Local Maps: New Insights into Mobile Agent Algorithms

In this paper, we study the complexity of computing with mobile agents having small local knowledge. In particular, we show that the number of mobile agents and the amount of local information given initially to agents can significantly influence the time complexity of resolving a distributed problem. Our results are based on a generic scheme allowing to transform a message passing algorithm, running on an $n$-node graph $G$, into a mobile agent one. By generic, we mean that the scheme is independent of both the message passing algorithm and the graph $G$. Our scheme, coupled with a well-chosen clustered representation of the graph, induces $\widetilde{O}(1) ratio between the time complexity of the obtained mobile agent algorithm and the time complexity of the original message passing counterpart, while using$\widetilde{O}(n)$mobile agents. If only$k$agents are allowed ($k$is an integer parameter), then we show that the time ratio is$O(n/\sqrt{k})$. As a consequence, we show that any global labeling function of$G$can be computed by exactly$n$mobile agents knowing their$n^{\epsilon}$-neighborhood in$\widetilde{O}(D)$time,$D$is the diameter of the graph and$\epsilon$is an arbitrary small constant. We apply our generic results for the fundamental problem of computing a leader (resp. a BFS tree) under the additional restriction of$\widetilde{O}(1)$(resp.$\widetilde{O}(n)$) memory bits per agent, and obtain$\widetilde{O}(D)$ time algorithms

Scalable Approximation Algorithm for Network Immunization

The problem of identifying important players in a given network is of pivotal importance for viral marketing, public health management, network security and various other fields of social network analysis. In this work we find the most important vertices in a graph G = (V;E) to immunize so as the chances of an epidemic outbreak is minimized. This problem is directly relevant to minimizing the impact of a contagion spread (e.g. flu virus, computer virus and rumor) in a graph (e.g. social network, computer network) with a limited budget (e.g. the number of available vaccines, antivirus software, filters). It is well known that this problem is computationally intractable (it is NP-hard). In this work we reformulate the problem as a budgeted combinational optimization problem and use techniques from spectral graph theory to design an efficient greedy algorithm to find a subset of vertices to be immunized. We show that our algorithm takes less time compared to the state of the art algorithm. Thus our algorithm is scalable to networks of much larger sizes than best known solutions proposed earlier. We also give analytical bounds on the quality of our algorithm. Furthermore, we evaluate the efficacy of our algorithm on a number of real world networks and demonstrate that the empirical performance of algorithm supplements the theoretical bounds we present, both in terms of approximation guarantees and computational efficiency

A Linear Fixed Parameter Tractable Algorithm for Connected Pathwidth

International audienceThe graph parameter of {\sl pathwidth} can be seen as a measure of the topological resemblance of a graph to a path. A popular definition of pathwidth is given in terms of {\sl node search} where we are given a system of tunnels (represented by a graph) that is contaminated by some infectious substance and we are looking for a search strategy that, at each step, either places a searcher on a vertex or removes a searcher from a vertex and where an edge is cleaned when both endpoints are simultaneously occupied by searchers. It was proved that the minimum number of searchers required for a successful cleaning strategy is equal to the pathwidth of the graph plus one.Two desired characteristics for a cleaning strategy is to be {\sl monotone} (no recontamination occurs) and {\sl connected} (clean territories always remain connected). Under these two demands, the number of searchers is equivalent to a variant of pathwidth called {\em connected pathwidth}. We prove that connected pathwidth is fixed parameter tractable, in particular we design a $2^{O(k^2)}\cdot n$ time algorithm that checks whether the connected pathwidth of $G$ is at most $k.$ This resolves an open question by [{\sl Dereniowski, Osula, and Rz{\k{a}}{\.{z}}ewski, Finding small-width connected path-decompositions in polynomial time. Theor. Comput. Sci., 794:85–100, 2019}\,]. For our algorithm, we enrich the {\sl typical sequence technique} that is able to deal with the connectivity demand. Typical sequences have been introduced in [{\sl Bodlaender and Kloks. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms, 21(2):358–402, 1996}\,] for the design of linear parameterized algorithms for treewidth and pathwidth. While this technique has been later applied to other parameters, none of its advancements was able to deal with the connectivity demand, as it is a ``global’’ demand that concerns an unbounded number of parts of the graph of unbounded size. The proposed extension is based on an encoding of the connectivity property that is quite versatile and may be adapted so to deliver linear parameterized algorithms for the connected variants of other width parameters as well. An immediate consequence of our result is a $2^{O(k^2)}\cdot n$ time algorithm for the monotone and connected version of the edge search number

