Symmetry in Computational Complexity

Periodic Reporting for period 4 - CoCoSym (Symmetry in Computational Complexity)

Reporting period: 2022-08-01 to 2023-01-31

The aim of CoCoSym was to better understand the complexity of computational problems. A computational problem is a task for a machine that specifies the type of an input and the required output; for example, "Given two numbers, find their product" is a computational problem. The complexity of such a problem is the amount of resources (e.g. time) that is needed to always solve the problem, i.e. we ask: how fast is the most efficient algorithm that returns correct answers to every possible input?

Computational complexity theory has identified robust ways to quantify complexity. In particular, the theory introduces complexity classes formed by computational problems of related complexity. Two of the most important complexity classes are P, which is intuitively the class of problems that can be efficiently solved, and NP, which is the class of problems where it is possible to efficiently verify that an output, given to us by an oracle, is correct. An example of a problem in P is the multiplication above. An important example of a problem in NP is the following, so-called k-coloring problem:

Given a list of objects and forbidden pairs of objects (such as a set of people with the relation "cannot be in one team"), find a 3-coloring, i.e. divide the objects into k groups so that there is no forbidden pair in any of the groups.

In fact, already the 3-coloring problem is, in a sense, the hardest problem in NP: if there exists an efficient algorithm for 3-coloring, then all problems in NP admit an efficient algorithm. It may seem intuitively obvious that NP is much larger than P, because checking whether a solution is correct seems to be a much easier process than finding a solution. However, we do not currently know whether this intuition is correct - this "P=NP?" problem is one of the seven Millennium Prize Problems in mathematics. One reason why it is so hard to find the answer is that we do not have a satisfactory understanding of what makes a computational problem easy or hard. Improving this understanding was the aim of CoCoSym.

The starting point was an exciting discovery from the end of the last millennium. T. Feder, M. Vardi, P. Jeavons, and other researchers discovered that the computational complexity within an important class of computational problems, so called fixed-template finite-domain constraint satisfaction problems (CSPs), is fully captured by symmetry: to every computational problem within the class one can assign a certain object (think of a picture) that consists of symmetries of that computational problem. The complexity of the original problem can then be determined by looking solely at the assigned object (the picture). While CSPs are a relatively tiny class from the global perspective, they include many important problems in computer science, e.g. the 3-coloring problem. After this discovery, the area of CSPs experienced rapid development, culminating in the complete classification of CSPs into problems in P and problems as hard as 3-coloring. This CSP dichotomy theorem was independently obtained by A. Bulatov and D. Zhuk in 2017.

The approach of studying computational complexity via symmetries was applied to larger classes of problems already before CoCoSym, but naturally hit significant barriers. The results of CoCoSym broke through some of these barriers. They expanded and applied the complexity-symmetry correspondence in much larger scope than CSPs and they provided novel methods that enable us to move from ad hoc, problem-specific arguments to a unified, systematic exploration of complexity. They make a step toward a satisfactory answer to a question that is fundamental for science and society: What makes a computational problem easy or hard?

CoCoSym enabled me to create a team that included top researchers in the area, Dmitriy Zhuk, a recepient of the Presburger Award in 2020 and a speaker at the International Congress of Mathematicians in 2022, and Antoine Mottet, a recepient of the Ackermann Award in 2019.

Dmitriy, as a CoCoSym team member, improved and simplified his proof of the CSP dichotomy theorem and published it in JACM. Apart from the dichotomy theorem, the five most significant results to which CoCoSym team members contributed are the following (the main contributions are discussed for two of them).

(1) Barto, Bulín, Krokhin, Opršal: Algebraic Approach to Promise Constraint Satisfaction, JACM

This paper and subsequent refinements show that (appropriately defined) symmetries govern computational complexity for a huge class of computational problems, contributing to one of the main questions of CoCoSym: what are the right objects ("the pictures") describing symmetries for classes of problems beyond CSPs? The paper is the foundation of the majority of recent developments in the area and it has already become a standard reference. The main value of the paper is the general theory it develops, nevertheless, concrete novel applications were given as well. For instance, the following "3-vs-5-coloring" problem was shown to be no easier than 3-coloring.

Given a list of objects and forbidden pairs of objects that admit a 3-coloring, find a 5-coloring.

(2) Barto, Kozik: Combinatorial Gap Theorem and Reductions between Promise CSPs, SODA

The previous paper shows the breadth of the approach to complexity via symmetries, this paper shows its depth. It contributes to another main question of CoCoSym which, using the picture metaphor, asks: from what "distance" do we need to look at the picture to still see the complexity? This paper substantially increase the distance and provides general methods to reduce one computational problem to another one, e.g. the 3-coloring can now be directly reduced to 3-vs-5-coloring. The results also suggests a novel, combinatorial approach to a major problem in theoretical computer science, the Unique Games Conjecture.

(3) Barto, Brady, Bulatov, Kozik, Zhuk: Minimal Taylor Algebras as a Common Framework for the Three Algebraic Approaches to the CSP, LICS
(4) Mottet, Pinsker: Smooth Approximations and CSPs over Finitely Bounded Homogeneous Structures, LICS
(5) Zhuk, Martin: QCSP monsters and the demise of the Chen Conjecture, JACM

Altogether, CoCoSym team has contributed to 29 published research papers, mostly in top journals (Journal of the ACM, SIAM Journal on Computing) and top conferences (STOC, SODA, LICS, ICALP). This number will increase as the more recent results of the project will only appear in print in the future. The team members have delivered more than 70 talks at conferences, workshops, and seminars, including 12 invited talks at conferences. We co-organized a yearly invitation-only intensive research workshop CWC, the CSP World Congress, which evolved into a central event for the complexity of CSPs.

All the scientific outputs of CoCoSym provide novel results by introducing novel concepts, methods, approaches, and research directions. In particular, I consider the five papers above as breakthtoughs that go well beyond the state of the art. The results of CoCoSym were among the starting points of a larger scale attack on understanding efficient computation which I designed together with M. Bodirsky (TU Dresden) and M. Pinsker (TU Wien). For this project, we obtained the ERC Synergy grant called POCOCOP (Polynomial-time computation: opening the blackboxes in constraint problems). It started in March 2023.

Ideal world

Periodic Reporting for period 4 - CoCoSym (Symmetry in Computational Complexity)

Share this page

Download