Julien Moncel

Codes identifying sets of vertices in random networks

In this paper we deal with codes identifying sets of vertices in random networks; that is, (1,=<@?)-identifying codes. These codes enable us to detect sets of faulty processors in a multiprocessor system, assuming that the maximum number of faulty processors is bounded by a fixed constant @?. The (1,=<1)-identifying codes are of special interest. For random graphs we use the model G(n,p), in which each one of the (n2) possible edges exists with probability p. We give upper and lower bounds on the minimum cardinality of a (1,=<@?)-identifying code in a random graph, as well as threshold functions for the property of admitting such a code. We derive existence results from probabilistic constructions. A connection between identifying codes and superimposed codes is also established.

Identifying codes of cycles

In this paper we deal with identifying codes in cycles. We show that for all r ≥ 1, any r-identifying code of the cycle Cn has cardinality at least gcd(2r + 1, n) ⌈n/2gcd(2r+ 1,n)⌉. This lower bound is enough to solve the case n even (which was already solved in [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (7) (2004) 969-987]), but the case n odd seems to be more complicated. An upper bound is given for the case n odd, and some special cases are solved. Furthermore, we give some conditions on n and r to attain the lower bound.

On graphs on n vertices having an identifying code of cardinality ⌈log 2 (n + 1)⇸

Identifying codes were defined to model fault diagnosis in multiprocessor systems. They are also used for the design of indoor detection systems based on wireless sensor networks. When designing such systems, one is usually interested in finding a network structure which minimizes the cardinality of such a code. Given a graph G on n vertices, it is easy to see that the minimum cardinality of an identifying code of G is at least ⌈log2(n + 1)⌉. In this paper, we provide a construction of all the optimal graphs for the identification of vertices, that is to say graphs on n vertices having an identifying code of cardinality ⌈log2(n + 1)⌉. We also compute various parameters of these graphs.

A linear algorithm for minimum 1-identifying codes in oriented trees

Consider an oriented graph G = (V, A), a subset of vertices C ⊆ V, and an integer r≥1; for any vertex v ∈ V, let Br- (v) denote the set of all vertices x such that there exists a path from x to v with at most r arcs. If for all vertices v ∈ V, the sets Br- (v)∩C are all nonempty and different, then we call C an r-identifying code. We describe a linear algorithm which gives a minimum I-identifying code in any oriented tree.

Identifying codes in some subgraphs of the square lattice

An identifying code of a graph is a subset of vertices C such that the sets B(v) ∩ C are all nonempty and different. In this paper, we investigate the problem of finding identifying codes of minimum cardinality in strips and finite grids. We first give exact values for the strips of height 1 and 2, then we give general bounds for strips and finite grids. Finally, we give a sublinear algorithm which finds the minimum cardinality of an identifying code in a restricted class of graphs which includes the grid.

New results on variants of covering codes in Sierpiński graphs

In this paper we study identifying codes, locating-dominating codes, and total-dominating codes in Sierpiński graphs. We compute the minimum size of such codes in Sierpiński graphs.

Adaptive identification in graphs

An adaptive version of identifying codes is introduced. These codes are motivated by various engineering applications. Bounds on adaptive identifying codes are given for regular graphs and torii in the square grid. The new codes are compared to the classical non-adaptive case.

Computational performances of a simple interchange heuristic for a scheduling problem with an availability constraint

This paper deals with a scheduling problem on a single machine with an availability constraint. The problem is known to be NP-complete and admits several approximation algorithms. In this paper we study the approximation scheme described in He et al. [Y. He, W. Zhong, H. Gu, Improved algorithms for two single machine scheduling problems, Theoretical Computer Science 363 (2006) 257-265]. We provide the computation of an improved relative error of this heuristic, as well as a proof that this new bound is tight. We also present some computational experiments to test this heuristic on random instances. These experiments include an implementation of the fully-polynomial time approximation scheme given in Kacem and Ridha Mahjoub [I. Kacem, A. Ridha Mahjoub, Fully polynomial time approximation scheme for the weighted flow-time minimization on a single machine with a fixed non-availability interval, Computers and Industrial Engineering 56 (2009) 1708-1712].

Dense and sparse graph partition

In a graph G=(V,E), the density is the ratio between the number of edges |E| and the number of vertices |V|. This criterion may be used to find communities in a graph: groups of highly connected vertices. We propose an optimization problem based on this criterion; the idea is to find the vertex partition that maximizes the sum of the densities of each class. We prove that this problem is NP-hard by giving a reduction from graph-k-colorability. Additionally, we give a polynomial time algorithm for the special case of trees.

A generalization of the pentomino exclusion problem: Dislocation of graphs

In this paper, we first investigate the pentomino exclusion problem, due to Golomb. We solve this problem on the 5xn grid and we give some lower and upper bounds for the kxn grid for all k and n. We then give a generalization of this problem in graphs, the @D-dislocation problem, which consists in finding the minimum number of vertices to be removed from a graph so as all the remaining connected components have cardinality at most @D. We investigate the algorithmic aspects of the @D-dislocation problem: we first prove the problem is NP-Complete, then we give a sublinear algorithm which solves the problem on a restricted class of graphs which includes the kxn grid graphs, provided k is not part of the input.