12 research outputs found
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
Ternary Syndrome Decoding with Large Weight
The Syndrome Decoding problem is at the core of many code-based
cryptosystems. In this paper, we study ternary Syndrome Decoding in large
weight. This problem has been introduced in the Wave signature scheme but has
never been thoroughly studied. We perform an algorithmic study of this problem
which results in an update of the Wave parameters. On a more fundamental level,
we show that ternary Syndrome Decoding with large weight is a really harder
problem than the binary Syndrome Decoding problem, which could have several
applications for the design of code-based cryptosystems
On Near-Linear-Time Algorithms for Dense Subset Sum
In the Subset Sum problem we are given a set of positive integers and a target and are asked whether some subset of sums to . Natural parameters for this problem that have been studied in the literature are and as well as the maximum input number and the sum of all input numbers . In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in . In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time . Our main question is: When can dense Subset Sum be solved in near-linear time ? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters for which dense Subset Sum is in time . For notational convenience we assume without loss of generality that (as larger numbers can be ignored) and (using symmetry). Then our dichotomy reads as follows: - By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP'91], we show that Subset Sum is in near-linear time if . - We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with , then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds
Privacy-Preserving Distributed Learning with Secret Gradient Descent
In many important application domains of machine learning, data is a
privacy-sensitive resource. In addition, due to the growing complexity of the
models, single actors typically do not have sufficient data to train a model on
their own. Motivated by these challenges, we propose Secret Gradient Descent
(SecGD), a method for training machine learning models on data that is spread
over different clients while preserving the privacy of the training data. We
achieve this by letting each client add temporary noise to the information they
send to the server during the training process. They also share this noise in
separate messages with the server, which can then subtract it from the
previously received values. By routing all data through an anonymization
network such as Tor, we prevent the server from knowing which messages
originate from the same client, which in turn allows us to show that breaking a
client's privacy is computationally intractable as it would require solving a
hard instance of the subset sum problem. This setup allows SecGD to work in the
presence of only two honest clients and a malicious server, and without the
need for peer-to-peer connections.Comment: 13 pages, 1 figur
Approximating Knapsack and Partition via Dense Subset Sums
Knapsack and Partition are two important additive problems whose fine-grained
complexities in the -approximation setting are not yet
settled. In this work, we make progress on both problems by giving improved
algorithms.
- Knapsack can be -approximated in time, improving the previous by Jin (ICALP'19). There is a known conditional
lower bound of based on -convolution
hypothesis.
- Partition can be -approximated in time, improving the previous by Bringmann and Nakos (SODA'21). There is a known
conditional lower bound of based on Strong
Exponential Time Hypothesis.
Both of our new algorithms apply the additive combinatorial results on dense
subset sums by Galil and Margalit (SICOMP'91), Bringmann and Wellnitz
(SODA'21). Such techniques have not been explored in the context of Knapsack
prior to our work. In addition, we design several new methods to speed up the
divide-and-conquer steps which naturally arise in solving additive problems.Comment: To appear in SODA 2023. Corrects minor mistakes in Lemma 3.3 and
Lemma 3.5 in the proceedings version of this pape