POCOCOP

Team Papers Conferences Visitors Visits Abstract

POCOCOP is an ERC Synergy Grant (ERC SyG No. 101071674) funded by the European Research Council (ERC)
POCOCOP starts 1 Mar 2023 and ends 28 Feb 2029

The goal of POCOCOP is to systematically explore polynomial-time tractability in the field of constraint satisfaction and its extensions, in particular promise CSPs, valued CSPs, and CSPs over infinite domains. The project is jointly led by three principal investigators: Manuel Bodirsky (TU Dresden), Michael Pinsker (TU Vienna), and Libor Barto (Charles University, Prague).

3rd POCOCOP meeting, Mar 2024

Team

For applications use the email jobs(at)acronym_of_the_grant(dot)eu. The applicants should have a strong background in at least one of the following fields: theoretical computer science, model theory, or universal algebra. Informal inquiries are very welcome.

PIs Libor Barto
CU Manuel Bodirsky
TU Dresden Michael Pinsker
TU Wien

postdocs Dmitriy Zhuk: since 1 Mar 2023 Santiago Guzmán Pro: since 22 May 2023 Paolo Marimon: since 1 Aug 2023

Alexey Barsukov: since 1 May 2023 Matěj Konečný: since 1 Oct 2023 Piotr Kawałek: since 1 Jan 2024

Jan Bok: since 1 Jun 2024 Andrew Moorhead: since 1 Oct 2023

PhD students Maximilian Hadek: since 15 Jan 2024 Sebastian Meyer: since 1 Mar 2023 Johanna Brunar: since 1 Oct 2023

Žaneta Semanišinová: since 1 May 2023 Roman Feller: since 15 Oct 2023

Philipp Grzywaczyk: since 1 Oct 2023

Past members Kaan Doğanay (PhD student): 1 May 2023 - 30 Apr 2024 Albert Vucaj (postdoc): 1 Oct 2023 - 30 Apr 2024

Albert Vucaj (PhD student): 1 May 2023 - 30 Sep 2023

Paolo Marimon (PhD student): 1 Jul 2023 - 31 Jul 2023

Piotr Kawałek (PhD student): 1 Sep 2023 - 31 Dec 2023

Papers

K. Asimi, L. Barto, V. Dalmau: The Complexity of Promise Constraint Satisfaction Problem Seen from the Other Side [arXiv]
L. Barto, S. Butti, V. Dalmau: The Sherali-Adams and Weisfeiler-Leman hierarchies in (Promise Valued) Constraint Satisfaction Problems [arXiv]
S. Guzmán Pro: Local expressions of hereditary classes [arXiv]
D. Bradley-Williams, P. J. Cameron, J. Hubička, M. Konečný: EPPA numbers of graphs [arXiv]
M. Bodirsky, S. Guzmán Pro: Forbidden Tournaments and the Orientation Completion Problem [arXiv]
L. Barto, A. Mottet: Finite Algebras with Hom-Sets of Polynomial Size [arXiv]
M. Pinsker, C. Schindler: The semigroup of increasing functions on the rational numbers has a unique Polish topology [arXiv]
A. Barsukov: Containment for Binary Guarded Monotone SNP [arXiv]
A. Moorhead: Mal’cev Complexes [arXiv]
D. Bradley-Williams, P. J. Cameron, J. Hubička, M. Konečný: EPPA numbers of graphs [arXiv]
P. Kawałek, A. Weiss: Violating Constant Degree Hypothesis Requires Breaking Symmetry [arXiv]
A. Mottet, T. Nagy, M. Pinsker: An order out of nowhere: a new algorithm for infinite-domain CSPs [arXiv]
T. Nagy, M. Pinsker: Strict width for Constraint Satisfaction Problems over homogeneous structures of finite duality [arXiv]
M. Bodirsky, S. Guzmán Pro: The Generic Circular Triangle-free Graph [arXiv]
R. Feller, M. Pinsker: An algebraic proof of the graph orientation problem dichotomy for forbidden tournaments [arXiv]
S. Guzmán Pro: GMSNP and Finite Structures [arXiv]
L. Barto, M. Kapytka: Multisorted Boolean Clones Determined by Binary Relations up to Minion Homomorphisms [arXiv]

