Domenico Parente

A 2-Approximation Algorithm for Finding an Optimum 3-Vertex-Connected Spanning Subgraph

The problem of finding a minimum weight k-vertex connected spanning subgraph in a graph G=(V,E) is considered. For k?2, this problem is known to be NP-hard. Combining properties of inclusion-minimal k-vertex connected graphs and of k-out-connected graphs (i.e., graphs which contain a vertex from which there exist k internally vertex-disjoint paths to every other vertex), we derive polynomial time algorithm for finding a (?k/2?+1)-connected subgraph with a weight at most twice the optimum to the original problem. In particular, we obtain a 2-approximation algorithm for the case k=3 of our problem. This improves the best previously known approximation ratio 3. The complexity of the algorithm is O(|V|3|E|)=O(|V|5).

Dynamic and static algorithms for optimal placement of resources in a tree

Abstract We consider the problem of optimally placing identical resources at the nodes of a weighted tree-shaped network of size N . The resources satisfy requests issued from the nodes of the network. The cost of a placement of the resources is the sum over all nodes of the network of the product of the node weight times the distance from the closest resource. The static problem consists in determining the minimal cost placement of the resources. The dynamic version consists in recomputing such a cost when a node weight has changed. We present a linear time algorithm that computes the optimal placement of two resources in a tree. For the dynamic version we give an O ( log N ) time algorithm for placing one resource and, for the case of complete binary trees, an O ( log N 3 ) time algorithm for two resources. The static algorithm is faster than the algorithms found in literature, while the dynamic algorithms are the first for this problem.

Synchronization of a line of identical processors at a given time

We are given a line of n identical processors (finite automata) that work synchronously. Each processor can transmit just one bit of information to the adjacent processors (if any) to the left and to the right. The computation starts at time 1 with the leftmost processor in an initial state and all other processors in a quiescent state. Given the time f(n), the problem is to set (synchronize) all the processors in a particular state for the first time, at the very same instant f(n). This problem is also known as the Firing Squad Synchronization Problem and was introduced by Moore in 1964. Mazoyer has given a minimal time solution with the least number of different states (six) and very recently he has given a minimal time solution for the constrained problem in which adjacent processors can exchange only one bit. In this paper we present solutions that synchronize the line at a given time, expressed as a function of n. In particular we give solutions that synchronize at the times nlogn, n√n, n 2 and 2 n. Moreover we also show how to compose solutions in such a way to obtain synchronizing solutions for all times expressed by polynomials with nonnegative coefficients. Clearly all such solutions work also in the general case when the bit constraint is relaxed.

A Linear Time Algorithm for the Feasibility of Pebble Motion on Trees

We consider the following generalization of the popular “15 puzzle.” Let T be a tree with n vertices and with k < n distinct pebbles numbered 1, ..., k on distinct vertices. A move consists in transferring a pebble from its current position to an adjacent unoccupied vertex. We ask the following question: Is a given arrangement of pebbles reachable from another?

Model checking coalitional games in shortage resource scenarios

Verification of multi-agents systems (MAS) has been recently studied taking into account the need of expressing resource bounds. Several logics for specifying properties of MAS have been presented in quite a variety of scenarios with bounded resources. In this paper, we study a different formalism, called Priced Resource-Bounded Alternating-time Temporal Logic (PRBATL), whose main novelty consists in moving the notion of resources from a syntactic level (part of the formula) to a semantic one (part of the model). This allows us to track the evolution of the resource availability along the computations and provides us with a formalisms capable to model a number of real-world scenarios. Two relevant aspects are the notion of global availability of the resources on the market, that are shared by the agents, and the notion of price of resources, depending on their availability. In a previous work of ours, an initial step towards this new formalism was introduced, along with an EXPTIME algorithm for the model checking problem. In this paper we better analyze the features of the proposed formalism, also in comparison with previous approaches. The main technical contribution is the proof of the EXPTIME-hardness of the the model checking problem for PRBATL, based on a reduction from the acceptance problem for Linearly-Bounded Alternating Turing Machines. In particular, since the problem has multiple parameters, we show two fixed-parameter reductions.

Better Algorithms for Minimum Weight Vertex-Connectivity Problems

Given a k vertex-connected graph with weighted edges, we study the problem of finding a minimum weight spanning subgraph which is k vertex-connected, for small values of k. The problem is known to be NP-hard for any k, even when edges have no weight.

A New Approach to Optimal Planning of Robot Motion on a Tree with Obstacles

In this paper we study the Graph Motion Planning of 1 Robot problem (GMP1R) on a tree. This problem consists in computing a minimum cost plan for moving a robot from one vertex to another in a tree whose vertices can have movable obstacles.

Placing Resources in a Tree: Dynamic and Static Algorithms

We study the classical problem of optimally placing resources in a tree. We give dynamic algorithms that recompute the optimal solution after a weight change in polylogarithmic time for the case of one resource in a general tree and for any constant number of resources in a complete tree. Our algorithms are the first dynamic algorithms for this problem. We also give linear-time algorithms for the static version of the problem for two resources. Previously known algorithms run in time quadratic in the number of vertices. We also discuss an on-line amortized constant time algorithm for placing any number of resources on a line.

Languages accepted by systolic Y-tree automata: structural characterizations

In this paper closure properties and decision problems for families of languages accepted by deterministic and nondeterministic systolic binary Y-tree automata are studied. Non closure results under basic language operations are stated by means of new nonacceptability criteria for these classes of automata. Necessary and sufficient conditions are given in terms of the shape of the underlying Y-tree, for the closure under λ-free regular substitution, concatenation, inverse homomorfism and for the closure under right concatenation with and quotient by finite sets. Moreover in the nondeterministic case necessary and sufficient conditions are given again in terms of the shape of the underlying Y-tree for the closure under right concatenation with regular sets and for the decidability of the problems of emptiness, finiteness, equivalence and co-emptiness. A sufficient condition is given for the decidability of the stability problem, in the deterministic case, while some undecidability results are proved in the nondeterministic case.

SYSTOLIC TREE WITH BASE AUTOMATA

Different systolic tree automata (STA) with base (T(b)−STA) are compared. This is a subclass of STA with interesting properties of modularity. We give a necessary and sufficient condition for the inclusion between classes of languages accepted by T(b)− STA, (L(T(b)−STA)), as b varies. We focus on T(b)−STA obtained by varying the base b in a natural way. We prove that for every base b within this framework there exists an a such that L(T(a)−STA) is not contained in L(T(b)−STA). We characterize the family of languages accepted by T(b)−STA when the input conditions are relaxed. Moreover we show that the emptiness problem is decidable for T(b)−STA.