Andrew J. Parkes

Setting the Research Agenda in Automated Timetabling: The Second International Timetabling Competition

The Second International Timetabling Competition (TTC2007) opened in August 2007. Building on the success of the first competition in 2002, this sequel aimed to further develop research activity in the area of educational timetabling. The broad aim of the competition was to create better understanding between researchers and practitioners by allowing emerging techniques to be developed and tested on real-world models of timetabling problems. To support this, a primary goal was to provide researchers with models of problems faced by practitioners through incorporating a significant number of real-world constraints. Another objective of the competition was to stimulate debate within the widening timetabling research community. The competition was divided into three tracks to reflect the important variations that exist in educational timetabling within higher education. Because these formulations incorporate an increased number of “real-world” issues, it is anticipated that the competition will now set the research agenda within the field. After finishing in January 2008, final results were made available in May 2008. Along with background to the competition, the competition tracks are described here along with a brief overview of the techniques used by the competition winners.

Finiteness in Rigid Supersymmetric Theories

Abstract The two-loop β function for an arbitrary rigid supersymmetric theory is given. The necessary and sufficient conditions for a rigid supersymmetric theory to be finite at one loop are derived. It is shown that these conditions also imply two-loop finiteness.

HyFlex: a benchmark framework for cross-domain heuristic search

This paper presents HyFlex, a software framework for the development of cross-domain search methodologies. The framework features a common software interface for dealing with different combinatorial optimisation problems and provides the algorithm components that are problem specific. In this way, the algorithm designer does not require a detailed knowledge of the problem domains and thus can concentrate his/her efforts on designing adaptive general-purpose optimisation algorithms. Six hard combinatorial problems are fully implemented: maximum satisfiability, one dimensional bin packing, permutation flow shop, personnel scheduling, traveling salesman and vehicle routing. Each domain contains a varied set of instances, including real-world industrial data and an extensive set of state-of-the-art problem specific heuristics and search operators. HyFlex represents a valuable new benchmark of heuristic search generality, with which adaptive cross-domain algorithms are being easily developed and reliably compared.This article serves both as a tutorial and a as survey of the research achievements and publications so far using HyFlex.

Decomposition, reformulation, and diving in university course timetabling

In many real-life optimisation problems, there are multiple interacting components in a solution. For example, different components might specify assignments to different kinds of resource. Often, each component is associated with different sets of soft constraints, and so with different measures of soft constraint violation. The goal is then to minimise a linear combination of such measures. This paper studies an approach to such problems, which can be thought of as multiphase exploitation of multiple objective-/value-restricted submodels. In this approach, only one computationally difficult component of a problem and the associated subset of objectives is considered at first. This produces partial solutions, which define interesting neighbourhoods in the search space of the complete problem. Often, it is possible to pick the initial component so that variable aggregation can be performed at the first stage, and the neighbourhoods to be explored next are guaranteed to contain feasible solutions. Using integer programming, it is then easy to implement heuristics producing solutions with bounds on their quality. Our study is performed on a university course timetabling problem used in the 2007 International Timetabling Competition (ITC), also known as the Udine Course Timetabling problem. The goal is to find an assignment of events to periods and rooms, so that the assignment of events to periods is a feasible bounded colouring of an associated conflict graph and the linear combination of the numbers of violations of four soft constraints is minimised. In the proposed heuristic, an objective-restricted neighbourhood generator produces assignments of periods to events, with decreasing numbers of violations of two period-related soft constraints. Those are relaxed into assignments of events to days, which define neighbourhoods that are easier to search with respect to all four soft constraints. Integer programming formulations for all subproblems are given and evaluated using ILOG CPLEX 11. The wider applicability of this approach is analysed and discussed.

A supernodal formulation of vertex colouring with applications in course timetabling

For many problems in scheduling and timetabling, the choice of a mathematical programming formulation is determined by the formulation of the graph colouring component. This paper briefly surveys seven known integer programming formulations of vertex colouring and introduces a new approach using “supernodes”.In the definition of George and McIntyre (SIAM J. Numer. Anal. 15(1):90–112, 1978), a “supernode” is a complete subgraph, within which every pair of vertices have the same neighbourhood outside of the subgraph. A polynomial-time algorithm for obtaining the best possible partition of an arbitrary graph into supernodes is given. This makes it possible to use any formulation of vertex multicolouring to encode vertex colouring. Results of empirical tests on benchmark instances in graph colouring (DIMACS) and timetabling (Udine Course Timetabling) are also provided and discussed.

Three Loop Results in Two Loop Finite Supersymmetric Gauge Theories

Abstract We investigate the three-loop infinity structure in the class of two-loop finite supersymmetric Yang-Mills theories. By means of explicit calculations of Feynman graphs in an N = 1 superfield context, we show that the β-function of the gauge coupling does not receive any contribution at the three-loop level. By use of the Adler-Bardeen theorem it is argued that this is a general result namely if the theory is finite at n loops then the gauge β-function will vanish at n + 1 loops. However, we show that infinities do arise at three loops in the renormalizations of the matter fields in at least some of these theories.

On covariant multi-loop superstring amplitudes

Abstract We present results for g-loop superstring amplitudes obtained by explicitly performing the sum over spin structures in the spinning string formulation. We do this by placing the picture changing operators at the zeroes of an arbitrary holomorphic one-form. Fays trisecant identity is used extensively to simplify the expressions until the Riemann identity can be applied. We analyze massless bosonic N-point functions, and our methods are powerful enough to treat g + N ⩽ 8 (and g ⩽ 2 for the ghost part); well beyond previous limits. For these cases we establish the vanishing of the cosmological constant and the non-renormalization theorem. Finally we evaluate the two-loop four-boson amplitude in theta-function language for the first time. All of our considerations are local in moduli space.

Search for a three-loop-finite chiral theory

Abstract Grand-unified theories have been constructed out of supersymmetric SU 5 theories which are finite at one and two loops. We investigate the three-loop divergences in these models and find that they can never be three-loop finite. We present an example based on E 6 of a three-loop-finite chiral theory.

A cubic action for self-dual Yang-Mills

Abstract We make a change of field variables in the J formulation of self-dual Yang-Mills theory. The field equations for the resulting algebra valued field are derivable form a simple cubic action. The cubic interaction vertex is different from that considered previously from the N =2 string, however, perturbation theory with this action shows that the only non-vanishing connected scattering amplitude is for three external particles just as for the string.

Three-loop finiteness conditions in N = 1 super-Yang-Mills

Abstract We consider the one-loop finite N = 1 Yang-Mills theories and by a mixture of direct calculation and deductions from the all-orders finiteness of some N = 2 theories we find an expression for the three-loop divergence in the matter field self-energy. We also discuss the effect of explicit supersymmetry breaking terms at the two-loop level.