Abstract: Consider a network vulnerable to viral infection. The system security software can guarantee
safety only to a limited part of the network. Such limitations result from economy costs or processing
costs. The problem raised is to which part of the network the security software should
be installed, so that to secure as much as possible the network. We model this practical network
scenario as a non-cooperative multi-player game on a graph, with two kinds of players, a set
of attackers and a protector player, representing the viruses and the system security software,
respectively. Each attacker player chooses a node of the graph (or a set of them, via a probability
distribution) to infect. The protector player chooses independently, in a basic case of the
problem, a simple path or an edge of the graph (or a set of them, via a probability distribution)
and cleans this part of the network from attackers. Each attacker wishes to maximize the probability
of escaping its cleaning by the protector. In contrast, the protector aims at maximizing
the expected number of cleaned attackers. We call the two games obtained from the two basic
cases considered, as the Path and the Edge model, respectively. For these two games, we are
interested in the associated Nash equilibria, where no network entity can unilaterally improve
its local objective. We obtain the following results:
• The problem of existence of a pure Nash equilibrium is NP-complete for the Path model.
This opposed to that, no instance of the Edge model possesses a pure Nash equilibrium,
proved in [7].
• In [7] a characterization of mixed Nash equilibria for the Edge model is provided. However,
that characterization only implies an exponential time algorithm for the general case.
Here, combining it with clever exploration of properties of various practical families of
graphs, we compute, in polynomial time, mixed Nash equilibria on corresponding graph
instances. These graph families include, regular graphs, graphs that can be decomposed, in
polynomially time, into vertex disjoint r-regular subgraphs, graphs with perfect matchings
and trees.
• We utilize the notion of social cost [6] for measuring system performance on such scenario;
here is defined to be the utility of the protector. We prove that the corresponding Price of
Anarchy in any mixed Nash equilibria of the game is upper and lower bounded by a linear
function of the number of vertices of the graph.
Abstract: We present new combinatorial approximation algorithms for k-set cover. Previous approaches are based on extending the greedy algorithm by efficiently handling small sets. The new algorithms further extend them by utilizing the natural idea of computing large packings of elements into sets of large size. Our results improve the previously best approximation bounds for the k-set cover problem for all values of k ≥ 6. The analysis technique could be of independent interest; the upper bound on the approximation factor is obtained by bounding the objective value of a factor-revealing linear program.
Abstract: We present new combinatorial approximation algorithms for
k-set cover. Previous approaches are based on extending the greedy al-
gorithm by e±ciently handling small sets. The new algorithms further
extend them by utilizing the natural idea of computing large packings
of elements into sets of large size. Our results improve the previously
best approximation bounds for the k-set cover problem for all values
of k ¸ 6. The analysis technique could be of independent interest; the
upper bound on the approximation factor is obtained by bounding the
objective value of a factor-revealing linear program.
Abstract: An intersection graph of n vertices assumes that each vertex is equipped with a subset of a global label set. Two vertices share an edge
when their label sets intersect. Random Intersection Graphs (RIGs) (as defined in [18, 31]) consider label sets formed by the following experiment:
each vertex, independently and uniformly, examines all the labels (m in total) one by one. Each examination is independent and the vertex
succeeds to put the label in her set with probability p. Such graphs nicely capture interactions in networks due to sharing of resources among nodes.
We study here the problem of efficiently coloring (and of finding upper bounds to the chromatic number) of RIGs. We concentrate in a range
of parameters not examined in the literature, namely: (a) m = n{\'a} for less than 1 (in this range, RIGs differ substantially from the Erd¨os- Renyi random graphs) and (b) the selection probability p is quite high
(e.g. at least ln2 n m in our algorithm) and disallows direct greedy colouring methods.
We manage to get the following results:
For the case mp ln n, for any constant < 1 − , we prove that np colours are enough to colour most of the vertices of the graph with high probability (whp). This means that even for quite dense
graphs, using the same number of colours as those needed to properly colour the clique induced by any label suffices to colour almost all of the vertices of the graph. Note also that this range of values of m, p
is quite wider than the one studied in [4].
