Recognizing sparse perfect elimination bipartite graphs

When applying Gaussian elimination to a sparse matrix, it is desirable to avoid turning zeros into non-zeros to preserve the sparsity. The class of perfect elimination bipartite graphs is closely related to square matrices that Gaussian elimination can be applied to without turning any zero into a non-zero. Existing literature on the recognition of this class and finding suitable pivots mainly focusses on time complexity. For $n \times n$ matrices with m non-zero elements, the currently best known algorithm has a time complexity of $O(n^3/\log n)$. However, when viewed from a practical perspective, the space complexity also deserves attention: it may not be worthwhile to look for a suitable set of pivots for a sparse matrix if this requires $\Omega(n^2)$ space. We present two new algorithms for the recognition of sparse instances: one with a $O(n m)$ time complexity in $\Theta(n^2)$ space and one with a $O(m^2)$ time complexity in $\Theta(m)$ space. Furthermore, if we allow only pivots on the diagonal, our second algorithm can easily be adapted to run in time $O(n m)$

Bisimplicial edges in bipartite graphs

Bisimplicial edges in bipartite graphs are closely related to pivots in Gaussian elimination that avoid turning zeroes into non-zeroes. We present a new deterministic algorithm to nd such edges in bipartite graphs. The expected time complexity of our new algorithm is $O(n^2 \log n)$ on random bipartite graphs in which each edge is present with a fixed probability p, a polynomial improvement over the fastest algorithm found in the existing literature

A note on perfect partial elimination

In Gaussian elimination it is often desirable to preserve existing zeros (sparsity). This is closely related to perfect elimination schemes on graphs. Such schemes can be found in polynomial time. Gaussian elimination uses a pivot for each column, so opportunities for preserving sparsity can be missed. In this paper we consider a more flexible process that selects a pivot for each nonzero to be eliminated and show that recognizing matrices that allow such perfect partial elimination schemes is NP-hard

Consistency of a system of equations: What does that mean?

The concept of (structural) consistency also called structural solvability is an important basic tool for analyzing the structure of systems of equations. Our aim is to provide a sound and practically relevant meaning to this concept. The implications of consistency are expressed in terms of explicit density and stability results. We also illustrate, by typical examples, the limitations of the concept

Bisimplicial edges in bipartite graphs

Bisimplicial edges in bipartite graphs are closely related to pivots in Gaussian elimination that avoid turning zeroes into non-zeroes. We present a new deterministic algorithm to find such edges in bipartite graphs. Our algorithm is very simple and easy to implement. Its running-time is $O(nm)$, where $n$ is the number of vertices and $m$ is the number of edges. Furthermore, for any fixed $p$ and random bipartite graphs in the $G_{n,n,p}$ model, the expected running-time of our algorithm is $O(n^2)$, which is linear in the input size

On bounded block decomposition problems for under-specified systems of equations

When solving a system of equations, it can be beneficial not to solve it in its entirety at once, but rather to decompose it into smaller subsystems that can be solved in order. Based on a bisimplicial graph representation we analyze the parameterized complexity of two problems central to such a decomposition: The Free Square Block problem related to finding smallest subsystems that can be solved separately, and the Bounded Block Decomposition problem related to determining a decomposition where the largest subsystem is as small as possible. We show both problems to be W[1]-hard. Finally we relate these problems to crown structures and settle two open questions regarding them using our results

Bipartite Graphs and the Decomposition of Systems of Equations

Solving large systems of equations is a problem often encountered in engineering disciplines. However, as such systems grow, the effort required for finding a solution to them increases as well. In order to be able to cope with ever larger systems of equations, some form of decomposition is needed. By decomposing a large system into smaller subsystems, the total effort required for finding a solution may decrease. However, whether this is really the case of course depends on the additional effort required for obtaining the decomposition itself. In this thesis several aspects of the difficulty of obtaining such decompositions are explored

Privacy-conscious threat intelligence using DNSBLoom

\u3cp\u3eThe Domain Name System (DNS) is an essential component of every interaction on the Internet. DNS translates human-readable names into machine readable IP addresses. Conversely, DNS requests provide a wealth of information about what goes on in the network. Malicious activity - such as phishing, malware and botnets - also makes use of the DNS. Thus, monitoring DNS traffic is essential for the security team's toolbox. Yet because DNS is so essential to Internet services, tracking DNS is also highly privacy-invasive, as what domain names a user requests reveals their Internet use. Therefore, in an age of comprehensive privacy legislation, such as Europe's GDPR, simply logging every DNS request is not acceptable.In this paper we present DNSBloom, a system that uses Bloom Filters as a privacy-enhancing technology to store DNS requests. Bloom Filters act as a probabilistic set, where a membership test either returns probable membership (with a small false positive probability), or certain non-membership. Because Bloom Filters do not store original information, and because DNSBloom aggregates queries from multiple users over fixed time periods, the system offers strong privacy guarantees while enabling security professionals to check with a high degree of confidence whether certain DNS queries associated with malicious activity have occurred. We validate DNSBloom through three case studies performed on the production DNS infrastructure of a major global research network, and release a working prototype, that integrates with popular DNS resolvers, in open source.\u3c/p\u3

Privacy-conscious threat intelligence using DNSBLoom

The Domain Name System (DNS) is an essential component of every interaction on the Internet. DNS translates human-readable names into machine readable IP addresses. Conversely, DNS requests provide a wealth of information about what goes on in the network. Malicious activity - such as phishing, malware and botnets - also makes use of the DNS. Thus, monitoring DNS traffic is essential for the security team's toolbox. Yet because DNS is so essential to Internet services, tracking DNS is also highly privacy-invasive, as what domain names a user requests reveals their Internet use. Therefore, in an age of comprehensive privacy legislation, such as Europe's GDPR, simply logging every DNS request is not acceptable.In this paper we present DNSBloom, a system that uses Bloom Filters as a privacy-enhancing technology to store DNS requests. Bloom Filters act as a probabilistic set, where a membership test either returns probable membership (with a small false positive probability), or certain non-membership. Because Bloom Filters do not store original information, and because DNSBloom aggregates queries from multiple users over fixed time periods, the system offers strong privacy guarantees while enabling security professionals to check with a high degree of confidence whether certain DNS queries associated with malicious activity have occurred. We validate DNSBloom through three case studies performed on the production DNS infrastructure of a major global research network, and release a working prototype, that integrates with popular DNS resolvers, in open source

Quarantine Net: design and application

Lately the world has seen an alarming increase in both the amount of new computer viruses and worms, as well as the speed at which they spread. Dealing with such outbreaks puts a heavy burden on end-users as well as operators of (enterprise) networks. In this paper we present an overview of, as well as operational experiences with Quarantine Net (QNET), a system that has been developed at the University of Twente. The goal of QNET is to limit the impact of computer viruses and worms outbreaks by quarantining infected systems in an automatic way. QNET not only reduces the speed at which viruses and worms spread, but also helps infected users to clean their systems and restore connectivity in a fast and easy way. Deployment at the University of Twente’s network has shown that QNET is an effective system to deal with outbreaks of computer viruses and worms, and sometimes achieves earlier results than popular antivirus tools