arXiv subjects

Michele Zito

Publications and source records attributed to Michele Zito.

3 recordsLinked to original sources

arXiv2019

Vertex-Connectivity Measures for Node Failure Identification in Boolean Network Tomography

In this paper we study the node failure identification problem in undirected graphs by means of Boolean Network Tomography. We argue that vertex connectivity plays a central role. We show tight bounds on the maximal identifiability in a particular class of graphs, the Line of Sight networks. We prove slightly weaker bounds on arbitrary networks. Finally we initiate the study of maximal identifiability in random networks. We focus on two models: the classical Erdős-Rényi model, and that of Random Regular graphs. The framework proposed in the paper allows a probabilistic analysis of the identifiability in random networks giving a tradeoff between the number of monitors to place and the maximal identifiability.

cs.NI↗

arXiv2018

Dynamic Programming Optimization in Line of Sight Networks

Line of Sight (LoS) networks were designed to model wireless communication in settings which may contain obstacles restricting node visibility. For fixed positive integer $d$, and positive integer $ω$, a graph $G=(V,E)$ is a ($d$-dimensional) LoS network with range parameter $ω$ if it can be embedded in a cube of side size $n$ of the $d$-dimensional integer grid so that each pair of vertices in $V$ are adjacent if and only if their embedding coordinates differ only in one position and such difference is less than $ω$. In this paper we investigate a dynamic programming (DP) approach which can be used to obtain efficient algorithmic solutions for various combinatorial problems in LoS networks. In particular DP solves the Maximum Independent Set (MIS) problem in LoS networks optimally for any $ω$ on {\em narrow} LoS networks (i.e. networks which can be embedded in a $n \times k \times k \ldots \times k$ region, for some fixed $k$ independent of $n$). In the unrestricted case it has been shown that the MIS problem is NP-hard when $ ω> 2$ (the hardness proof goes through for any $ω=O(n^{1-δ})$, for fixed $0<δ<1$). We describe how DP can be used as a building block in the design of good approximation algorithms. In particular we present a 2-approximation algorithm and a fast polynomial time approximation scheme for the MIS problem in arbitrary $d$-dimensional LoS networks. Finally we comment on how the approach can be adapted to solve a number of important optimization problems in LoS networks.

cs.DS↗

arXiv2011

The Complexity of the Empire Colouring Problem

We investigate the computational complexity of the empire colouring problem (as defined by Percy Heawood in 1890) for maps containing empires formed by exactly $r > 1$ countries each. We prove that the problem can be solved in polynomial time using $s$ colours on maps whose underlying adjacency graph has no induced subgraph of average degree larger than $s/r$. However, if $s \geq 3$, the problem is NP-hard even if the graph is a forest of paths of arbitrary lengths (for any $r \geq 2$, provided $s < 2r - \sqrt(2r + 1/4+ 3/2)$. Furthermore we obtain a complete characterization of the problem's complexity for the case when the input graph is a tree, whereas our result for arbitrary planar graphs fall just short of a similar dichotomy. Specifically, we prove that the empire colouring problem is NP-hard for trees, for any $r \geq 2$, if $3 \leq s \leq 2r-1$ (and polynomial time solvable otherwise). For arbitrary planar graphs we prove NP-hardness if $s<7$ for $r=2$, and $s < 6r-3$, for $r \geq 3$. The result for planar graphs also proves the NP-hardness of colouring with less than 7 colours graphs of thickness two and less than $6r-3$ colours graphs of thickness $r \geq 3$.

cs.CC↗