� We propose and analyze an algorithm CliqueColour for finding a proper colouring of a random instance of Gn,m,p, for any mp >=ln2 n. The algorithm uses information of the label sets assigned to the
vertices of Gn,m,p and runs in O (n2mp2/ln n) time, which is polynomial in n and m. We also show by a reduction to the uniform random
intersection graphs model that the number of colours required by the algorithm are of the correct order of magnitude with the actual
chromatic number of Gn,m,p.
⋆ This work was partially supported by the ICT Programme of the European Union under contract number ICT-2008-215270 (FRONTS). Also supported by Research Training Group GK-693 of the Paderborn Institute for Scientific Computation
(PaSCo).
� We finally compare the problem of finding a proper colouring for Gn,m,p to that of colouring hypergraphs so that no edge is monochromatic.We show how one can find in polynomial time a k-colouring of the vertices of Gn,m,p, for any integer k, such that no clique induced by only one label in Gn,m,p is monochromatic. Our techniques are novel and try to exploit as much as possible the hidden structure of random intersection graphs in this interesting range.
Abstract: In this work, we overview some results concerning communication combinatorial properties in random intersection graphs and uniform random intersection graphs. These properties relate crucially to algorithmic design for important problems (like secure communication and frequency assignment) in distributed networks characterized by dense, local interactions and resource limitations, such as sensor networks. In particular, we present and discuss results concerning the existence of large independentsets of vertices whp in random instances of each of these models. As the main contribution of our paper, we introduce a new, general model, which we denote G(V, χ, f). In this model, V is a set of vertices and χ is a set of m vectors in ℝm. Furthermore, f is a probability distribution over the powerset 2χ of subsets of χ. Every vertex selects a random subset of vectors according to the probability f and two vertices are connected according to a general intersection rule depending on their assigned set of vectors. Apparently, this new general model seems to be able to simulate other known random graph models, by carefully describing its intersection rule.
Abstract: We consider the problem of computing minimum congestion,
fault-tolerant, redundant assignments of messages to faulty parallel de-
livery channels. In particular, we are given a set M of faulty channels,
each having an integer capacity ci and failing independently with proba-
bility fi. We are also given a set of messages to be delivered over M, and
a fault-tolerance constraint (1), and we seek a redundant assignment
that minimizes congestion Cong( ), i.e. the maximum channel load,
subject to the constraint that, with probability no less than (1 ), all
the messages have a copy on at least one active channel. We present a
4-approximation algorithm for identical capacity channels and arbitrary
message sizes, and a 2l ln(jMj=)
ln(1=fmax)m-approximation algorithm for related
capacity channels and unit size messages.
Both algorithms are based on computing a collection of disjoint chan-
nel subsets such that, with probability no less than (1 ), at least one
channel is active in each subset. The objective is to maximize the sum of
the minimum subset capacities. Since the exact version of this problem
is NP-complete, we present a 2-approximation algorithm for identical
capacities, and a (8 + o(1))-approximation algorithm for arbitrary ca-
pacities.
Abstract: We investigate the existence and efficient algorithmic
construction of close to optimal independentsets in random models
of intersection graphs. In particular, (a) we propose \emph{a new model} for random intersection graphs
($G_{n, m, \vec{p}}$) which includes the model of
\cite{RIG} (the ``uniform" random intersection graphs model) as an
important special case. We also define an interesting variation of
the model of random intersection graphs, similar in spirit to
random regular graphs. (b) For this model we derive \emph{exact formulae} for the mean
and variance of the number of independentsets of size $k$ (for
any $k$) in the graph. (c) We then propose and analyse \emph{three algorithms} for the
efficient construction of large independentsets in this model.
The first two are variations of the greedy technique while the
third is a totally new algorithm. Our algorithms are analysed for
the special case of uniform random intersection graphs.
Our analyses show that these algorithms succeed in finding
\emph{close to optimal} independentsets for an interesting range
of graph parameters.
