17 research outputs found
The Densest k-Subhypergraph Problem
The Densest -Subgraph (DS) problem, and its corresponding minimization
problem Smallest -Edge Subgraph (SES), have come to play a central role
in approximation algorithms. This is due both to their practical importance,
and their usefulness as a tool for solving and establishing approximation
bounds for other problems. These two problems are not well understood, and it
is widely believed that they do not an admit a subpolynomial approximation
ratio (although the best known hardness results do not rule this out).
In this paper we generalize both DS and SES from graphs to hypergraphs.
We consider the Densest -Subhypergraph problem (given a hypergraph ,
find a subset of vertices so as to maximize the number of
hyperedges contained in ) and define the Minimum -Union problem (given a
hypergraph, choose of the hyperedges so as to minimize the number of
vertices in their union). We focus in particular on the case where all
hyperedges have size 3, as this is the simplest non-graph setting. For this
case we provide an -approximation (for arbitrary constant )
for Densest -Subhypergraph and an -approximation for
Minimum -Union. We also give an -approximation for Minimum
-Union in general hypergraphs. Finally, we examine the interesting special
case of interval hypergraphs (instances where the vertices are a subset of the
natural numbers and the hyperedges are intervals of the line) and prove that
both problems admit an exact polynomial time solution on these instances.Comment: 21 page
Balancing spreads of influence in a social network
The personalization of our news consumption on social media has a tendency to
reinforce our pre-existing beliefs instead of balancing our opinions. This
finding is a concern for the health of our democracies which rely on an access
to information providing diverse viewpoints. To tackle this issue from a
computational perspective, Garimella et al. (NIPS'17) modeled the spread of
these viewpoints, also called campaigns, using the well-known independent
cascade model and studied an optimization problem that aims at balancing
information exposure in a social network when two opposing campaigns propagate
in the network. The objective in their -hard optimization problem is to
maximize the number of people that are exposed to either both or none of the
viewpoints. For two different settings, one corresponding to a model where
campaigns spread in a correlated manner, and a second one, where the two
campaigns spread in a heterogeneous manner, they provide constant ratio
approximation algorithms. In this paper, we investigate a more general
formulation of this problem. That is, we assume that different campaigns
propagate in a social network and we aim to maximize the number of people that
are exposed to either or none of the campaigns, where . We
provide dedicated approximation algorithms for both the correlated and
heterogeneous settings. Interestingly, for the heterogeneous setting with
, we give a reduction leading to several approximation hardness
results. Maybe most importantly, we obtain that the problem cannot be
approximated within a factor of for any assuming
Gap-ETH, denoting with the number of nodes in the social network. For , there is no -approximation algorithm if a certain class
of one-way functions exists, where is a given constant which
depends on
On the Concrete Security of Goldreichâs Pseudorandom Generator
International audienceLocal pseudorandom generators allow to expand a short random string into a long pseudo-random string, such that each output bit depends on a constant number d of input bits. Due to its extreme efficiency features, this intriguing primitive enjoys a wide variety of applications in cryptography and complexity. In the polynomial regime, where the seed is of size n and the output of size n s for s > 1, the only known solution, commonly known as Goldreich's PRG, proceeds by applying a simple d-ary predicate to public random sized subsets of the bits of the seed. While the security of Goldreich's PRG has been thoroughly investigated, with a variety of results deriving provable security guarantees against class of attacks in some parameter regimes and necessary criteria to be satisfied by the underlying predicate, little is known about its concrete security and efficiency. Motivated by its numerous theoretical applications and the hope of getting practical instantiations for some of them, we initiate a study of the concrete security of Goldreich's PRG, and evaluate its resistance to cryptanalytic attacks. Along the way, we develop a new guess-and-determine-style attack, and identify new criteria which refine existing criteria and capture the security guarantees of candidate local PRGs in a more fine-grained way
On the Computational Complexity of Measuring Global Stability of Banking Networks
Threats on the stability of a financial system may severely affect the
functioning of the entire economy, and thus considerable emphasis is placed on
the analyzing the cause and effect of such threats. The financial crisis in the
current and past decade has shown that one important cause of instability in
global markets is the so-called financial contagion, namely the spreading of
instabilities or failures of individual components of the network to other,
