Paolo Penna

On the power assignment problem in radio networks

Given a finite set S of points (i.e. the stations of a radio network) on a d-dimensional Euclidean space and a positive integer 1≤h≤|S|−1, the MIN DD H-RANGE ASSIGNMENT problem consists of assigning transmission ranges to the stations so as to minimize the total power consumption, provided that the transmission ranges of the stations ensure the communication between any pair of stations in at most h hops.Two main issues related to this problem are considered in this paper: the trade-off between the power consumption and the number of hops; the computational complexity of the MIN DD H-RANGE ASSIGNMENT problem.As for the first question, we provide a lower bound on the minimum power consumption of stations on the plane for constant h. The lower bound is a function of |S|, h and the minimum distance over all the pairs of stations in S. Then, we derive a constructive upper bound as a function of |S|, h and the maximum distance over all pairs of stations in S (i.e. the diameter of S). It turns out that when the minimum distance between any two stations is “not too small” (i.e. well spread instances) the upper bound matches the lower bound. Previous results for this problem were known only for very special 1-dimensional configurations (i.e., when points are arranged on a line at unitary distance) [Kirousis, Kranakis, Krizanc and Pelc, 1997].As for the second question, we observe that the tightness of our upper bound implies that MIN 2D H-RANGE ASSIGNMENT restricted to well spread instances admits a polynomial time approximation algorithm. Then, we also show that the same approximation result can be obtained for random instances. On the other hand, we prove that for h=|S|−1 (i.e. the unbounded case) MIN 2D H-RANGE ASSIGNMENT is NP-hard and MIN 3D H-RANGE ASSIGNMENT is APX-complete.

Hardness Results for the Power Range Assignmet Problem in Packet Radio Networks

The minimum range assignment problem consists of assigning transmission ranges to the stations of a multi-hop packet radio network so as to minimize the total power consumption provided that the transmission range assigned to the stations ensures the strong connectivity of the network (i.e. each station can communicate with any other station by multi-hop transmission). The complexity of this optimization problem was studied by Kirousis, Kranakis, Krizanc, and Pelc (1997). In particular, they proved that, when the stations are located in a 3-dimensional Euclidean space, the problem is NP-hard and admits a 2-approximation algorithm. On the other hand, they left the complexity of the 2-dimensional case as an open problem.

On the Complexity of Computing Minimum Energy Consumption Broadcast Subgraphs

We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broadcast operation. In particular, we consider this problem as a special case of the following more general combinatorial optimization problem, called Minimum Energy Consumption Broadcast Subgraph (in short, MECBS): Given a weighted directed graph and a specified source node, find a minimum cost range assignment to the nodes, whose corresponding transmission graph contains a spanning tree rooted at the source node. We first prove that MECBS is not approximable within a sub-logarithmic factor (unless P=NP). We then consider the restriction of MECBS to wireless networks and we prove several positive and negative results, depending on the geometric space dimension and on the distance-power gradient. The main result is a polynomial-time approximation algorithm for the NP-hard case in which both the dimension and the gradient are equal to 2: This algorithm can be generalized to the case in which the gradient is greater than or equal to the dimension.

On-line algorithms for the channel assignment problem in cellular networks

We consider the on-line channel assignment problem in the case of cellular networks and we formalize this problem as an on-line load balancing problem for temporary tasks with restricted assignment. For the latter problem, we provide a general solution (denoted as the cluster algorithm ) and we characterize its competitive ratio in terms of the combinatorial properties of the graph representing the network. We then compare the cluster algorithm with the greedy one when applied to the channel assignment problem: it turns out that the competitive ratio of the cluster algorithm is strictly better than the competitive ratio of the greedy algorithm. The cluster method is general enough to be applied to other on-line load balancing problems and, for some topologies, it can be proved to be optimal.

On the approximation ratio of the MST based heuristic for the energy-efficient broadcast problem in static ad-hoc radio networks

We present a technique to evaluate the approximation ratio on random instances of the minimum energy broadcast problem in ad-hoc radio networks which is known to be NP-hard and approximable within 12. Our technique relies on polynomial-time computable lower bound on the optimal cost of any instance. The main result of this paper is that the approximation ratio has never achieved a value greater than 6.4. Furthermore, the worst values of this ratio are achieved for small network sizes. We also provide a clear geometrical motivation of such good approximation results.

On Computing Ad-hoc Selective Families

We study the problem of computing ad-hoc selective families: Given a collection F of subsets of [n] = {1, 2, . . ., n}, a selective family for F is a collection S of subsets of [n] such that for any F ? F there exists S ∈ S such that |F ∩ S| = 1. We first provide a polynomialtime algorithm that, for any instance F, returns a selective family of size O((1 + log(Δmax/Δmin)) ċ log |F|) where Δmax and Δmin denote the maximal and the minimal size of a subset in F, respectively. This result is applied to the problem of broadcasting in radio networks with known topology. We indeed develop a broadcasting protocol which completes any broadcast operation within O(DlogΔlog n/D) time-slots, where n, D and Δ denote the number of nodes, the maximal eccentricity, and the maximal in-degree of the network, respectively. Finally, we consider the combinatorial optimization problem of computing broadcasting protocols with minimal completion time and we prove some hardness results regarding the approximability of this problem.

More powerful and simpler cost-sharing methods

We provide a new technique to derive group strategyproof mechanisms for the cost-sharing problem. Our technique is simpler and provably more powerful than the existing one based on so called cross-monotonic cost-sharing methods given by Moulin and Shenker [1997]. Indeed, our method yields the first polynomial-time mechanism for the Steiner tree game which is group strategyproof, budget balance and also meets other standard requirements (No Positive Transfer, Voluntary Participation and Consumer Sovereignty). A known result by Megiddo [1978] implies that this result cannot be achieved with cross-monotonic cost-sharing methods, even if using exponential-time mechanisms.

Energy consumption in radio networks: Selfish agents and rewarding mechanisms

We consider the range assignment problem in ad-hoc wireless networks in the context of selfish agents: a network manager aims in assigning transmission ranges to the stations so to achieve a suitable network with a minimal overall energy; stations are not directly controlled by the manager and may refuse to transmit with a certain transmission range because this results in a power consumption proportional to that range.

Noisy Data Make the Partial Digest Problem NP -hard

The problem to find the coordinates of n points on a line such that the pairwise distances of the points form a given multi-set of \(n \choose 2\) distances is known as Partial Digest problem, which occurs for instance in DNA physical mapping and de novo sequencing of proteins. Although Partial Digest was – as a combinatorial problem – already proposed in the 1930’s, its computational complexity is still unknown.

Server Placements, Roman Domination and other Dominating Set Variants

Dominating sets in their many variations model a wealth of optimization problems like facility location or distributed file sharing. For instance, when a request can occur at any node in a graph and requires a server at that node, a minimum dominating set represents a minimum set of servers that serve an arbitrary single request by moving a server along at most one edge. This paper studies domination problems for two requests. For the problem of placing a minimum number of servers such that two requests at different nodes can be served with two different servers (called win-win), we present a logarithmic approximation, and we prove that nothing better is possible. We show that the same is true for Roman domination, the well studied problem variant that asks for each vertex to either possess its own server or to have a neighbor with two servers. Still the same is true if each idle server can move along one edge while the first of both requests is being served. For planar graphs, we propose a PTAS for Roman domination (and show that nothing better exists), and we get a constant approximation for win-win.