Abstract: We extend here the Population Protocol model of Angluin et al. [2004] in order to model more powerful networks of very small resource-limited artefacts (agents) that are possibly mobile. Communication can happen only between pairs of artefacts. A communication graph (or digraph) denotes the permissible pairwise interactions. The main feature of our extended model is to allow edges of the communication graph, G, to have states that belong to a constant size set. We also allow edges to have readable only costs, whose values also belong to a constant size set. We then allow the protocol rules for pairwise interactions to modify the corresponding edge state. Thus, our protocol specifications are still independent of the population size and do not use agent ids, i.e. they preserve scalability, uniformity and anonymity. Our Mediated Population Protocols (MPP) can stably compute graph properties of the communication graph. We show this for the properties of maximal matchings (in undirected communication graphs), also for finding the transitive closure of directed graphs and for finding all edges of small cost. We demonstrate that our mediated protocols are stronger than the classical population protocols, by presenting a mediated protocol that stably computes the product of two positive integers, when G is the complete graph. This is not a semilinear predicate. To show this fact, we state and prove a general Theorem about the Composition of two stably computing mediated population protocols. We also show that all predicates stably computable in our model are (non-uniformly) in the class NSPACE(m), where m is the number of edges of the communication graph. We also define Randomized MPP and show that, any Peano predicate accepted by a MPP, can be verified in deterministic Polynomial Time.
Abstract: We consider the problem of computing minimum congestion, fault-tolerant, redundant assignments of messages to faulty, parallel delivery channels. In particular, we are given a set K of faulty channels, each having an integer capacity ci and failing independently with probability fi. We are also given a set M of messages to be delivered over K, and a fault-tolerance constraint (1 - {\aa}), and we seek a redundant assignment {\"o} that minimizes congestion Cong({\"o}), i.e. the maximum channel load, subject to the constraint that, with probability no less than (1 - e), all the messages have a copy on at least one active channel. We present a polynomial-time 4-approximation algorithm for identical capacity channels and arbitrary message sizes, and a 2[ln(|K|/{\aa})/ln(1/fmax)]-approximation algorithm for related capacity channels and unit size messages. Both algorithms are based on computing a collection (K1,., K{\'i}} of disjoint channel subsets such that, with probability no less than (1 - {\aa}), at least one channel is active in each subset. The objective is to maximize the sum of the minimum subset capacities. Since the exact version of this problem is NP-complete, we provide a 2-approximation algorithm for identical capacities, and a polynomial-time (8+o(1))-approximation algorithm for arbitrary capacities.
Abstract: In this work we consider temporal graphs, i.e. graphs, each edge of which isassigned a set of discrete time-labels drawn from a set of integers. The labelsof an edge indicate the discrete moments in time at which the edge isavailable. We also consider temporal paths in a temporal graph, i.e. pathswhose edges are assigned a strictly increasing sequence of labels. Furthermore,we assume the uniform case (UNI-CASE), in which every edge of a graph isassigned exactly one time label from a set of integers and the time labelsassigned to the edges of the graph are chosen randomly and independently, withthe selection following the uniform distribution. We call uniform randomtemporal graphs the graphs that satisfy the UNI-CASE. We begin by deriving theexpected number of temporal paths of a given length in the uniform randomtemporal clique. We define the term temporal distance of two vertices, which isthe arrival time, i.e. the time-label of the last edge, of the temporal paththat connects those vertices, which has the smallest arrival time amongst alltemporal paths that connect those vertices. We then propose and study twostatistical properties of temporal graphs. One is the maximum expected temporaldistance which is, as the term indicates, the maximum of all expected temporaldistances in the graph. The other one is the temporal diameter which, looselyspeaking, is the expectation of the maximum temporal distance in the graph. Wederive the maximum expected temporal distance of a uniform random temporal stargraph as well as an upper bound on both the maximum expected temporal distanceand the temporal diameter of the normalized version of the uniform randomtemporal clique, in which the largest time-label available equals the number ofvertices. Finally, we provide an algorithm that solves an optimization problemon a specific type of temporal (multi)graphs of two vertices.
Abstract: Consider an information network with harmful procedures called attackers (e.g., viruses); each attacker uses a probability distribution to choose a node of the network to damage. Opponent to the attackers is the system protector scanning and cleaning from attackers some part of the network (e.g., an edge or a path), which it chooses independently using another probability distribution. Each attacker wishes to maximize the probability of escaping its cleaning by the system protector; towards a conflicting objective, the system protector aims at maximizing the expected number of cleaned attackers.
We model this network scenario as a non-cooperative strategic game on graphs. We focus on the special case where the protector chooses a single edge. We are interested in the associated Nash equilibria, where no network entity can unilaterally improve its local objective. We obtain the following results:
{\^a}{\"i}¿½{\"i}¿½ No instance of the game possesses a pure Nash equilibrium.
