Computational Complexity (U01954)

? Credit Points : 10 ? SCQF Level : 10 ? Acronym : INF-4-CMC

The module extends a line of study, begun in CS3 Computability and Intractability, in which computational problems are classified according to their intrinsic difficulty or ``complexity.'' We formalise the notion of complexity of a problem as the amount of time (or space) required to solve the problem on a simple universal computing device, namely the Turing machine. We study some fundamental features of computation in this model, such as time and space hierarchies, the relationship between time and space, and between determinism and non-determinism. We introduce a number of natural complexity classes, which are essentially independent of the Turing machine model, and characterise these classes by identifying some of their complete (i.e., hardest) problems. We then introduce a computational model based on Boolean circuits that allows us to classify problems according to their parallel complexities; as with sequential computation, we are able to separate those problems that can be solved efficiently on a parallel computer from those that (apparently) cannot. Next, we examine the role of randomisation (allowing occasional incorrect answers) in making apparently intractable problems easier. We meet a surprising characterisation of the class NP in terms of ``probabilistically checkable proofs,'' and make an equally surprising connection between this new view of NP and non-approximability of combinatorial optimisation problems. Finally, we investigate some really hard problems that are provably intractable.

Entry Requirements

? Pre-requisites : Successful completion of Year 3 of an Informatics Single or Combined Honours Degree, or equivalent by permission of the School. Participants should have some facility with mathematical modes of reasoning.

Subject Areas

Home subject area

Theoretical Computer Science, (School of Informatics, Schedule O)

Delivery Information

? Normal year taken : 4th year

? Delivery Period : Not being delivered

? Contact Teaching Time : 2 hour(s) per week for 10 weeks

Summary of Intended Learning Outcomes

- Students will be able to formulate models of sequential, randomised and parallel compution, and be able to describe the relationships between these models.
- They will be able to quantify the resources employed by these models, such as time, space and circuit size/depth.
- Students will be able to analyse computational problems from a complexity perspective, and so locate them within the complexity landscape (a landscape which is much refined from that described in Computability and Intractability).
- In particular, they will further develop their skill in conducting a completeness proof, which is in a sense a practical skill.
- Students will be able to apply mathematical skills and knowledge from earlier years (e.g., from probability theory and logic) to concrete problems in computational complexity.
- Students will study the topic in sufficient depth as to gain an appreciation of some of the challenging issues in computer science today (e.g., P =? NP).