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 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: We study the on-line versions of two fundamental graph problems, maximumindependentset 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: 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 maximumindependentset 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.