{\^a}{\"i}¿½{\"i}¿½Every mixed Nash equilibrium enjoys a graph-theoretic structure, which enables a (typically exponential) algorithm to compute it.
{\^a}{\"i}¿½{\"i}¿½ We coin a natural subclass of mixed Nash equilibria, which we call matching Nash equilibria, for this game on graphs. Matching Nash equilibria are defined using structural parameters of graphs, such as independentsets and matchings.
{\^a}{\"i}¿½{\"i}¿½We derive a characterization of graphs possessing matching Nash equilibria. The characterization enables a linear time algorithm to compute a matching Nash equilibrium on any such graph with a given independentset and vertex cover.
{\^a}{\"i}¿½{\"i}¿½ Bipartite graphs are shown to satisfy the characterization. So, using a polynomial-time algorithm to compute a perfect matching in a bipartite graph, we obtain, as our main result, an efficient graph-theoretic algorithm to compute a matching Nash equilibrium on any instance of the game with a bipartite graph.
Abstract: Random Intersection Graphs is a new class of random graphs introduced in [5], in which each of n vertices randomly and independently chooses some elements from a universal set, of cardinality m. Each element is chosen with probability p. Two vertices are joined by an edge iff their chosen element sets intersect. Given n, m so that m=n{\'a}, for any real {\'a} different than one, we establish here, for the first time, tight lower bounds p0(n,m), on the value of p, as a function of n and m, above which the graph Gn,m,p is almost certainly Hamiltonian, i.e. it contains a Hamilton Cycle almost certainly. Our bounds are tight in the sense that when p is asymptotically smaller than p0(n,m) then Gn,m,p almost surely has a vertex of degree less than 2. Our proof involves new, nontrivial, coupling techniques that allow us to circumvent the edge dependencies in the random intersection model. Interestingly, Hamiltonicity appears well below the general thresholds, of [4], at which Gn,m,p looks like a usual random graph. Thus our bounds are much stronger than the trivial bounds implied by those thresholds.
Our results strongly support the existence of a threshold for Hamiltonicity in Gn,m,p.
Abstract: We study the on-line versions of two fundamental graph problems, maximum independentset and minimum coloring, for the case of disk graphs which are graphs resulting from intersections of disks on the plane. In particular, we investigate whether randomization can be used to break known lower bounds for deterministic on-line independentset algorithms and present new upper and lower bounds; we also present an improved upper bound for on-line coloring.
Abstract: In this work we extend the population protocol model of Angluin et al., in
order to model more powerful networks of very small resource limited
artefacts (agents) that is possible to follow some unpredictable passive
movement. These agents communicate in pairs according to the commands of
an adversary scheduler. A directed (or undirected) communication graph
encodes the following information: each edge (u,\~{o}) denotes that during the
computation it is possible for an interaction between u and \~{o} to happen in
which u is the initiator and \~{o} the responder. The new characteristic of
the proposed mediated population protocol model is the existance of a
passive communication provider that we call mediator. The mediator is a
simple database with communication capabilities. Its main purpose is to
maintain the permissible interactions in communication classes, whose
number is constant and independent of the population size. For this reason
we assume that each agent has a unique identifier for whose existence the
agent itself is not informed and thus cannot store it in its working
memory. When two agents are about to interact they send their ids to the
mediator. The mediator searches for that ordered pair in its database and
if it exists in some communication class it sends back to the agents the
state corresponding to that class. If this interaction is not permitted to
the agents, or, in other words, if this specific pair does not exist in
the database, the agents are informed to abord the interaction. Note that
in this manner for the first time we obtain some control on the safety of
the network and moreover the mediator provides us at any time with the
network topology. Equivalently, we can model the mediator by communication
links that are capable of keeping states from a edge state set of constant
cardinality. This alternative way of thinking of the new model has many
advantages concerning the formal modeling and the design of protocols,
since it enables us to abstract away the implementation details of the
mediator. Moreover, we extend further the new model by allowing the edges
to keep readable only costs, whose values also belong to a constant size
set. We then allow the protocol rules for pairwise interactions to modify
the corresponding edge state by also taking into account the costs. Thus,
our protocol descriptions are still independent of the population size and
do not use agent ids, i.e. they preserve scalability, uniformity and
anonymity. The proposed Mediated Population Protocols (MPP) can stably
compute graph properties of the communication graph. We show this for the
properties of maximal matchings (in undirected communication graphs), also
for finding the transitive closure of directed graphs and for finding all
edges of small cost. We demonstrate that our mediated protocols are
stronger than the classical population protocols. First of all we notice
an obvious fact: the classical model is a special case of the new model,
that is, the new model can compute at least the same things with the
classical one. We then present a mediated protocol that stably computes
the product of two nonnegative integers in the case where G is complete
directed and connected. Such kind of predicates are not semilinear and it
has been proven that classical population protocols in complete graphs can
compute precisely the semilinear predicates, thus in this manner we show
that there is at least one predicate that our model computes and which the
classical model cannot compute. To show this fact, we state and prove a
general Theorem about the composition of two mediated population
protocols, where the first one has stabilizing inputs. We also show that
all predicates stably computable in our model are (non-uniformly) in the
class NSPACE(m), where m is the number of edges of the communication
graph. Finally, we define Randomized MPP and show that, any Peano
predicate accepted by a Randomized MPP, can be verified in deterministic
polynomial time.
