Reconstructing Set Partitions

Colloque avec actes et comité de lecture.We study the following combinatorial search problem: Reconstruct an unknown partition of the set [n]={1,...,n} into at most K disjoint non-empty subsets (classes) by making queries about subsets $Q_i \subseteq [n]$ such that the query returns the number of classes represented in $Q_i$. The goal is to reconstruct the whole partition with as few queries as possible. We also consider a variant of the problem where a representative of each class should be found without necessarily reconstructing the whole partition. Besides its theoretical interest, the problem has practical applications. Paper \cite{SBAW-RECOMB97} considers a physical mapping method based on hybridizing probes to chromosomes using the FISH technology. Under some reasonable assumptions, the method amounts to finding a partition of probes determined by the chromosome they hybridize to, and actually corresponds to the above formalization

Optimal Reconstruction of Graphs under the Additive Model

Article dans revue scientifique avec comité de lecture.We study the problem of reconstructing unknown graphs under the additive combinatorial search model. The main result concerns the reconstruction of {\em bounded degree} \/graphs, i.e.\ graphs with the degree of all vertices bounded by a constant $d$. We show that such graphs can be reconstructed in $O(dn)$ non--adaptive queries, which matches the information--theoretic lower bound. The proof is based on the technique of se\-pa\-rating matrices. A central result here is a new upper bound for a general class of separating matrices. As a particular case, we obtain a tight upper bound for the class of $d$--separating matrices, which settles an open question stated by Lindström in~\cite{Lindstrom75}. Finally, we consider several particular classes of graphs. We show how an optimal non--adaptive solution of $O(n^2/\log n)$ queries for general graphs can be obtained. We also prove that trees with unbounded vertex degree can be reconstructed in a linear number of queries by a non--adaptive algorithm

On the Power of Additive Combinatorial Search Model

We consider two generic problems of combinatorial search under the additive model. The first one is the problem of reconstructing bounded-weight vectors. We establish an optimal upper bound and observe that it unifies many known results for coin-weighing problems. The developed technique provides a basis for the graph reconstruction problem. Optimal upper bound is proven for the class of k-degenerate graphs

Recherche combinatoire : problèmes de pesage, reconstruction de graphes et applications

Texte intégral accessible uniquement aux membres de l'Université de LorraineNot availableLa recherche combinatoire est une branche de l'algorithmique combinatoire qui est étroitement liée avec d'autres domaines des mathématiques et de l'informatique, tels que l'étude de la complexité d'algorithmes, la théorie des graphes, la théorie des nombres, la théorie des ensembles extrémaux. En termes très généraux, la recherche combinatoire étudié les problèmes d'identification d'un objet inconnu dans un ensemble d'objets à l'aide de questions indirectes sur cet objet. Les méthodes de la recherche combinatoire trouvent de nombreuses applications pratiques dans les domaines de la biologie, la médecine, la conception de réseaux, et d'autres. Le thème central de la thèse est le modèle additif en recherche combinatoire. Nous étudions la puissance de ce modèle en l'appliquant à quelques problèmes classiques de recherche combinatoire. Trois familles de problèmes sont considérées : les problèmes de pesage de monnaies, le problème de reconstruction de graphes d'une classe donnée et le problème de reconstruction de partitions. Pour les problèmes de reconstruction de graphes et de pesage, nous examinons les résultats connus et démontrons de nouveaux résultats plus généraux. Un des résultats les plus importants est l'obtention d'une borne supérieure pour le problème de reconstruction de vecteurs à poids borné. Nous introduisons également le problème de reconstruction de partitions, pour lequel nous développons des algorithmes efficaces dont nous analysons la complexit

Reconstructing a Hamiltonian Circuit by Querying the Graph: Application to DNA Physical Mapping

This paper studies three mathematical models of the multiplex PCR method of genome physical mapping described in [12]. The models are expressed as combinatorial group testing problems of finding an unknown Hamiltonian circuit in the complete graph by means of queries of different type. For each model, an efficient algorithm is proposed that matches asymptotically the information-theoretic lower bound. Keywords: genome physical mapping, combinatorial group testing, algorithms, asymptotic complexity. 1 Introduction 1.1 Biological Motivation This paper studies several mathematical models of the multiplex PCR method of genome physical mapping described in [12]. Physical mapping is the central stage in genome exploration which consists in creating some landmarks or tags throughout the DNA molecule. These landmarks are some specific nucleotide sequences associated with their positions on the molecule. They are usually obtained through the process of cloning which allows to extract some fra..

Optimal Reconstruction of Graphs Under the Additive Model

We study the problem of combinatorial search for graphs under the additive model. The main result concerns the reconstruction of bounded degree graphs, i.e. graphs with the degree of all vertices bounded by a constant d. We show that such graphs can be reconstructed in O(dn) non-adaptive queries, that matches the information-theoretic lower bound. The proof is based on the technique of separating matrices. In particular, a new upper bound is obtained for d-separating matrices, that settles an open question stated by Lindstr#m in [16]. Finally, we consider several particular classes of graphs. We show how an optimal non-adaptive solution of O(n²/log n) queries for general graphs can be obtained

Reconstructing a Hamiltonian Cycle by Querying the Graph: Application to DNA Physical Mapping

Article dans revue scientifique avec comité de lecture.This paper studies four mathematical models of the multiplex PCR method of genome physical mapping described in \cite{Sorokin1}. The models are expressed as combinatorial group testing problems of finding an unknown Hamiltonian cycle in the complete graph by means of queries of different type. For each model, an efficient algorithm is proposed that matches asymptotically the information-theoretic lower bound

Optimal Query Bounds for Reconstructing a Hamiltonian Cycle in Complete Graphs (extended abstract)

This paper studies four combinatorial search models of reconstructing a fixed unknown Hamiltonian cycle in the complete graph by means of queries about subgraphs. For each model, an efficient algorithm is proposed that matches asymptotically the information-theoretic lower bound. The problem is motivated by an application to genome physical mapping. 1. Introduction 1.1. Combinatorial Search Combinatorial Search can be informally defined as determining an unknown object of a certain class through indirect queries about this object. The goal of combinatorial search is to identify the unknown object with as little cost as possible. While the cost measure may vary, it is often defined as the number of queries made by the search algorithm. We refer to [1] for a systematic presentation of the field. Group Testing is probably the oldest and most wellknown subfield of combinatorial search. In group testing, we are given a set of items some of which are "defective". We want to determine the ..