We show that in the same online setting, applying the proposed sequential approach yields a (1/6.99)-competitive randomized algorithm for GAP. Again, our proposed algorithm outperforms the current best result of competitive ratio 1/8.06 [Kesselheim et al. SIAM J. Comp. 47(5)]
Polymatroid Prophet Inequalities
Consider a gambler and a prophet who observe a sequence of independent,
non-negative numbers. The gambler sees the numbers one-by-one whereas the
prophet sees the entire sequence at once. The goal of both is to decide on
fractions of each number they want to keep so as to maximize the weighted
fractional sum of the numbers chosen.
The classic result of Krengel and Sucheston (1977-78) asserts that if both
the gambler and the prophet can pick one number, then the gambler can do at
least half as well as the prophet. Recently, Kleinberg and Weinberg (2012) have
generalized this result to settings where the numbers that can be chosen are
subject to a matroid constraint.
In this note we go one step further and show that the bound carries over to
settings where the fractions that can be chosen are subject to a polymatroid
constraint. This bound is tight as it is already tight for the simple setting
where the gambler and the prophet can pick only one number. An interesting
application of our result is in mechanism design, where it leads to improved
results for various problems
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
Online Independent Set Beyond the Worst-Case: Secretaries, Prophets, and Periods
We investigate online algorithms for maximum (weight) independent set on
graph classes with bounded inductive independence number like, e.g., interval
and disk graphs with applications to, e.g., task scheduling and spectrum
allocation. In the online setting, it is assumed that nodes of an unknown graph
arrive one by one over time. An online algorithm has to decide whether an
arriving node should be included into the independent set. Unfortunately, this
natural and practically relevant online problem cannot be studied in a
meaningful way within a classical competitive analysis as the competitive ratio
on worst-case input sequences is lower bounded by .
As a worst-case analysis is pointless, we study online independent set in a
stochastic analysis. Instead of focussing on a particular stochastic input
model, we present a generic sampling approach that enables us to devise online
algorithms achieving performance guarantees for a variety of input models. In
particular, our analysis covers stochastic input models like the secretary
model, in which an adversarial graph is presented in random order, and the
prophet-inequality model, in which a randomly generated graph is presented in
adversarial order. Our sampling approach bridges thus between stochastic input
models of quite different nature. In addition, we show that our approach can be
applied to a practically motivated admission control setting.
Our sampling approach yields an online algorithm for maximum independent set
with competitive ratio with respect to all of the mentioned
stochastic input models. for graph classes with inductive independence number
. The approach can be extended towards maximum-weight independent set by
losing only a factor of in the competitive ratio with denoting
the (expected) number of nodes
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