Submodular Maximization Subject to Matroid Intersection on the Fly

Despite a surge of interest in submodular maximization in the data stream model, there remain significant gaps in our knowledge about what can be achieved in this setting, especially when dealing with multiple constraints. In this work, we nearly close several basic gaps in submodular maximization subject to k matroid constraints in the data stream model. We present a new hardness result showing that super polynomial memory in k is needed to obtain an o(k/(log k))-approximation. This implies near optimality of prior algorithms. For the same setting, we show that one can nevertheless obtain a constant-factor approximation by maintaining a set of elements whose size is independent of the stream size. Finally, for bipartite matching constraints, a well-known special case of matroid intersection, we present a new technique to obtain hardness bounds that are significantly stronger than those obtained with prior approaches. Prior results left it open whether a 2-approximation may exist in this setting, and only a complexity-theoretic hardness of 1.91 was known. We prove an unconditional hardness of 2.69

Towards Tight Bounds for the Streaming Set Cover Problem

We consider the classic Set Cover problem in the data stream model. For $n$ elements and $m$ sets ($m\geq n$) we give a $O(1/\delta)$-pass algorithm with a strongly sub-linear $\tilde{O}(mn^{\delta})$ space and logarithmic approximation factor. This yields a significant improvement over the earlier algorithm of Demaine et al. [DIMV14] that uses exponentially larger number of passes. We complement this result by showing that the tradeoff between the number of passes and space exhibited by our algorithm is tight, at least when the approximation factor is equal to $1$. Specifically, we show that any algorithm that computes set cover exactly using $({1 \over 2\delta}-1)$ passes must use $\tilde{\Omega}(mn^{\delta})$ space in the regime of $m=O(n)$. Furthermore, we consider the problem in the geometric setting where the elements are points in $\mathbb{R}^2$ and sets are either discs, axis-parallel rectangles, or fat triangles in the plane, and show that our algorithm (with a slight modification) uses the optimal $\tilde{O}(n)$ space to find a logarithmic approximation in $O(1/\delta)$ passes. Finally, we show that any randomized one-pass algorithm that distinguishes between covers of size 2 and 3 must use a linear (i.e., $\Omega(mn)$) amount of space. This is the first result showing that a randomized, approximate algorithm cannot achieve a space bound that is sublinear in the input size. This indicates that using multiple passes might be necessary in order to achieve sub-linear space bounds for this problem while guaranteeing small approximation factors.Comment: A preliminary version of this paper is to appear in PODS 201

Secure and Efficient Multiparty Private Set Intersection Cardinality

The article of record as published may be found at http://dx.doi.org/10.3934/amc.2020071In the field of privacy preserving protocols, Private Set Intersection (PSI) plays an important role. In most of the cases, PSI allows two parties to securely determine the intersection of their private input sets, and no other information. In this paper, employing a Bloom filter, we propose a Multiparty Private Set Intersection Cardinality (MPSI-CA), where the number of participants in PSI is not limited to two. The security of our scheme is achieved in the standard model under the Decisional Diffie-Hellman (DDH) assumption against semi-honest adversaries. Our scheme is flexible in the sense that set size of one participant is independent from that of the others. We consider the number of modular exponentiations in order to determine computational complexity. In our construction, communication and computation overheads of each participant is O(v max k) except that the complexity of the designated party is O(v1), where v max is the maximum set size, v1 denotes the set size of the designated party and k is a security parameter. Particularly, our MSPI-CA is the first that incurs linear complexity in terms of set size, namely O(nv max k), where n is the number of participants. Further, we extend our MPSI-CA to MPSI retaining all the security attributes and other properties. As far as we are aware of, there is no other MPSI so far where individual computational cost of each participant is independent of the number of participants. Unlike MPSI-CA, our MPSI does not require any kind of broadcast channel as it uses star network topology in the sense that a designated party communicates with everyone else

Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams

We study the Maximum Independent Set problem for geometric objects given in the data stream model. A set of geometric objects is said to be independent if the objects are pairwise disjoint. We consider geometric objects in one and two dimensions, i.e., intervals and disks. Let $\alpha$ be the cardinality of the largest independent set. Our goal is to estimate $\alpha$ in a small amount of space, given that the input is received as a one-pass stream. We also consider a generalization of this problem by assigning weights to each object and estimating $\beta$, the largest value of a weighted independent set. We initialize the study of this problem in the turnstile streaming model (insertions and deletions) and provide the first algorithms for estimating $\alpha$ and $\beta$. For unit-length intervals, we obtain a $(2+\epsilon)$-approximation to $\alpha$ and $\beta$ in poly$(\frac{\log(n)}{\epsilon})$ space. We also show a matching lower bound. Combined with the $3/2$-approximation for insertion-only streams by Cabello and Perez-Lanterno [CP15], our result implies a separation between the insertion-only and turnstile model. For unit-radius disks, we obtain a $\left(\frac{8\sqrt{3}}{\pi}\right)$-approximation to $\alpha$ and $\beta$ in poly$(\log(n), \epsilon^{-1})$ space, which is closely related to the hexagonal circle packing constant. We provide algorithms for estimating $\alpha$ for arbitrary-length intervals under a bounded intersection assumption and study the parameterized space complexity of estimating $\alpha$ and $\beta$, where the parameter is the ratio of maximum to minimum interval length.Comment: The lower bound for arbitrary length intervals in the previous version contains a bug, we are updating the submission to reflect thi

Breaking two PSI-CA protocols in polynomial time

Private Set Intersection Cardinality(PSI-CA) is a type of secure two-party computation. It enables two parties, each holding a private set, to jointly compute the cardinality of their intersection without revealing any other private information about their respective sets. In this paper, we manage to break two PSI-CA protocols by recovering the specific intersection items in polynomial time. Among them, the PSI-CA protocol proposed by De Cristofaro et al. in 2012 is the most popular PSI-CA protocol based on the Google Scholar search results and it is still deemed one of the most efficient PSI-CA protocols. In this paper, we also propose several solutions to these protocols\u27 security problems

Secure and efficient multiparty private set intersection cardinality

17 USC 105 interim-entered record; under review.The article of record as published may be found at http://dx.doi.org/10.3934/amc.2020071In the field of privacy preserving protocols, Private Set Intersection (PSI) plays an important role. In most of the cases, PSI allows two parties to securely determine the intersection of their private input sets, and no other information. In this paper, employing a Bloom filter, we propose a Multiparty Private Set Intersection Cardinality (MPSI-CA), where the number of participants in PSI is not limited to two. The security of our scheme is achieved in the standard model under the Decisional Diffie-Hellman (DDH) assumption against semi-honest adversaries. Our scheme is flexible in the sense that set size of one participant is independent from that of the others. We consider the number of modular exponentiations in order to determine computational complexity. In our construction, communication and computation overheads of each participant is O(vmaxk) except that the complexity of the designated party is O(v1), where vmax is the maximum set size, v1 denotes the set size of the designated party and k is a security parameter. Particularly, our MSPI-CA is the first that incurs linear complexity in terms of set size, namely O(nvmaxk), where n is the number of participants. Further, we extend our MPSI-CA to MPSI retaining all the security attributes and other properties. As far as we are aware of, there is no other MPSI so far where individual computational cost of each participant is independent of the number of participants. Unlike MPSI-CA, our MPSI does not require any kind of broadcast channel as it uses star network topology in the sense that a designated party communicates with everyone else

Crypto'Graph: Leveraging Privacy-Preserving Distributed Link Prediction for Robust Graph Learning

Graphs are a widely used data structure for collecting and analyzing relational data. However, when the graph structure is distributed across several parties, its analysis is particularly challenging. In particular, due to the sensitivity of the data each party might want to keep their partial knowledge of the graph private, while still willing to collaborate with the other parties for tasks of mutual benefit, such as data curation or the removal of poisoned data. To address this challenge, we propose Crypto'Graph, an efficient protocol for privacy-preserving link prediction on distributed graphs. More precisely, it allows parties partially sharing a graph with distributed links to infer the likelihood of formation of new links in the future. Through the use of cryptographic primitives, Crypto'Graph is able to compute the likelihood of these new links on the joint network without revealing the structure of the private individual graph of each party, even though they know the number of nodes they have, since they share the same graph but not the same links. Crypto'Graph improves on previous works by enabling the computation of a certain number of similarity metrics without any additional cost. The use of Crypto'Graph is illustrated for defense against graph poisoning attacks, in which it is possible to identify potential adversarial links without compromising the privacy of the graphs of individual parties. The effectiveness of Crypto'Graph in mitigating graph poisoning attacks and achieving high prediction accuracy on a graph neural network node classification task is demonstrated through extensive experimentation on a real-world dataset