Self-duality of bounded monotone boolean functions and related problems

AbstractIn this paper we examine the problem of determining the self-duality of a monotone boolean function in disjunctive normal form (DNF). We show that the self-duality of monotone boolean functions with n disjuncts such that each disjunct has at most k literals can be determined in O(2k2k2n) time. This implies an O(n2logn) algorithm for determining the self-duality of logn-DNF functions. We also consider the version where any two disjuncts have at most c literals in common. For this case we give an O(n4(c+1)) algorithm for determining self-duality

PTAS for Ordered Instances of Resource Allocation Problems

We consider the problem of fair allocation of indivisible goods where we are given a set I of m indivisible resources (items) and a set P of n customers (players) competing for the resources. Each resource j in I has a same value vj > 0 for a subset of customers interested in j and it has no value for other customers. The goal is to find a feasible allocation of the resources to the interested customers such that in the Max-Min scenario (also known as Santa Claus problem) the minimum utility (sum of the resources) received by each of the customers is as high as possible and in the Min-Max case (also known as R||C_max problem), the maximum utility is as low as possible. In this paper we are interested in instances of the problem that admit a PTAS. These instances are not only of theoretical interest but also have practical applications. For the Max-Min allocation problem, we start with instances of the problem that can be viewed as a convex bipartite graph; there exists an ordering of the resources such that each customer is interested (has positive evaluation) in a set of consecutive resources and we demonstrate a PTAS. For the Min-Max allocation problem, we obtain a PTAS for instances in which there is an ordering of the customers (machines) and each resource (job) is adjacent to a consecutive set of customers (machines). Next we show that our method for the Max-Min scenario, can be extended to a broader class of bipartite graphs where the resources can be viewed as a tree and each customer is interested in a sub-tree of a bounded number of leaves of this tree (e.g. a sub-path)

A revised nitrogen budget for the Arabian Sea

Despite its importance for the global oceanic nitrogen (N) cycle, considerable uncertainties exist about the N fluxes of the Arabian Sea. On the basis of our recent measurements during the German Arabian Sea Process Study as part of the Joint Global Ocean Flux Study (JGOFS) in 1995 and 1997, we present estimates of various N sources and sinks such as atmospheric dry and wet depositions of N aerosols, pelagic denitrification, nitrous oxide (N2O) emissions, and advective N input from the south. Additionally, we estimated the N burial in the deep sea and the sedimentary shelf denitrification. On the basis of our measurements and literature data, the N budget for the Arabian Sea was reassessed. It is dominated by the N loss due to denitrification, which is balanced by the advective input of N from the south. The role of N fixation in the Arabian Sea is still difficult to assess owing to the small database available; however, there are hints that it might be more important than previously thought. Atmospheric N depositions are important on a regional scale during the intermonsoon in the central Arabian Sea; however, they play only a minor role for the overall N cycling. Emissions of N2O and ammonia, deep-sea N burial, and N inputs by rivers and marginal seas (i.e., Persian Gulf and Red Sea) are of minor importance. We found that the magnitude of the sedimentary denitrification at the shelf might be ∼17% of the total denitrification in the Arabian Sea, indicating that the shelf sediments might be of considerably greater importance for the N cycling in the Arabian Sea than previously thought. Sedimentary and pelagic denitrification together demand ∼6% of the estimated particulate organic nitrogen export flux from the photic zone. The main northward transport of N into the Arabian Sea occurs in the intermediate layers, indicating that the N cycle of the Arabian Sea might be sensitive to variations of the intermediate water circulation of the Indian Ocean

The arithmetic of maximal planes

The geometry of shapes made up of finite planes is considered in detail. Algorithms on maximal planes for performing shape arithmetic are developed

The maximal representation of shape

The maximal representation of a shape is defined and algorithms for shape arithmetic are develope