On the Distance Identifying Set Meta-Problem and Applications to the Complexity of Identifying Problems on Graphs

Numerous problems consisting in identifying vertices in graphs using distances are useful in domains such as network verification and graph isomorphism. Unifying them into a meta-problem may be of main interest. We introduce here a promising solution named Distance Identifying Set. The model contains Identifying Code (IC), Locating Dominating Set (LD) and their generalizations $r$-IC and $r$-LD where the closed neighborhood is considered up to distance $r$. It also contains Metric Dimension (MD) and its refinement $r$-MD in which the distance between two vertices is considered as infinite if the real distance exceeds $r$. Note that while IC = 1-IC and LD = 1-LD, we have MD = $\infty$-MD; we say that MD is not local In this article, we prove computational lower bounds for several problems included in Distance Identifying Set by providing generic reductions from (Planar) Hitting Set to the meta-problem. We mainly focus on two families of problem from the meta-problem: the first one, called bipartite gifted local, contains $r$-IC, $r$-LD and $r$-MD for each positive integer $r$ while the second one, called 1-layered, contains LD, MD and $r$-MD for each positive integer $r$. We have: - the 1-layered problems are NP-hard even in bipartite apex graphs, - the bipartite gifted local problems are NP-hard even in bipartite planar graphs, - assuming ETH, all these problems cannot be solved in $2^{o(\sqrt{n})}$ when restricted to bipartite planar or apex graph, respectively, and they cannot be solved in $2^{o(n)}$ on bipartite graphs, - even restricted to bipartite graphs, they do not admit parameterized algorithms in $2^{O(k)}.n^{O(1)}$ except if W[0] = W[2]. Here $k$ is the solution size of a relevant identifying set. In particular, Metric Dimension cannot be solved in $2^{o(n)}$ under ETH, answering a question of Hartung in 2013

An Agent-Based Algorithm exploiting Multiple Local Dissimilarities for Clusters Mining and Knowledge Discovery

We propose a multi-agent algorithm able to automatically discover relevant regularities in a given dataset, determining at the same time the set of configurations of the adopted parametric dissimilarity measure yielding compact and separated clusters. Each agent operates independently by performing a Markovian random walk on a suitable weighted graph representation of the input dataset. Such a weighted graph representation is induced by the specific parameter configuration of the dissimilarity measure adopted by the agent, which searches and takes decisions autonomously for one cluster at a time. Results show that the algorithm is able to discover parameter configurations that yield a consistent and interpretable collection of clusters. Moreover, we demonstrate that our algorithm shows comparable performances with other similar state-of-the-art algorithms when facing specific clustering problems

A parallel genetic algorithm for the Steiner Problem in Networks

This paper presents a parallel genetic algorithm to the Steiner Problem in Networks. Several previous papers have proposed the adoption of GAs and others metaheuristics to solve the SPN demonstrating the validity of their approaches. This work differs from them for two main reasons: the dimension and the characteristics of the networks adopted in the experiments and the aim from which it has been originated. The reason that aimed this work was namely to build a comparison term for validating deterministic and computationally inexpensive algorithms which can be used in practical engineering applications, such as the multicast transmission in the Internet. On the other hand, the large dimensions of our sample networks require the adoption of a parallel implementation of the Steiner GA, which is able to deal with such large problem instances

A resource-frugal probabilistic dictionary and applications in (meta)genomics

Genomic and metagenomic fields, generating huge sets of short genomic sequences, brought their own share of high performance problems. To extract relevant pieces of information from the huge data sets generated by current sequencing techniques, one must rely on extremely scalable methods and solutions. Indexing billions of objects is a task considered too expensive while being a fundamental need in this field. In this paper we propose a straightforward indexing structure that scales to billions of element and we propose two direct applications in genomics and metagenomics. We show that our proposal solves problem instances for which no other known solution scales-up. We believe that many tools and applications could benefit from either the fundamental data structure we provide or from the applications developed from this structure.Comment: Submitted to PSC 201

Unifying Sparsest Cut, Cluster Deletion, and Modularity Clustering Objectives with Correlation Clustering

Graph clustering, or community detection, is the task of identifying groups of closely related objects in a large network. In this paper we introduce a new community-detection framework called LambdaCC that is based on a specially weighted version of correlation clustering. A key component in our methodology is a clustering resolution parameter, $\lambda$, which implicitly controls the size and structure of clusters formed by our framework. We show that, by increasing this parameter, our objective effectively interpolates between two different strategies in graph clustering: finding a sparse cut and forming dense subgraphs. Our methodology unifies and generalizes a number of other important clustering quality functions including modularity, sparsest cut, and cluster deletion, and places them all within the context of an optimization problem that has been well studied from the perspective of approximation algorithms. Our approach is particularly relevant in the regime of finding dense clusters, as it leads to a 2-approximation for the cluster deletion problem. We use our approach to cluster several graphs, including large collaboration networks and social networks

Multiscale approach for the network compression-friendly ordering

We present a fast multiscale approach for the network minimum logarithmic arrangement problem. This type of arrangement plays an important role in a network compression and fast node/link access operations. The algorithm is of linear complexity and exhibits good scalability which makes it practical and attractive for using on large-scale instances. Its effectiveness is demonstrated on a large set of real-life networks. These networks with corresponding best-known minimization results are suggested as an open benchmark for a research community to evaluate new methods for this problem