Andrea E. F. Clementi

On the hardness of approximating optimum schedule problems in store and forward networks

Problems related to message communication and traffic control have been assuming more and more importance due the massive use of computer networks. Scheduling a set of messages in a store and forward network means assigning to them network resources in order to deliver each message to its respective destination. The goal of typical scheduling problems is to devise strategies of assignments that minimize the delivery time of all messages or, alternatively, their total end-to-end delay. Both problems have been proven to be network performance (NP)-hard even under very restrictive hypothesis. We study the computational complexity of approximating the minimum end-to-end delay. Unfortunately, it turns out that finding an approximated solution with approximation ratio smaller than k/sup 1/10/ is as difficult as finding the optimal solution (where k is the number of messages in the network). More precisely, approximating the optimum delay is NP-hard even when a nonconstant approximation ratio is allowed. This result holds also in the case of layered networks. We then prove that if we consider a particular class of local schedules (i.e., distributed strategies that can only use information about limited regions of the network) then the approximation error cannot be bounded by any sublinear function in k. Such results also provide a lower bound on the approximation of the minimum delivery time problem. Thus, if the attention is restricted to polynomial-time algorithms, the only possibility is designing heuristics that behave well on average. As a first step to this aim, we evaluate the expected approximation error of some simple heuristics using several experimental tests.

Optimum schedule problems in store and forward networks

Store and forward networks are a convenient model to represent problems in the areas of message communication and traffic control. The goal of a typical scheduling problem is to devise an optimal strategy to send messages from their source sites into their sink ones following paths in the network. In this paper the computational complexity of such a problem is analyzed in the cases of fixed and dynamically variable paths. Unfortunately, it turns out that both of the versions of the considered problem are NP-complete even under very restrictive hypothesis. Moreover, they are not approximable, that is, they can be solved only by polynomial-time heuristic algorithms such that the distance between the exact and the approximate solution is not bounded by any fixed value.<<ETX>>

The reachability problem for finite cellular automata

Abstract We investigate the complexity of the Configuration REachability Problem (CREP) for two classes of finite weakly predictable cellular automata: the invertible and the additive ones. In both cases we prove that CREP belongs to the “Arthur-Merlin” class CoAM[2] 1 .

Optimal Bounds on the Approximation of Boolean Functions with Consequences on the Concept of Hardware

We prove an optimal bound for the function L(n, m, e) that gives the worst-case circuit-size complexity to approximate partial boolean functions having n inputs and domain size m within degree at least e. Our bound applies to any partial boolean function and any approximation degree, completing the study of boolean function approximation introduced in [15]. We also provide the approximation degree (i.e. the value e) achieved by polynomial size circuits on a ‘random’ boolean function. Our results give a new upper bound for the hardness function h(f), the function denoting the minimum value l for which there exists a circuit of size at most l that approximates a boolean function f with degree at least 1/l [14]. The contribution in the proof of the upper bound for L(n, m, e) can be viewed as a set of technical results that globally show how boolean linear operators are “well” distributed over the class of 4-regular domains. We show how to apply this property to approximate partial boolean functions on general domains.