Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

Breaks, cuts, and patterns

Wegeneralize the concept of a break by considering pairs of arbitrary rounds.Weshow that a set of homeaway patterns minimizing the number of generalized breaks cannot be found in polynomial time, unless P = NP. When all teams have the same break set, the decision version becomes easy; optimizing remains NP-hard.status: publishe

Dynamic Monopolies in Colored Tori

The {\em information diffusion} has been modeled as the spread of an information within a group through a process of social influence, where the diffusion is driven by the so called {\em influential network}. Such a process, which has been intensively studied under the name of {\em viral marketing}, has the goal to select an initial good set of individuals that will promote a new idea (or message) by spreading the "rumor" within the entire social network through the word-of-mouth. Several studies used the {\em linear threshold model} where the group is represented by a graph, nodes have two possible states (active, non-active), and the threshold triggering the adoption (activation) of a new idea to a node is given by the number of the active neighbors. The problem of detecting in a graph the presence of the minimal number of nodes that will be able to activate the entire network is called {\em target set selection} (TSS). In this paper we extend TSS by allowing nodes to have more than two colors. The multicolored version of the TSS can be described as follows: let $G$ be a torus where every node is assigned a color from a finite set of colors. At each local time step, each node can recolor itself, depending on the local configurations, with the color held by the majority of its neighbors. We study the initial distributions of colors leading the system to a monochromatic configuration of color $k$, focusing on the minimum number of initial $k$-colored nodes. We conclude the paper by providing the time complexity to achieve the monochromatic configuration

Distributed Computing with Adaptive Heuristics

We use ideas from distributed computing to study dynamic environments in which computational nodes, or decision makers, follow adaptive heuristics (Hart 2005), i.e., simple and unsophisticated rules of behavior, e.g., repeatedly "best replying" to others' actions, and minimizing "regret", that have been extensively studied in game theory and economics. We explore when convergence of such simple dynamics to an equilibrium is guaranteed in asynchronous computational environments, where nodes can act at any time. Our research agenda, distributed computing with adaptive heuristics, lies on the borderline of computer science (including distributed computing and learning) and game theory (including game dynamics and adaptive heuristics). We exhibit a general non-termination result for a broad class of heuristics with bounded recall---that is, simple rules of behavior that depend only on recent history of interaction between nodes. We consider implications of our result across a wide variety of interesting and timely applications: game theory, circuit design, social networks, routing and congestion control. We also study the computational and communication complexity of asynchronous dynamics and present some basic observations regarding the effects of asynchrony on no-regret dynamics. We believe that our work opens a new avenue for research in both distributed computing and game theory.Comment: 36 pages, four figures. Expands both technical results and discussion of v1. Revised version will appear in the proceedings of Innovations in Computer Science 201

Multicolored Dynamos on Toroidal Meshes

Detecting on a graph the presence of the minimum number of nodes (target set) that will be able to "activate" a prescribed number of vertices in the graph is called the target set selection problem (TSS) proposed by Kempe, Kleinberg, and Tardos. In TSS's settings, nodes have two possible states (active or non-active) and the threshold triggering the activation of a node is given by the number of its active neighbors. Dealing with fault tolerance in a majority based system the two possible states are used to denote faulty or non-faulty nodes, and the threshold is given by the state of the majority of neighbors. Here, the major effort was in determining the distribution of initial faults leading the entire system to a faulty behavior. Such an activation pattern, also known as dynamic monopoly (or shortly dynamo), was introduced by Peleg in 1996. In this paper we extend the TSS problem's settings by representing nodes' states with a "multicolored" set. The extended version of the problem can be described as follows: let G be a simple connected graph where every node is assigned a color from a finite ordered set C = {1, . . ., k} of colors. At each local time step, each node can recolor itself, depending on the local configurations, with the color held by the majority of its neighbors. Given G, we study the initial distributions of colors leading the system to a k monochromatic configuration in toroidal meshes, focusing on the minimum number of initial k-colored nodes. We find upper and lower bounds to the size of a dynamo, and then special classes of dynamos, outlined by means of a new approach based on recoloring patterns, are characterized