A personal account of Turing’s imprint on the development of computer science

The rst part of the XX century saw the development of the digital computer and the eld of computer science. In the present paper, we sketch our vision of that period and of the role that Alan Turing and some of his contemporary peers played in that development.Preprin

H-colorings of large degree graphs

We consider the H-coloring problem on graphs with vertices of large degree. We prove that for H an odd cycle, the problem belongs to P. We also study the phase transition of the problem, for an infinite family of graphs of a given chromatic number, i.e. the threshold density value for which the problem changes from P to NP-complete. We extend the result for the case that the input graph has a logarithmic size of small degree vertices.As a corollary, we get a new result on the chromatic number; a new family of graphs, for which computing the chromatic number can be done in polynomial time.Postprint (published version

Linear orderings of random geometric graphs (extended abstract)

In random geometric graphs, vertices are randomly distributed on [0,1]^2 and pairs of vertices are connected by edges whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a certain measure is minimized. In this paper, we study several layout problems on random geometric graphs: Bandwidth, Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. We first prove that some of these problems remain \NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold with high probability on random geometric graphs. Finally, we characterize the probabilistic behavior of the lexicographic ordering for our layout problems on the class of random geometric graphs.Postprint (published version

Convergence theorems for some layout measures on random lattice and random geometric graphs

This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. For full square lattices, we give optimal layouts for the problems still open. Our convergence theorems can be viewed as an analogue of the Beardwood, Halton and Hammersley theorem for the Euclidian TSP on random points in the $d$-dimensional cube. As the considered layout measures are non-subadditive, we use percolation theory to obtain our results on random lattices and random geometric graphs. In particular, we deal with the subcritical regimes on these class of graphs.Postprint (published version

On the stability of generalized second price auctions with budgets

The Generalized Second Price (GSP) auction used typically to model sponsored search auctions does not include the notion of budget constraints, which is present in practice. Motivated by this, we introduce the different variants of GSP auctions that take budgets into account in natural ways. We examine their stability by focusing on the existence of Nash equilibria and envy-free assignments. We highlight the differences between these mechanisms and find that only some of them exhibit both notions of stability. This shows the importance of carefully picking the right mechanism to ensure stable outcomes in the presence of budgets.Peer ReviewedPostprint (author's final draft

Theoretical aspects of graph models for MANETS

We survey the main theoretical aspects of models for Mobile Ad Hoc Networks (MANETs). We present theoretical characterizations of mobile network structural properties, di erent dynamic graph models of MANETs, and nally we give detailed summaries of a few selected articles. In particular, we focus on articles dealing with connectivity of mobile networks, and on articles which show that mobility can be used to propagate information between nodes of the network while at the same time maintaining small transmission distances, and thus saving energy.Preprin

Sequential and parallel complexity of learning DFA

It is known that the class of deterministic finite automata is polynomial time learnable by using membership and equivalence queries. We investigate the query complexity of learning deterministic finite automata, i.e., the number of membership and equivalence queries made during the process of learning. We prove lower bounds on the number of alternations between membership and equivalence queries, and also show that a trade-off exists, allowing us to reduce the number of equivalence queries at the price of increasing the number of membership queries. Finally, we study learning in a parallel model, the CRCW PRAM. We prove a lower bound on the parallel time needed for learning and design an algorithm that asymptotically achieves this bound

Algorithmically efficient syntactic characterization of possibility domains

In the field of Judgment Aggregation, a domain, that is a subset of a Cartesian power of {0, 1}, is considered to reflect abstract rationality restrictions on vectors of two-valued judgments on a number of issues. We are interested in the ways we can aggregate the positions of a set of individuals, whose positions over each issue form vectors of the domain, by means of unanimous (idempotent) functions, whose output is again an element of the domain. Such functions are called non-dictatorial, when their output is not simply the positions of a single individual. Here, we consider domains admitting various kinds of non-dictatorial aggregators, which reflect various properties of majority aggregation: (locally) non-dictatorial, generalized dictatorships, anonymous, monotone, StrongDem and systematic. We show that interesting and, in some sense, democratic voting schemes are always provided by domains that can be described by propositional formulas of specific syntactic types we define. Furthermore, we show that we can efficiently recognize such formulas and that, given a domain, we can both efficiently check if it is described by such a formula and, in case it is, construct it. Our results fall in the realm of classical results concerning the syntactic characterization of domains with specific closure properties, like domains closed under logical AND which are the models of Horn formulas. The techniques we use to obtain our results draw from judgment aggregation as well as propositional logic and universal algebra.The first two authors’ research was partially supported by TIN2017-86727-C2-1-R, GRAMM. The research of the second author was carried out while visiting the Computer Science Department of the Universitat Politècnica de Catalunya.Peer ReviewedPostprint (published version

Reconstruction of random geometric graphs: breaking the Ω(r) distortion barrier

Embedding graphs in a geographical or latent space, i.e. inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We consider the classic model of random geometric graphs where n points are scattered uniformly in a square of area n, and two points have an edge between them if and only if their Euclidean distance is less than r. The reconstruction problem then consists of inferring the vertex positions, up to the symmetries of the square, given only the adjacency matrix of the resulting graph. We give an algorithm that, if r = n α for any 0 < α < 1/2, with high probability reconstructs the vertex positions with a maximum error of O(n β ) where β = 1/2−(4/3)α, until α ≥ 3/8 where β = 0 and the error becomes O( √ log n). This improves over earlier results, which were unable to reconstruct with error less than r. Our method estimates Euclidean distances using a hybrid of graph distances and short-range estimates based on the number of common neighbors. We extend our results to the surface of the sphere in R 3 and to hypercubes in any constant fixed dimension.Josep Díaz: partially supported by PID-2020-112581GB-C21 (MOTION). Cristopher Moore: partially supported by National Science Foundation grant IIS-1838251.Peer ReviewedPostprint (published version

Communication tree problems

In this paper, we consider random communication requirements and several cost measures for a particular model of tree routing on a complete network. First we show that a random tree does not give any approximation. Then give approximation algorithms for the case for two random models of requirements.Postprint (published version