Vinícius Gusmão Pereira de Sá

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

The pair completion algorithm for the homogeneous set sandwich problem

k \leq 2

On the geodetic Radon number of grids

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.

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

A homogeneous set is a non-trivial module of a graph, i.e., a non-empty, non-unitary, proper vertex subset such that all its elements present the same outer neighborhood. Given two graphsG1 (V, E1) and G2(V, E2), the Homogeneous Set Sandwich Problem (HSSP) asks whether there exists a graph GS(V, ES), E1 ⊆ ES ⊆ E2, which has a homogeneous set. This paper presents an algorithm that uses the concept of bias graph [S. Tang, F. Yeh, Y. Wang, An efficient algorithm for solving the homogeneous set sandwich problem, Inform. Process. Lett. 77 (2001) 17-22] to solve the problem in O(n min{|E1|,|E2|} log n) time, thus outperforming the other known HSSP deterministic algorithms for inputs where max{|E1|, |E2|} = Ω (n log n).

Note on the Homogeneous Set Sandwich Problem

Abstract It is NP-hard to determine the Radon number of graphs in the geodetic convexity. However, for certain classes of graphs, this well-known convexity parameter can be determined efficiently. In this paper, we focus on geodetic convexity spaces built upon d -dimensional grids, which are the Cartesian products of d paths. After revisiting a result of Eckhoff concerning the Radon number of R d in the convexity defined by Manhattan distance, we present a series of theoretical findings that disclose some very nice combinatorial aspects of the problem for grids. We also give closed expressions for the Radon number of the product of P 2 ’s and the product of P 3 ’s, as well as computer-aided results covering the Radon number of all possible Cartesian products of d paths for d ≤ 9 .

Near-linear-time algorithm for the geodetic Radon number ofźgrids

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.

On the recognition of unit disk graphs and the Distance Geometry Problem with Ranges

A homogeneous set is a non-trivial module of a graph, i.e., a non-unitary, proper subset H of a graphs vertices such that all vertices in H have the same neighbors outside H. Given two graphs G1(V, E1), G2(V, E2), the Homogeneous Set Sandwich Problem asks whether there exists a sandwich graph GS(V, ES), E1 ⊆ ES ⊆ E2, which has a homogeneous set. Recently, Tang et al. [Inform. Process. Lett. 77 (2001) 17-22] proposed an interesting O(Δ1 ċ n2) algorithm for this problem, which has been considered its most efficient algorithm since. We show the incorrectness of their algorithm by presenting three counterexamples.

Algorithms for the Homogeneous Set Sandwich Problem

The Radon number of a graph is the minimum integer r such that all sets of at least r of its vertices can be partitioned into two subsets whose convex hulls intersect. Determining the Radon number of general graphs in the geodetic convexity is NP-hard. In this paper, we show the problem is polynomial for d -dimensional grids, for all d ź 1 . The proposed algorithm runs in near-linear O ( d ( log d ) 1 / 2 ) time for grids of arbitrary sizes, and in sub-linear O ( log d ) time when all grid dimensions have the same size.

Faster Deterministic and Randomized Algorithms on the Homogeneous Set Sandwich Problem

We introduce a method to decide whether a graph G admits a realization on the plane in which two vertices lie within unitary distance from one another exactly if they are neighbors in G . Such graphs are called unit disk graphs, and their recognition is a known NP-hard problem. By iteratively discretizing the plane, we build a sequence of geometrically defined trigraphs-graphs with mandatory, forbidden and optional adjacencies-until either we find a realization of G or the interruption of such a sequence certifies that no realization exists. Additionally, we consider the proposed method in the scope of the more general Distance Geometry Problem with Ranges, where arbitrary intervals of pairwise distances are allowed.