Parameterized Analysis of the Cops and Robber Game

Pursuit-evasion games have been intensively studied for several decades due to their numerous applications in artificial intelligence, robot motion planning, database theory, distributed computing, and algorithmic theory. Cops and Robber (CnR) is one of the most well-known pursuit-evasion games played on graphs, where multiple cops pursue a single robber. The aim is to compute the cop number of a graph, k, which is the minimum number of cops that ensures the capture of the robber. From the viewpoint of parameterized complexity, CnR is W[2]-hard parameterized by k [Fomin et al., TCS, 2010]. Thus, we study structural parameters of the input graph. We begin with the vertex cover number (vcn). First, we establish that k ? vcn/3+1. Second, we prove that CnR parameterized by vcn is FPT by designing an exponential kernel. We complement this result by showing that it is unlikely for CnR parameterized by vcn to admit a polynomial compression. We extend our exponential kernels to the parameters cluster vertex deletion number and deletion to stars number, and design a linear vertex kernel for neighborhood diversity. Additionally, we extend all of our results to several well-studied variations of CnR

Connected searching of weighted trees

AbstractIn this paper we consider the problem of connected edge searching of weighted trees. Barrière et al. claim in [L. Barrière, P. Flocchini, P. Fraigniaud, N. Santoro, Capture of an intruder by mobile agents, in: SPAA’02: Proceedings of the Fourteenth Annual ACM Symposium on Parallel Algorithms and Architectures, ACM, New York, NY, USA, 2002, pp. 200–209] that there exists a polynomial-time algorithm for finding an optimal search strategy, that is, a strategy that minimizes the number of used searchers. However, due to some flaws in their algorithm, the problem turns out to be open. It is proven in this paper that the considered problem is strongly NP-complete even for node-weighted trees (the weight of each edge is 1) with one vertex of degree greater than 2. It is also shown that there exists a polynomial-time algorithm for finding an optimal connected search strategy for a given bounded degree tree with arbitrary weights on the edges and on the vertices. This is an FPT algorithm with respect to the maximum degree of a tree

Almost Universal Anonymous Rendezvous in the Plane

Two mobile agents represented by points freely moving in the plane and starting at two distinct positions, have to meet. The meeting, called rendezvous, occurs when agents are at distance at most $r$ of each other and never move after this time, where $r$ is a positive real unknown to them, called the visibility radius. Agents are anonymous and execute the same deterministic algorithm. Each agent has a set of private attributes, some or all of which can differ between agents. These attributes are: the initial position of the agent, its system of coordinates (orientation and chirality), the rate of its clock, its speed when it moves, and the time of its wake-up. If all attributes (except the initial positions) are identical and agents start at distance larger than $r$ then they can never meet. However, differences between attributes make it sometimes possible to break the symmetry and accomplish rendezvous. Such instances of the rendezvous problem (formalized as lists of attributes), are called feasible. Our contribution is three-fold. We first give an exact characterization of feasible instances. Thus it is natural to ask whether there exists a single algorithm that guarantees rendezvous for all these instances. We give a strong negative answer to this question: we show two sets $S_1$ and $S_2$ of feasible instances such that none of them admits a single rendezvous algorithm valid for all instances of the set. On the other hand, we construct a single algorithm that guarantees rendezvous for all feasible instances outside of sets $S_1$ and $S_2$. We observe that these exception sets $S_1$ and $S_2$ are geometrically very small, compared to the set of all feasible instances: they are included in low-dimension subspaces of the latter. Thus, our rendezvous algorithm handling all feasible instances other than these small sets of exceptions can be justly called almost universal

An Analysis of Multiple Layered Networks

Current infrastructure network models of single functionality do not typically account for the interdependent nature of infrastructure networks. Infrastructure networks are generally modeled individually, as an isolated network or with minimal recognition of interactions. This research develops a methodology to model the individual infrastructure network types while explicitly modeling their interconnected effects. The result is a formulation built with two sets of variables (the original set to model infrastructure characteristics and an additional set representing cuts of interdependent elements). This formulation is decomposed by variable type using Benders Partitioning and solved to optimality using a Benders Partitioning algorithm. Current infrastructure network models of single functionality do not typically account for the interdependent nature of infrastructure networks, Infrastructure networks are generally modeled individually, as an isolated network or with minimal recognition of interactions, This research develops a methodology to model the individual infrastructure network types while explicitly modeling their interconnected effects, The result is a formulation built with two sets of variables (the original set to model infrastructure characteristics and an additional set representing cuts of interdependent elements) This formulation is decomposed by variable type using Benders\u27 Partitioning and solved to optimality using a Benders\u27 Partitioning algorithm