EPSRC Reference: 
EP/R034516/1 
Title: 
The Complexity of Promise Constraint Satisfaction 
Principal Investigator: 
Krokhin, Professor A 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Computer Science 
Organisation: 
Durham, University of 
Scheme: 
Standard Research 
Starts: 
20 June 2018 
Ends: 
19 December 2021 
Value (£): 
441,210

EPSRC Research Topic Classifications: 
Fundamentals of Computing 


EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 
Panel Date  Panel Name  Outcome 
01 Mar 2018

EPSRC ICT Prioritisation Panel March 2018

Announced


Summary on Grant Application Form 
Why is it that some computational problems admit algorithms that always work fast, i.e. scale up well with the size of data to be processed, while other computational problems are not like this and (appear to) admit only algorithms that scale up exponentially? Answering this question is one of the fundamental goals of theoretical computer science. Computational complexity theory formalises the two kinds of problems as tractable and NPhard, respectively. So we can rephrase the above question as follows: What kind of inherent mathematical structure makes a computational problem tractable? This very general question is known to be extremely difficult. The Constraint Satisfaction Problem (CSP) and its variants are extensively used towards answering this question for two reasons: on the one hand, the CSP framework is very general and includes a wide variety of computational problems, and on the other hand, this framework has very rich mathematical structure providing an excellent laboratory both for complexity classification methods and for algorithmic techniques.
The socalled algebraic approach to the CSP has been very successful in this quest for understanding tractability. The idea of this approach is that nontrivial algebraic structure (which can viewed roughly as multidimensional symmerties) in problem instances leads to tractability, while the absence of such structure leads to NPhardness. This approach has already provided very deep insights and delivered very strong complexity classification results. It is a common perception that the power of this approach comes largely from the fact that symmetries can be composed, i.e. cascaded, to form another symmetry. The proposed researh will challenge this perception and extend the algebraic approach significantly beyond this seemingly indispensable property. Thus, we will provide further very strong evidence to the thesis that tractable problems posess algebraic structure. We will also apply our new theory to resolve longstanding open questions about some classical NPhard optimisation problems, specifically how much the optimality demand must be relaxed there to guarantee tractability.

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