Periodic Reporting for period 4 - PowAlgDO (Power of Algorithms in Discrete Optimisation)
Reporting period: 2021-07-01 to 2022-06-30
Convex relaxations, such as linear and semidefinite programming, constitute one of the most powerful techniques for designing efficient algorithms, and have been studied in theoretical computer science, operational research, and applied mathematics. The project seeks to establish the power convex relaxations through the lens of, and with the extensions of methods designed for, Constraint Satisfaction Problems (CSPs).
The goal of the project is twofold. First, to provide precise characterisations of the applicability of convex relaxations such as which problems can be solved by linear programming relaxations. Secondly, to derive computational complexity consequences such as for which classes of problems the considered algorithms are optimal in that they solve optimally everything that can be solved in polynomial time. For optimisation problems, we aim to characterise the limits of linear and semidefinite programming relaxations for exact and approximate solvability.
This is a theoretical project. While answering questions on the power of convex relaxations and delineated the tractability boundary for large classes of computational problems are of fundamental nature, they may also have practical consequence. In particular, any progress on algorithmic techniques enlarges the scope of our algorithms. Moreover, computational intractability indicates for which problems a more relaxed notions of tractability or heuristics are needed.
The goal of the project is twofold. First, to provide precise characterisations of the applicability of convex relaxations such as which problems can be solved by linear programming relaxations. Secondly, to derive computational complexity consequences such as for which classes of problems the considered algorithms are optimal in that they solve optimally everything that can be solved in polynomial time. For optimisation problems, we aim to characterise the limits of linear and semidefinite programming relaxations for exact and approximate solvability.
This is a theoretical project. While answering questions on the power of convex relaxations and delineated the tractability boundary for large classes of computational problems are of fundamental nature, they may also have practical consequence. In particular, any progress on algorithmic techniques enlarges the scope of our algorithms. Moreover, computational intractability indicates for which problems a more relaxed notions of tractability or heuristics are needed.
The main results of the projects include characterisations of convex relaxations for fragments of constraint satisfaction problems. The highlights of the project, published in top computer science conferences and journals, include the following results:
* The power of Sherali-Adams linear programming relaxations for general-valued CSPs. In particular, showing that the third level of the Sherali-Adams linear programming relaxation is equivalent in power to any constant level of the same hierarchy. The condition characterising the power is linked to other important algorithms, namely consistency methods in artificial intelligence.
* The power of convex relaxations for general-valued CSPs. This result establishes an algorithmic dichotomy. Among all possible convex relaxations, for the class of non-uniform general-valued CSPs, either the third level of the Sherali-Adams linear programming relaxation finds an optimal solution or no convex relaxation does.
* The power of Sherali-Adams linear programming relaxations fro general-valued CSPs parameterised by the structure of the instance. This result links linear programming relaxations to the well-studied notion of treewidth, and also implies interesting results in database theory, namely the feasibility of evaluation of conjunctive queries over the tropical semiring.
* The power of the combined basic linear programming and affine integer programming relaxation. This paper proposes a simpler yet powerful algorithm that solves, in a unified way, all tractable non-uniform CSPs on the Boolean domain.
* A structural condition that implies the existence of a polynomial-time approximation scheme for Max-CSPs that unifies problems in sparse graph (such as planar graphs) and dense graphs.
* The power of Sherali-Adams linear programming relaxations for general-valued CSPs. In particular, showing that the third level of the Sherali-Adams linear programming relaxation is equivalent in power to any constant level of the same hierarchy. The condition characterising the power is linked to other important algorithms, namely consistency methods in artificial intelligence.
* The power of convex relaxations for general-valued CSPs. This result establishes an algorithmic dichotomy. Among all possible convex relaxations, for the class of non-uniform general-valued CSPs, either the third level of the Sherali-Adams linear programming relaxation finds an optimal solution or no convex relaxation does.
* The power of Sherali-Adams linear programming relaxations fro general-valued CSPs parameterised by the structure of the instance. This result links linear programming relaxations to the well-studied notion of treewidth, and also implies interesting results in database theory, namely the feasibility of evaluation of conjunctive queries over the tropical semiring.
* The power of the combined basic linear programming and affine integer programming relaxation. This paper proposes a simpler yet powerful algorithm that solves, in a unified way, all tractable non-uniform CSPs on the Boolean domain.
* A structural condition that implies the existence of a polynomial-time approximation scheme for Max-CSPs that unifies problems in sparse graph (such as planar graphs) and dense graphs.
In addition to the already mentioned results, on rather unexpected outcome of this grant is making progress on so-called promise constraint satisfaction problems (PCSPs). Promise CSPs are a recently introduced generalisation of CSPs that allows for capturing problems with perfect satisfiability. In detail, there are two main ways of dealing with computational hardness. One way is to relax the goal of the problem. Rather than trying to find an optimal solution to a given problems (say, maximising the number of satisfied constraints), we only require that we satisfy at least a half of the optimal number of satisfiable constraints. This idea leads to Max-CSPs, that were studied in this grant, as envisaged in the proposal. The other way is to relax the constraints but still insist on satisfying all of them. A canonical example of this is graph colouring. Given a graph that is 3-colourable, it is well known and goes back to Karp's result in the 1970s that finding a 3-colouring is NP-hard. However, what if more colours are allowed? Can a 6-colouring be found efficiently for a graph that is promised to be 3-colourable? This problem is still open. The idea of promise CSPs formalised the framework of problems of this kind and allows for a uniform treatment of such problems, often via deep links to different branches of mathematics. Results obtained in the final stages of this project include establishing computational complexity classifications for fragments of PCSPs, identifying new reasons for computational hardness, and improving the state-of-the-art on the above-mentioned graph colouring problem.