Abstract: In Descriptive Complexity, there is a vast amount of literature on decision problems, and their classes such as \bf{P, NP, L and NL}. However, research on the descriptive complexity of optimisation problems has been limited. In a previous paper, we characterised the optimisation versions of \bf{P} via expressions in second order logic, using universal Horn formulae with successor relations. In this talk, we look at the syntactic hierarchy within the class of polynomially bound maximisation problems. We extend the result in the previous paper by showing that the class of polynomially-bound \bf{NP} (not just \bf{P}) maximisation problems can be expressed in second-order logic using Horn formulae with successor relations. Finally, we provide an application -- we show that the Bin Packing problem with online and item-size constraints can be approximated to within a $\Theta(\log n)$ bound, by providing a syntactic characterisation for this problem.

Short Bio: Prabhu Manyem obtained his BS and MS degrees at the Indian Institute of Technology, and then his PhD (Operations Research), with a minor in Computer Science, at North Carolina State University, Raleigh, NC in 1996. After working in the US industry for three years, he joined the University of South Australia at Adelaide (Australia) as a Research Fellow. In February 2005, he moved to the University of Ballarat (Australia) as an Assistant Professor in Mathematics. His interests are in Complexity, Approximation Algorithms and Discrete Optimisation.