perhaps healthier, components. This leads to a natural question of whether the
regulatory authorities could have predicted and perhaps mitigated the current
economic crisis by effective computations of some stability measure of the
banking networks. Motivated by such observations, we consider the problem of
defining and evaluating stabilities of both homogeneous and heterogeneous
banking networks against propagation of synchronous idiosyncratic shocks given
to a subset of banks. We formalize the homogeneous banking network model of
Nier et al. and its corresponding heterogeneous version, formalize the
synchronous shock propagation procedures, define two appropriate stability
measures and investigate the computational complexities of evaluating these
measures for various network topologies and parameters of interest. Our results
and proofs also shed some light on the properties of topologies and parameters
of the network that may lead to higher or lower stabilities.Comment: to appear in Algorithmic
Pattern Masking for Dictionary Matching:Theory and Practice
Data masking is a common technique for sanitizing sensitive data maintained in database systems which is becoming increasingly important in various application areas, such as in record linkage of personal data. This work formalizes the Pattern Masking for Dictionary Matching (PMDM) problem: given a dictionary D of d strings, each of length â, a query string q of length â, and a positive integer z, we are asked to compute a smallest set Kâ{1, âŠ, â}, so that if q[i] is replaced by a wildcard for all iâK, then q matches at least z strings from D. Solving PMDM allows providing data utility guarantees as opposed to existing approaches. We first show, through a reduction from the well-known k-Clique problem, that a decision version of the PMDM problem is NP-complete, even for binary strings. We thus approach the problem from a more practical perspective. We show a combinatorial O((dâ)|K|/3+dâ)-time and O(dâ)-space algorithm for PMDM for |K|=O(1). In fact, we show that we cannot hope for a faster combinatorial algorithm, unless the combinatorial k-Clique hypothesis fails (Abboud et al. in SIAM J Comput 47:2527â2555, 2018; Lincoln et al., in: 29th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2018). Our combinatorial algorithm, executed with small |K|, is the backbone of a greedy heuristic that we propose. Our experiments on real-world and synthetic datasets show that our heuristic finds nearly-optimal solutions in practice and is also very efficient. We also generalize this algorithm for the problem of masking multiple query strings simultaneously so that every string has at least z matches in D. PMDM can be viewed as a generalization of the decision version of the dictionary matching with mismatches problem: by querying a PMDM data structure with string q and z=1, one obtains the minimal number of mismatches of q with any string from D. The query time or space of all known data structures for the more restricted problem of dictionary matching with at most k mismatches incurs some exponential factor with respect to k. A simple exact algorithm for PMDM runs in time O(2âd). We present a data structure for PMDM that answers queries over D in time O(2â/2(2â/2+Ï)â) and requires space O(2âd2/Ï2+2â/2d), for any parameter Ïâ[1, d]. We complement our results by showing a two-way polynomial-time reduction between PMDM and the Minimum Union problem [ChlamtĂĄÄ et al., ACM-SIAM Symposium on Discrete Algorithms (SODA) 2017]. This gives a polynomial-time O(d1/4+Ï”)-approximation algorithm for PMDM, which is tight under a plausible complexity conjecture. This is an extended version of a paper that was presented at International Symposium on Algorithms and Computation (ISAAC) 2021
Revisiting the Concrete Security of Goldreich's Pseudorandom Generator
Local pseudorandom generators are a class of fundamental cryptographic
primitives having very broad applications in theoretical cryptography.
Following Couteau et al.'s work in ASIACRYPT 2018, this paper further studies
the concrete security of one important class of local pseudorandom generators,
i.e., Goldreich's pseudorandom generators. Our first attack is of the
guess-and-determine type. Our result significantly improves the
state-of-the-art algorithm proposed by Couteau et al., in terms of both
asymptotic and concrete complexity, and breaks all the challenge parameters
they proposed. For instance, for a parameter set suggested for 128 bits of
security, we could solve the instance faster by a factor of about ,
thereby destroying the claimed security completely. Our second attack further
exploits the extremely sparse structure of the predicate and combines
ideas from iterative decoding. This novel attack, named guess-and-decode,
substantially improves the guess-and-determine approaches for
cryptographic-relevant parameters. All the challenge parameter sets proposed in
Couteau et al.'s work in ASIACRYPT 2018 aiming for 80-bit (128-bit) security
levels can be solved in about () operations. We suggest new
parameters for achieving 80-bit (128-bit) security with respect to our attacks.
We also extend the attack to other promising predicates and investigate their
resistance.Comment: 20 pages, 9 figure