Abstract: We consider in this paper the problem of scheduling a set of independent
parallel tasks (jobs) with respect to two criteria, namely,
the makespan (time of the last finishing job) and the minsum (average
completion time). There exist several algorithms with a good
performance guaranty for one of these criteria. We are interested
here in studying the optimization of both criteria simultaneously.
The numerical values are given for the moldable task model, where
the execution time of a task depends on the number of processors
alloted to it. The main result of this paper is to derive explicitly
a family of algorithms guaranteed for both the minsum and the
makespan. The performance guaranty of these algorithms is better
than the best algorithms known so far. The Guaranty curve
of the family is the set of all points (x; y) such that there is an
algorithm with guarantees x on makespan and y on the minsum.
When the ratio on the minsum increases, the curve tends to the
best ratio known for the makespan for moldable tasks (3=2). One
extremal point of the curves is a (3;6)-approximation algorithm.
Finally a randomized version is given, which improves this results
to (3;4.08).
Abstract: The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graph G(V,E) is an assignment function View the MathML source such that |{\"E}(u)−{\"E}(v)|greater-or-equal, slanted2, when u,v are neighbors in G, and |{\"E}(u)−{\"E}(v)|greater-or-equal, slanted1 when the distance of u,v in G is two. The discrete number of frequencies used is called order and the range of frequencies used, span. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span (min span RCP) or the order (min order RCP).
In this paper, we deal with an interesting, yet not examined until now, variation of the radiocoloring problem: that of satisfying frequency assignment requests which exhibit some periodic behavior. In this case, the interference graph (modelling interference between transmitters) is some (infinite) periodic graph. Infinite periodic graphs usually model finite networks that accept periodic (in time, e.g. daily) requests for frequency assignment. Alternatively, they can model very large networks produced by the repetition of a small graph.
A periodic graph G is defined by an infinite two-way sequence of repetitions of the same finite graph Gi(Vi,Ei). The edge set of G is derived by connecting the vertices of each iteration Gi to some of the vertices of the next iteration Gi+1, the same for all Gi. We focus on planar periodic graphs, because in many cases real networks are planar and also because of their independent mathematical interest.
We give two basic results:
• We prove that the min span RCP is PSPACE-complete for periodic planar graphs.
• We provide an O(n({\"A}(Gi)+{\'o})) time algorithm (where|Vi|=n, {\"A}(Gi) is the maximum degree of the graph Gi and {\'o} is the number of edges connecting each Gi to Gi+1), which obtains a radiocoloring of a periodic planar graph G that approximates the minimum span within a ratio which tends to View the MathML source as {\"A}(Gi)+{\'o} tends to infinity.
We remark that, any approximation algorithm for the min span RCP of a finite planar graph G, that achieves a span of at most {\'a}{\"A}(G)+constant, for any {\'a} and where {\"A}(G) is the maximum degree of G, can be used as a subroutine in our algorithm to produce an approximation for min span RCP of asymptotic ratio {\'a} for periodic planar graphs.
