Raphael C. S. Machado

Efficient sub-5 approximations for minimum dominating sets in unit disk graphs

Abstract A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their applicability in wireless ad-hoc networks. Because the minimum dominating set problem for unit disk graphs is NP-hard, numerous approximation algorithms have been proposed in the literature, including some PTASs. However, since the proposal of a linear-time 5-approximation algorithm in 1995, the lack of efficient algorithms attaining better approximation factors has aroused attention. We introduce an O ( n + m ) algorithm that takes the usual adjacency representation of the graph as input and outputs a 44/9-approximation. This approximation factor is also attained by a second algorithm, which takes the geometric representation of the graph as input and runs in O ( n log n ) time regardless of the number of edges. Additionally, we propose a 43/9-approximation which can be obtained in O ( n 2 m ) time given only the graphs adjacency representation. It is noteworthy that the dominating sets obtained by our algorithms are also independent sets.

Towards a Provably Resilient Scheme for Graph-Based Watermarking

Digital watermarks have been considered a promising way to fight software piracy. Graph-based watermarking schemes encode authorship/ownership data as control-flow graph of dummy code. In 2012, Chroni and Nikolopoulos developed an ingenious such scheme which was claimed to withstand attacks in the form of a single edge removal. We extend the work of those authors in various aspects. First, we give a formal characterization of the class of graphs generated by their encoding function. Then, we formulate a linear-time algorithm which recovers from ill-intentioned removals of

DCJ-indel and DCJ-substitution distances with distinct operation costs

k \leq 2

DCJ-indel distance with distinct operation costs

edges, therefore proving their claim. Furthermore, we provide a simpler decoding function and an algorithm to restore watermarks with an arbitrary number of missing edges whenever at all possible. By disclosing and improving upon the resilience of Chroni and Nikolopouloss watermark, our results reinforce the interest in regarding it as a possible solution to numerous applications.

Decompositions for edge-coloring join graphs and cobipartite graphs

BackgroundClassical approaches to compute the genomic distance are usually limited to genomes with the same content and take into consideration only rearrangements that change the organization of the genome (i.e. positions and orientation of pieces of DNA, number and type of chromosomes, etc.), such as inversions, translocations, fusions and fissions. These operations are generically represented by the double-cut and join (DCJ) operation. The distance between two genomes, in terms of number of DCJ operations, can be computed in linear time. In order to handle genomes with distinct contents, also insertions and deletions of fragments of DNA – named indels – must be allowed. More powerful than an indel is a substitution of a fragment of DNA by another fragment of DNA. Indels and substitutions are called content-modifying operations. It has been shown that both the DCJ-indel and the DCJ-substitution distances can also be computed in linear time, assuming that the same cost is assigned to any DCJ or content-modifying operation.ResultsIn the present study we extend the DCJ-indel and the DCJ-substitution models, considering that the content-modifying cost is distinct from and upper bounded by the DCJ cost, and show that the distance in both models can still be computed in linear time. Although the triangular inequality can be disrupted in both models, we also show how to efficiently fix this problem a posteriori.

Edge-colouring and total-colouring chordless graphs

The double-cut and join (DCJ) is a genomic operation that generalizes the typical mutations to which genomes are subject. The distance between two genomes, in terms of number of DCJ operations, can be computed in linear time. More powerful is the DCJ-indel model, which handles genomes with unequal contents, allowing, besides the DCJ operations, the insertion and/or deletion of pieces of DNA --- named indel operations. It has been shown that the DCJ-indel distance can also be computed in linear time, assuming that the same cost is assigned to any DCJ or indel operation. In the present work we consider a new DCJ-indel distance in which the indel cost is distinct from and upper bounded by the DCJ cost. Considering that the DCJ cost is equal to 1, we set the indel cost equal to a positive constant w≤1 and show that the distance can still be computed in linear time. This new distance generalizes the previous DCJ-indel distance considered in the literature (which uses the same cost for both types of operations).

Software evaluation of smart meters within a Legal Metrology perspective: A Brazilian case

An edge-coloring is an association of colors to the edges of a graph, in such a way that no pair of adjacent edges receive the same color. A graph G is Class 1 if it is edge-colorable with a number of colors equal to its maximum degree @D(G). To determine whether a graph G is Class 1 is NP-complete [I. Holyer, The NP-completeness of edge-coloring, SIAM J. Comput. 10 (1981) 718-720]. First, we propose edge-decompositions of a graph G with the goal of edge-coloring G with @D(G) colors. Second, we apply these decompositions for identifying new subsets of Class 1 join graphs and cobipartite graphs. Third, the proposed technique is applied for proving that the chromatic index of a graph is equal to the chromatic index of its semi-core, the subgraph induced by the maximum degree vertices and their neighbors. Finally, we apply these decomposition tools to a classical result [A.J.W. Hilton, Z. Cheng, The chromatic index of a graph whose core has maximum degree 2, Discrete Math. 101 (1992) 135-147] that relates the chromatic index of a graph to its core, the subgraph induced by the maximum degree vertices.

Linear Time Approximation for Dominating Sets and Independent Dominating Sets in Unit Disk Graphs

Abstract A graph G is chordless if no cycle in G has a chord. In the present work we investigate the chromatic index and total chromatic number of chordless graphs. We describe a known decomposition result for chordless graphs and use it to establish that every chordless graph of maximum degree Δ ≥ 3 has chromatic index Δ and total chromatic number Δ + 1 . The proofs are algorithmic in the sense that we actually output an optimal colouring of a graph instance in polynomial time.

Clique-Colouring and biclique-colouring unichord-free graphs

Induced by a very unfavorable scenario of nontechnical losses (electricity theft), some electrical distribution companies in Brazil have moved from the traditional one meter per residence measurement process to a centralized multi-residences meter with two-way communication features, enabling both automatic reads and remote connect/disconnect of energy supplies. Those improvements can be seen as a natural transition toward an effective Smart grid, and have brought together the needs of a deep revision on regulatory procedures for the electrical meters evaluation conducted by the Brazilian Metrology Office. This paper shows that the increasing complexity from those new electrical grid equipments, particularly conveyed by additional software procedures, is a major issue and needs to be rapidly assimilated by metrological controls to avoid the appearance of stepping stones in the smart grid pathway. This paper aims to describe the whole set of relevant aspects revealed in this particular Brazilian experience, discussing the real scene and the respective challenges opposed.

Restricted DCJ-indel model: sorting linear genomes with DCJ and indels

A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their application in wireless ad-hoc networks. Since the minimum dominating set problem for unit disk graphs is NP-hard, several approximation algorithms with different merits have been proposed in the literature. On one extreme, there is a linear time 5-approximation algorithm. On another extreme, there are two PTAS whose running times are polynomials of very high degree. We introduce a linear time approximation algorithm that takes the usual adjacency representation of the graph as input and attains a 44/9 approximation factor. This approximation factor is also attained by a second algorithm we present, which takes the geometric representation of the graph as input and runs in O(n logn) time, regardless of the number of edges. The analysis of the approximation factor of the algorithms, both of which are based on local improvements, exploits an assortment of results from discrete geometry to prove that certain graphs cannot be unit disk graphs. It is noteworthy that the dominating sets obtained by our algorithms are also independent sets.