Section: New Results

Complexity issues in distributed graph algorithms

Do unique node identifiers help in deciding whether a network $G$ has a prescribed property $P$? We study this question in the context of distributed local decision, where the objective is to decide whether $G\in P$ by having each node run a constant-time distributed decision algorithm. If $G\in P$, all the nodes should output yes; if $G\notin P$, at least one node should output no. A recent work (Fraigniaud et al., OPODIS 2012) studied the role of identifiers in local decision and gave several conditions under which identifiers are not needed. In this article [21] , we answer their original question. More than that, we do so under all combinations of the following two critical variations on the underlying model of distributed computing: ($B$): the size of the identifiers is bounded by a function of the size of the input network; as opposed to ($\neg B$): the identifiers are unbounded. ($C$): the nodes run a computable algorithm; as opposed to ($\neg C$): the nodes can compute any, possibly uncomputable function. While it is easy to see that under ($\neg B,\neg C$) identifiers are not needed, we show that under all other combinations there are properties that can be decided locally if and only if identifiers are present. Our constructions use ideas from classical computability theory.

A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. Our paper [7] , introduces several classes of distributed decision problems, proves separation among them and presents some complete problems. More specifically, we consider the standard LOCAL model of computation and define LD (for local decision) as the class of decision problems that can be solved in constant number of communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD, and ask whether LD=BPLD. We provide a partial answer to this question by showing that in many cases, randomization does not help for deciding hereditary languages. In addition, we define the notion of local many-one reductions, and introduce the (nondeterministic) class NLD of decision problems for which there exists a certificate that can be verified in constant number of communication rounds. We prove that there exists an NLD-complete problem. We also show that there exist problems not in NLD. On the other hand, we prove that the class NLD#n, which is NLD assuming that each processor can access an oracle that provides the number of nodes in the network, contains all (decidable) languages. For this class we provide a natural complete problem as well.

The paper [9] ,studies notions of locality that are inherent to the specification of distributed tasks, and independent of the computing model, by identifying fundamental relationships between the various scales of computation, from the individual process to the whole system. A locality property called projection-closed is identified. This property completely characterizes tasks that are wait-free checkable, where a task $T=(ℐ,𝒪,\Delta )$ is said to be checkable if there exists a distributed algorithm that, given $s\in ℐ$ and $t\in 𝒪$, determines whether $t\in \Delta \left(s\right)$, i.e., whether $t$ is a valid output for $s$ according to the specification of $T$. Projection-closed tasks are proved to form a rich class of tasks. In particular, determining whether a projection-closed task is wait-free solvable is shown to be undecidable. A stronger notion of locality is identified by considering tasks whose outputs ”look identical” to the inputs at every process: a task $T=(ℐ,𝒪,\Delta )$ is said to be locality-preserving if $𝒪$ is a covering complex of $ℐ$. We show that this topological property yields obstacles for wait-free solvability different in nature from the classical impossibility results. On the other hand, locality-preserving tasks are projection-closed, and thus they are wait-free checkable. A classification of locality-preserving tasks in term of their relative computational power is provided. This is achieved by defining a correspondence between subgroups of the edgepath group of an input complex and locality-preserving tasks. This correspondence enables to demonstrate the existence of hierarchies of locality-preserving tasks, each one containing, at the top, the universal task (induced by the universal covering complex), and, at the bottom, the trivial identity task.

The aim of rendezvous in a graph is meeting of two mobile agents at some node of an unknown anonymous connected graph. In this paper [8] , we focus on rendezvous in trees, and, analogously to the efforts that have been made for solving the exploration problem with compact automata, we study the size of memory of mobile agents that permits to solve the rendezvous problem deterministically. We assume that the agents are identical, and move in synchronous rounds. We first show that if the delay between the starting times of the agents is arbitrary, then the lower bound on memory required for rendezvous is $\Omega (logn)$ bits, even for the line of length $n$. This lower bound meets a previously known upper bound of $O(logn)$ bits for rendezvous in arbitrary graphs of size at most $n$. Our main result is a proof that the amount of memory needed for rendezvous with simultaneous start depends essentially on the number $\ell$ of leaves of the tree, and is exponentially less impacted by the number $n$ of nodes. Indeed, we present two identical agents with $O(log\ell +loglogn)$ bits of memory that solve the rendezvous problem in all trees with at most $n$ nodes and at most $\ell$ leaves. Hence, for the class of trees with polylogarithmically many leaves, there is an exponential gap in minimum memory size needed for rendezvous between the scenario with arbitrary delay and the scenario with delay zero. Moreover, we show that our upper bound is optimal by proving that $\Omega (log\ell +loglogn)$ bits of memory are required for rendezvous, even in the class of trees with degrees bounded by 3.

Internet is a large-scale and highly competitive economic ecosystem. In order to make fair decisions, while preventing the economic actors from manipulating the natural outcome of the decision process, game theory is a natural framework, and voting systems represent an interesting alternative that, to our knowledge, has not yet being considered. They allow competing entities to decide among different options. In this paper [20] , we investigate their use for end-to-end path selection in multi-operator networks, analyzing their manipulability by tactical voting and their economic efficiency.We show that Instant Runoff Voting is much more efficient and resistant to tactical voting than the natural system which tries to get the economic optimum.