459,382 research outputs found
Multidimensional subset sum problem
This thesis explores new modifications to the successful LLL approach to solving the Subset Sum problem. This work is an optimization of the matrix representation of an instance. Traditionally, the basis matrix contained only one column with set elements and the sum. In this thesis we suggest having several data columns (thus introducing multidimensionality). This allows us to reduce the size of coulmn entries which changes the complexity of the problem. Splitting the data into multiple columns greatly simplifies the task of solving the Subset Sum problem. However, other problems arise when we try to generate multiple columns. Here we try to find the optimal way to do the split and present the results. Our main goal was to try to solve the current hardest Subset Sum problem instances: the ones with density slightly greater than 1. Dramatic improvement in the rate of success was observed (up to 1500 %) compared to one- dimensional implementations
A Practical Approach for Successive Omniscience
The system that we study in this paper contains a set of users that observe a
discrete memoryless multiple source and communicate via noise-free channels
with the aim of attaining omniscience, the state that all users recover the
entire multiple source. We adopt the concept of successive omniscience (SO),
i.e., letting the local omniscience in some user subset be attained before the
global omniscience in the entire system, and consider the problem of how to
efficiently attain omniscience in a successive manner. Based on the existing
results on SO, we propose a CompSetSO algorithm for determining a complimentary
set, a user subset in which the local omniscience can be attained first without
increasing the sum-rate, the total number of communications, for the global
omniscience. We also derive a sufficient condition for a user subset to be
complimentary so that running the CompSetSO algorithm only requires a lower
bound, instead of the exact value, of the minimum sum-rate for attaining global
omniscience. The CompSetSO algorithm returns a complimentary user subset in
polynomial time. We show by example how to recursively apply the CompSetSO
algorithm so that the global omniscience can be attained by multi-stages of SO
Fast Monotone Summation over Disjoint Sets
We study the problem of computing an ensemble of multiple sums where the
summands in each sum are indexed by subsets of size of an -element
ground set. More precisely, the task is to compute, for each subset of size
of the ground set, the sum over the values of all subsets of size that are
disjoint from the subset of size . We present an arithmetic circuit that,
without subtraction, solves the problem using arithmetic
gates, all monotone; for constant , this is within the factor
of the optimal. The circuit design is based on viewing the summation as a "set
nucleation" task and using a tree-projection approach to implement the
nucleation. Applications include improved algorithms for counting heaviest
-paths in a weighted graph, computing permanents of rectangular matrices,
and dynamic feature selection in machine learning
Improved Low-qubit Hidden Shift Algorithms
Hidden shift problems are relevant to assess the quantum security of various
cryptographic constructs. Multiple quantum subexponential time algorithms have
been proposed. In this paper, we propose some improvements on a polynomial
quantum memory algorithm proposed by Childs, Jao and Soukharev in 2010. We use
subset-sum algorithms to significantly reduce its complexity. We also propose
new tradeoffs between quantum queries, classical time and classical memory to
solve this problem
Structure-Aware Sampling: Flexible and Accurate Summarization
In processing large quantities of data, a fundamental problem is to obtain a
summary which supports approximate query answering. Random sampling yields
flexible summaries which naturally support subset-sum queries with unbiased
estimators and well-understood confidence bounds.
Classic sample-based summaries, however, are designed for arbitrary subset
queries and are oblivious to the structure in the set of keys. The particular
structure, such as hierarchy, order, or product space (multi-dimensional),
makes range queries much more relevant for most analysis of the data.
Dedicated summarization algorithms for range-sum queries have also been
extensively studied. They can outperform existing sampling schemes in terms of
accuracy on range queries per summary size. Their accuracy, however, rapidly
degrades when, as is often the case, the query spans multiple ranges. They are
also less flexible - being targeted for range sum queries alone - and are often
quite costly to build and use.
In this paper we propose and evaluate variance optimal sampling schemes that
are structure-aware. These summaries improve over the accuracy of existing
structure-oblivious sampling schemes on range queries while retaining the
benefits of sample-based summaries: flexible summaries, with high accuracy on
both range queries and arbitrary subset queries
A Note on the Density of the Multiple Subset Sum Problems
It is well known that the general subset sum problem is NP-complete. However, almost all subset sum problems with density less than can be solved in polynomial time with an oracle that can find the shortest vector in a special lattice. In this paper, we give a similar result for the multiple subset sum problems which has subset sum problems with the same solution. Some extended versions of the multiple subset sum problems are also considered. In addition, a modified lattice is involved to make the analysis much simpler than before
Subset sum phase transitions and data compression
We propose a rigorous analysis approach for the subset sum problem in the
context of lossless data compression, where the phase transition of the subset
sum problem is directly related to the passage between ambiguous and
non-ambiguous decompression, for a compression scheme that is based on
specifying the sequence composition. The proposed analysis lends itself to
straightforward extensions in several directions of interest, including
non-binary alphabets, incorporation of side information at the decoder
(Slepian-Wolf coding), and coding schemes based on multiple subset sums. It is
also demonstrated that the proposed technique can be used to analyze the
critical behavior in a more involved situation where the sequence composition
is not specified by the encoder.Comment: 14 pages, submitted to the Journal of Statistical Mechanics: Theory
and Experimen
Multiple Subset Problem as an encryption scheme for communication
Using well-known mathematical problems for encryption is a widely used
technique because they are computationally hard and provide security against
potential attacks on the encryption method. The subset sum problem (SSP) can be
defined as finding a subset of integers from a given set, whose sum is equal to
a specified integer. The classic SSP has various variants, one of which is the
multiple-subset problem (MSSP). In the MSSP, the goal is to select items from a
given set and distribute them among multiple bins, en-suring that the capacity
of each bin is not exceeded while maximizing the total weight of the selected
items. This approach addresses a related problem with a different perspective.
Here a related different kind of problem is approached: given a set of sets
A={A1, A2..., An}, find an integer s for which every subset of the given sets
is summed up to, if such an integer exists. The problem is NP-complete when
considering it as a variant of SSP. However, there exists an algorithm that is
relatively efficient for known pri-vate keys. This algorithm is based on
dispensing non-relevant values of the potential sums. In this paper we present
the encryption scheme based on MSSP and present its novel usage and
implementation in communication
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