Periodic Reporting for period 1 - InfCSP (Descriptive Complexity of Infinite Domain Constraint Satisfaction Problems)
Okres sprawozdawczy: 2018-07-01 do 2020-06-30
The constraint satisfaction problem is a computational problem where the input consists of a set of variables and a set of constraints, and the goal is to decide whether there exists an assignment of values to the variables satisfying all the constraints. This simple framework captures many computational problems such as satisfiability, graph colouring or solving systems of equations. The unified formulation allows for analysing such problems globally, instead of studying each problem in isolation. Intense efforts to understand the complexity of constraint satisfaction problems with a finite set of values culminated recently in the confirmation of the famous Dichotomy Conjecture - it has been shown that every problem of this kind is either NP-complete or solvable in polynomial time. However, for many problems which appear naturally in different areas of computer science, such as combinatorial optimisation, artificial intelligence, scheduling and computational biology, the scenario where the set of possible values is finite is too restricted. The overall objective of this project was to exploit the consequences of symmetries to understand the power of logic-based approaches to the infinite domain constraint satisfaction problem - a version of the constraint satisfaction problem where the set of possible values is infinite. The project, taking place at the interface of mathematics and computer science, advanced the research on symmetric computation and strengthened collaborations between the University of Cambridge and other world-leading institutions in the field.
Linear programming is a powerful and widely-used tool for studying combinatorial optimisation problems. It can be formalised as an infinite domain constraint satisfaction problem where the set of values is the set of rational numbers, and the constraints are specified by linear inequalities. In collaboration with the supervisor Prof. Anuj Dawar and Prof. Albert Atserias from Universitat Politècnica de Catalunya the researcher studied symmetric linear programs that decide properties of graphs in the sense that, for each size of graph, there is a linear program defining a polyhedral lift that separates the integer points corresponding to graphs with the property from those corresponding to graphs without the property. The main results include:
1. establishing a tight three-way correspondence between symmetric Boolean threshold circuits, symmetric linear programs, and bounded-variable formulas of counting logic;
2. using this connection to obtain upper and lower bounds on the power of symmetric linear programs to recognise some key graph properties, for example Hamiltonicity and perfect matching;
3. observing that if the well-studied planted-clique problem can be solved in polynomial time, then it is solvable by polynomial sized symmetric linear programs.
A paper containing those results was published at LICS - the main conference on logic in computer science. Its extended version is submitted to the Journal of the ACM - a top computer science journal. Prior to publication the results were disseminated in the form of a preliminary report under an Open Access license at the public e-Print archive arXiv and made available at the project’s website.
1. establishing a tight three-way correspondence between symmetric Boolean threshold circuits, symmetric linear programs, and bounded-variable formulas of counting logic;
2. using this connection to obtain upper and lower bounds on the power of symmetric linear programs to recognise some key graph properties, for example Hamiltonicity and perfect matching;
3. observing that if the well-studied planted-clique problem can be solved in polynomial time, then it is solvable by polynomial sized symmetric linear programs.
A paper containing those results was published at LICS - the main conference on logic in computer science. Its extended version is submitted to the Journal of the ACM - a top computer science journal. Prior to publication the results were disseminated in the form of a preliminary report under an Open Access license at the public e-Print archive arXiv and made available at the project’s website.
The results obtained during the project go beyond the state of the art in the research directions that they address. This is witnessed in particular by the fact that they were accepted for presentation at a highly competitive peer-reviewed international conference and invited for submission to a journal which provides coverage of the most significant work on principles of computer science.
The project led to a significant progress in the study of symmetric computation. The researcher greatly benefited from Prof. Dawar’s expertise in this area to acquire new skills and knowledge which are now being transferred primarily to the scientific community in France where the researcher was offered a permanent position. The results discussed above were presented in particular as an invited talk at a workshop Complexité et Algorithmes - a yearly meeting of a working group gathering French researchers working in the area of algorithms and complexity. The project also facilitated the transfer of knowledge to the host institution. Most importantly, a weekly reading group on constraint satisfaction problems allowed several researchers and PhD students at the University of Cambridge to get familiar with this research area.
Finally, the project contributed to establishing the researcher’s position as a leading young researcher in logic in computer science. The reinforced position in the field and expanded international network allow the researcher to serve the scientific community via program committee work and organisation of international events, such as summer schools and specialised workshops.
The project led to a significant progress in the study of symmetric computation. The researcher greatly benefited from Prof. Dawar’s expertise in this area to acquire new skills and knowledge which are now being transferred primarily to the scientific community in France where the researcher was offered a permanent position. The results discussed above were presented in particular as an invited talk at a workshop Complexité et Algorithmes - a yearly meeting of a working group gathering French researchers working in the area of algorithms and complexity. The project also facilitated the transfer of knowledge to the host institution. Most importantly, a weekly reading group on constraint satisfaction problems allowed several researchers and PhD students at the University of Cambridge to get familiar with this research area.
Finally, the project contributed to establishing the researcher’s position as a leading young researcher in logic in computer science. The reinforced position in the field and expanded international network allow the researcher to serve the scientific community via program committee work and organisation of international events, such as summer schools and specialised workshops.