83,332 research outputs found
Maximizing the number of nonnegative subsets
Given a set of real numbers, if the sum of elements of every subset of
size larger than is negative, what is the maximum number of subsets of
nonnegative sum? In this note we show that the answer is , settling a problem of Tsukerman.
We provide two proofs, the first establishes and applies a weighted version of
Hall's Theorem and the second is based on an extension of the nonuniform
Erd\H{o}s-Ko-Rado Theorem
Solving Medium-Density Subset Sum Problems in Expected Polynomial Time: An Enumeration Approach
The subset sum problem (SSP) can be briefly stated as: given a target integer
and a set containing positive integer , find a subset of
summing to . The \textit{density} of an SSP instance is defined by the
ratio of to , where is the logarithm of the largest integer within
. Based on the structural and statistical properties of subset sums, we
present an improved enumeration scheme for SSP, and implement it as a complete
and exact algorithm (EnumPlus). The algorithm always equivalently reduces an
instance to be low-density, and then solve it by enumeration. Through this
approach, we show the possibility to design a sole algorithm that can
efficiently solve arbitrary density instance in a uniform way. Furthermore, our
algorithm has considerable performance advantage over previous algorithms.
Firstly, it extends the density scope, in which SSP can be solved in expected
polynomial time. Specifically, It solves SSP in expected time
when density , while the previously best
density scope is . In addition, the overall
expected time and space requirement in the average case are proven to be
and respectively. Secondly, in the worst case, it
slightly improves the previously best time complexity of exact algorithms for
SSP. Specifically, the worst-case time complexity of our algorithm is proved to
be , while the previously best result is .Comment: 11 pages, 1 figur
Near-optimal mean value estimates for multidimensional Weyl sums
We obtain sharp estimates for multidimensional generalisations of
Vinogradov's mean value theorem for arbitrary translation-dilation invariant
systems, achieving constraints on the number of variables approaching those
conjectured to be the best possible. Several applications of our bounds are
discussed
Percentile Queries in Multi-Dimensional Markov Decision Processes
Markov decision processes (MDPs) with multi-dimensional weights are useful to
analyze systems with multiple objectives that may be conflicting and require
the analysis of trade-offs. We study the complexity of percentile queries in
such MDPs and give algorithms to synthesize strategies that enforce such
constraints. Given a multi-dimensional weighted MDP and a quantitative payoff
function , thresholds (one per dimension), and probability thresholds
, we show how to compute a single strategy to enforce that for all
dimensions , the probability of outcomes satisfying is at least . We consider classical quantitative payoffs from
the literature (sup, inf, lim sup, lim inf, mean-payoff, truncated sum,
discounted sum). Our work extends to the quantitative case the multi-objective
model checking problem studied by Etessami et al. in unweighted MDPs.Comment: Extended version of CAV 2015 pape
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