Abstract: The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. The Radiocoloring (RC) of a graph G(V,E) is an assignment function {\"O}: V → IN such that ∣{\"O}(u) - {\"O}(v)∣ ≥2, when u, v are neighbors in G, and ∣{\"O}(u) - {\"O}(v)∣ ≥1 when the distance of u, v in G is two. The range of frequencies used is called span. Here, we consider the optimization version of the Radiocoloring Problem (RCP) of finding a radiocoloring assignment of minimum span, called min span RCP. In this paper, we deal with a variation of RCP: that of satisfying frequency assignment requests with some periodic behavior. In this case, the interference graph is an (infinite) periodic graph. Infinite periodic graphs model finite networks that accept periodic (in time, e.g. daily) requests for frequency assignment. Alternatively, they may model very large networks produced by the repetition of a small graph. A periodic graph G is defined by an infinite two-way sequence of repetitions of the same finite graph G i (V i ,E i ). The edge set of G is derived by connecting the vertices of each iteration G i to some of the vertices of the next iteration G i +1, the same for all G i . The model of periodic graphs considered here is similar to that of periodic graphs in Orlin [13], Marathe et al [10]. We focus on planar periodic graphs, because in many cases real networks are planar and also because of their independent mathematical interest. We give two basic results: - We prove that the min span RCP is PSPACE-complete for periodic planar graphs. - We provide an O(n({\"A}(G i ) + {\'o})) time algorithm, (where ∣V i ∣ = n, {\"A}(G i ) is the maximum degree of the graph G i and {\'o} is the number of edges connecting each G i to G i +1), which obtains a radiocoloring of a periodic planar graph G that approximates the minimum span within a ratio which tends to 2 as {\"A}(Gi) + {\'o} tends to infinity.
Abstract: We study the on-line version of the maximum independentset problem, for the case of disk graphs which are graphs resulting
from intersections of disks on the plane. In particular, we investigate whether randomization can be used to break known lower
bounds for deterministic on-line independentset algorithms and present new upper and lower bounds.
Abstract: In recent years there has been signi1cant interest in the study of random k-SAT formulae. For
a given set of n Boolean variables, let Bk denote the set of all possible disjunctions of k distinct,
non-complementary literals from its variables (k-clauses). A random k-SAT formula Fk (n;m) is
formed by selectinguniformly and independently m clauses from Bk and takingtheir conjunction.
Motivated by insights from statistical mechanics that suggest a possible relationship between the
?order? of phase transitions and computational complexity, Monasson and Zecchina (Phys. Rev.
E 56(2) (1997) 1357) proposed the random (2+p)-SAT model: for a given p ¸ [0; 1], a random
(2 + p)-SAT formula, F2+p(n;m), has m randomly chosen clauses over n variables, where pm
clauses are chosen from B3 and (1 − p)m from B2. Usingthe heuristic ?replica method? of
statistical mechanics, Monasson and Zecchina gave a number of non-rigorous predictions on the
behavior of random (2 + p)-SAT formulae. In this paper we give the 1rst rigorous results for
random (2 + p)-SAT, includingthe followingsurprisingfact: for p 6 2=5, with probability
1 − o(1), a random (2 + p)-SAT formula is satis1able i@ its 2-SAT subformula is satis1able.
That is, for p 6 2=5, random (2 + p)-SAT behaves like random 2-SAT.
Abstract: Random Intersection Graphs, Gn,m,p, is a class of random graphs introduced in Karoński (1999) [7] where each of the n vertices chooses independently a random subset of a universal set of m elements. Each element of the universal sets is chosen independently by some vertex with probability p. Two vertices are joined by an edge iff their chosen element sets intersect. Given n, m so that m=left ceilingn{\'a}right ceiling, for any real {\'a} different than one, we establish here, for the first time, a sharp threshold for the graph property “Contains a Hamilton cycle”. Our proof involves new, nontrivial, coupling techniques that allow us to circumvent the edge dependencies in the random intersection graph model.
Abstract: We explore the capability of a network of extremely limited
computational entities to decide properties about any of its subnetworks.
We consider that the underlying network of the interacting
entities (devices, agents, processes etc.) is modeled by a complete in-
teraction graph and we devise simple graph protocols that can decide
properties of some input subgraph provided by some preprocessing on
the network. The agents are modeled as nite-state automata and run
the same global graph protocol. Each protocol is a xed size grammar,
that is, its description is independent of the size (number of agents) of
the network. This size is not known by the agents. We propose a simple
model, the Mediated Graph Protocol (MGP) model, similar to the Population
Protocol model of Angluin et al., in which each network link is
characterized by a state taken from a nite set. This state can be used
and updated during each interaction between the corresponding agents.
