Measure based metrics for aggregated data

Aggregated data arises commonly from surveys and censuses where groups of individuals are studied as coherent entities. The aggregated data can take many forms including sets, intervals, distributions and histograms. The data analyst needs to measure the similarity between such aggregated data items and a range of metrics are reported in the literature to achieve this (e.g. the Jaccard metric for sets and the Wasserstein metric for histograms). In this paper, a unifying theory based on measure theory is developed that establishes not only that known metrics are essentially similar but also suggests new metrics

Jamming Model for the Extremal Optimization Heuristic

Extremal Optimization, a recently introduced meta-heuristic for hard optimization problems, is analyzed on a simple model of jamming. The model is motivated first by the problem of finding lowest energy configurations for a disordered spin system on a fixed-valence graph. The numerical results for the spin system exhibit the same phenomena found in all earlier studies of extremal optimization, and our analytical results for the model reproduce many of these features.Comment: 9 pages, RevTex4, 7 ps-figures included, as to appear in J. Phys. A, related papers available at http://www.physics.emory.edu/faculty/boettcher

Polynomial-time approximation schemes for scheduling problems with time lags

We identify two classes of machine scheduling problems with time lags that possess Polynomial-Time Approximation Schemes (PTASs). These classes together, one for minimizing makespan and one for minimizing total completion time, include many well-studied time lag scheduling problems. The running times of these approximation schemes are polynomial in the number of jobs, but exponential in the number of machines and the ratio between the largest time lag and the smallest positive operation time. These classes constitute the first PTAS results for scheduling problems with time lags

NP-Completeness of a combinator optimisation problem

We consider a deterministic rewrite system for combinatory logic over combinators S, K, I, B, C, S', B' and C'. Terms will be represented by graphs so that reduction of a duplicator will cause the duplicated expression to be "shared" rather than copied. To each normalising term we assign a weighting which is the number of reduction steps necessary to reduce the expression to normal form. A lambda expression may be represented by several distinct expressions in combinatory logic, and two combinatory logic expressions are considered equivalent if they represent the same lambda expression (up to beta-eta-equivalence). The problem of minimising the number of reduction steps over equivalent combinator expressions (i.e. the problem of finding the "fastest running" combinator representation for a specific lambda expression) is proved to be NP-complete by reduction from the "Hitting Set" problem