We provide some interesting properties of the MGP model among which
is the ability to decide properties on stabilizing (initially changing for a
nite number of steps) input graphs and we show that the MGP model
has the ability to decide properties of disconnected input graphs. We
show that the computational power within the connected components is
fairly restricted. Finally, we give an exact characterization of the class
GMGP, of graph languages decidable by the MGP model: it is equal
to the class of graph languages decidable by a nondeterministic Turing
Machine of linear space that receives its input graph by its adjacency
matrix representation.
Abstract: We explore the capability of a network of extremely limited computational entities to decide properties about itself or any of its subnetworks. We consider that the underlying network of the interacting entities (devices, agents, processes etc.) is modeled by an interaction graph that reflects the network’s connectivity. We examine the following two cases: First, we consider the case where the input graph is the whole interaction graph and second where it is some subgraph of the interaction graph given by some preprocessing on the network. In each case, we devise simple graph protocols that can decide properties of the input graph. The computational entities, that are called agents, are modeled as finite-state automata and run the same global graph protocol. Each protocol is a fixed size grammar, that is, its description is independent of the size (number of agents) of the network. This size is not known by the agents. We present two simple models (one for each case), the Graph Decision Mediated Population Protocol (GDMPP) and the Mediated Graph Protocol (MGP) models, similar to the Population Protocol model of Angluin et al., where each network link (edge of the interaction graph) is characterized by a state taken from a finite set. This state can be used and updated during each interaction between the corresponding agents. We provide some example protocols and some interesting properties for the two models concerning the computability of graph languages in various settings (disconnected input graphs, stabilizing input graphs). We show that the computational power within the family of all (at least) weakly-connected input graphs is fairly restricted. Finally, we give an exact characterization of the class of graph languages decidable by the MGP model in the case of complete interaction graphs: it is equal to the class of graph languages decidable by a nondeterministic Turing Machine of linear space that receives its input graph by its adjacency matrix representation.
Abstract: We investigate the existence and efficient algorithmic construction
of close to optimal independentsets in random models of intersection
graphs. In particular, (a) we propose a new model for random
intersection graphs (Gn,m,p) which includes the model of [10] (the “uniform”
random intersection graphs model) as an important special case.
We also define an interesting variation of the model of random intersection
graphs, similar in spirit to random regular graphs. (b) For this
model we derive exact formulae for the mean and variance of the number
of independentsets of size k (for any k) in the graph. (c) We then propose
and analyse three algorithms for the efficient construction of large
independentsets in this model. The first two are variations of the greedy
technique while the third is a totally new algorithm. Our algorithms are
analysed for the special case of uniform random intersection graphs.
Our analyses show that these algorithms succeed in finding close to optimal
independentsets for an interesting range of graph parameters.
Abstract: We investigate the existence and efficient algorithmic construction of close to opti-
mal independentsets in random models of intersection graphs. In particular, (a) we
propose a new model for random intersection graphs (Gn,m,~p) which includes the
model of [10] (the “uniform” random intersection graphs model) as an important
special case. We also define an interesting variation of the model of random intersec-
tion graphs, similar in spirit to random regular graphs. (b) For this model we derive
exact formulae for the mean and variance of the number of independentsets of size
k (for any k) in the graph. (c) We then propose and analyse three algorithms for
the efficient construction of large independentsets in this model. The first two are
variations of the greedy technique while the third is a totally new algorithm. Our
algorithms are analysed for the special case of uniform random intersection graphs.
Our analyses show that these algorithms succeed in finding close to optimal in-
dependent sets for an interesting range of graph parameters.
Abstract: In this paper, we discuss the integration of Wireless
Sensor Networks (WSN) and smart objects with the Web. We present a set of research challenges which we believe are the most important ones rising from this integration and propose a prototype system, Uberdust, which addresses such challenges. Uberdust is a brokerage web service for connecting smart objects to the Internet of Things, providing storage, sharing and discovery of real-time and historical data from smart objects, devices & building installations around the world via the Web. Our system provides high-level language-independent APIs so IoT application developers may choose their favorite programming or scripting languages.