L. Barto, S. Butti, A. Kazda, C. Viola, S. Živný: Algebraic Approach to Approximation, LICS'24, pp. 10:1-10:14 [DOI] [arXiv]
J. Rydval, Ž. Semanišinová, M. Wrona: Identifying Tractable Quantified Temporal Constraints within Ord-Horn, ICALP'24, pp. 151:1-151:20 [DOI] [arXiv]
M. Bodirsky, Ž. Semanišinová, C. Lutz: The Complexity of Resilience Problems via Valued Constraint Satisfaction Problems, LICS'24, pp. 14:1-14:14 [DOI] [arXiv]
A. Barsukov, M. M. Kanté: Generalisations of matrix partitions: Complexity and obstructions, Theoretical Computer Science, Volume 1007 (2024), Article 114652 [DOI] [arXiv]
P. Grzywaczyk, A. Winterhof: Primitive Elements of finite fields F_q^r avoiding affine hyperplanes for q=4 and q=5, Finite Fields and Their Applications, Volume 96 (2024), Article 102416 [DOI] [arXiv]
L. Barto, Z. Brady, A. Bulatov, M. Kozik, D. Zhuk, Unifying the Three Algebraic Approaches to the CSP via Minimal Taylor Algebras, TheoretiCS, Volume 3 (2024), Article 14 [DOI] [arXiv]
A. Vucaj, D. Zhuk: Submaximal clones over a three-element set up to minor-equivalence, Algebra Universalis, Volume 85 (2024), Article 22 [DOI] [arXiv]
P. Kawałek, M. Kompatscher, J. Krzaczkowski: Circuit Equivalence in 2-Nilpotent Algebras, STACS'24, pp. 45:1-45:17 [DOI] [arXiv]
M. Pinsker, C. Schindler: On the Zariski topology on endomorphism monoids of omega-categorical structures, Journal of Symbolic Logic (2023) [DOI] [arXiv]
L. Elliott, J. Jonušas, J. D. Mitchell, Y. Péresse, M. Pinsker: Polish topologies on endomorphism monoids of relational structures, Advances in Mathematics, Volume 431 (2023), Article 109214 [DOI] [arXiv]
M. Bodirsky, S. Knaeuer: Network Satisfaction Problems Solvable by k-Consistency, ICALP'23, pp. 116:1-116:20 [DOI] [arXiv]
M. Bodirsky, A. Vucaj, D. Zhuk: The lattice of clones of self-dual operations collapsed, International Journal of Algebra and Computation, Volume 30 (2023), Issue 4, pp. 717-749 [DOI] [arXiv]
M. Bodirsky, J. Bulín, F. Starke, M. Wernthaler: The smallest hard trees, Constraints, Volume 28 (2023), pp. 105-137 [DOI] [arXiv]

Conferences, workshops, seminars

6-12 Jul 2024, Ž. Semanišinová attending LICS, ICALP, and collocated workshops, Tallinn, Estonia
1. Invited talk Identifying Tractable Quantified Temporal Constraints (6 Jul) at Trends in Arithmetic Theories 2024 [Slides]
2. Talk Classification Transfer for CSPs via Algebraic Products (9 Jul) at Women in Logic 2024 [Slides]
3. Conference talk The Complexity of Resilience Problems via Valued Constraint Satisfaction Problems (11 Jul) at LICS 2024 [Slides]
8-12 Jul 2024, M. Pinsker, invited talk Reconstructing the topology of algebraic structures: how and why at 38th Summer Conference on Topology and its Applications, University of Coimbra, Portugal [Slides]
1-6 Jul 2024, A. Moorhead, conference talk When are bounded arity polynomials enough? at TACL 2024, Universitat de Barcelona, Spain [Slides]
24-28 Jun 2024, M. Pinsker, attending Workshop on Measurable Combinatorics at Erdős center, Budapest, Hungary
25-28 Jun 2024, M. Pinsker, invited talk The power of polymorphisms: the triangle of universal algebra, model theory, and theoretical computer science at TACL 2024 Summer School, Barcelona, Spain [Slides]
14 Jun 2024, R. Feller, seminar talk The graph orientation problem for forbidden tournaments at Forschungsseminar der Algebra-Gruppe, TU Wien, Austria
3 Jun 2024, M. Hadek, talk Algebraic Methods for the Complexity of Constraint Satisfaction Problems at Day of doctoral students - school of mathematics, Charles University, Prague, Czechia [Slides]
31 May - 2 Jun 2023, L. Barto, M. Bodirsky, M. Pinsker, D. Zhuk, A. Barsukov, A. Moorhead, P. Marimon, P. Kawałek, M. Hadek, S. Meyer, Ž. Semanišinová, P. Grzywaczyk, J. Brunar, R. Feller, C. Spiess attending AAA 105, Prague, Czechia
1. A. Barsukov: talk Edge-colourings and constraint satisfaction problems (31 May) [Slides]
2. S. Meyer: talk Polymorphisms in CSPs, Topology and Social Choices (31 May) [Slides]
3. D. Zhuk: talk Strong subalgebras: version 4 (31 May) [Slides]
4. P. Marimon: talk Minimal operations over permutation groups (31 May) [Slides]
5. Ž. Semanišinová: talk Identifying Tractable Quantified Temporal Constraints within Ord-Horn (1 Jun) [Slides]
15 May 2024, A. Moorhead, conference talk When are bounded arity polynomials enough? at ASL 2024 meeting, Iowa State University, USA [Slides]
6 May 2024, M. Bodirsky, invited conference talk at DMV-Tagung, Passau, Germany
1 May 2024, M. Bodirsky, invited seminar talk at FU Berlin, Germany
22 Apr 2024, M. Konečný, seminar talk Finding dense locally finite subgroups in oligomorphic groups at Logic Seminar, Cornell University, USA (Zoom) [Slides]
11 Apr 2024, M. Pinsker, outreach talk Warum fallen Sudokus auch Computern schwer? at TUForMath, TU Wien, Austria
10 Apr 2024, P. Marimon, seminar talk When measures don’t care about structures (and when they do) at Model theory seminar, KGRC, University of Vienna, Austria [Slides]
2-5 Apr 2024, M. Pinsker, jury member at Vienna Mathematics Competition at University of Vienna, Austria
14 Mar 2024, P. Marimon, seminar talk Minimal operations over permutation groups at Algebra seminar, KIAS (Korean Institute of advanced Studies), Korea [Slides]
8 Mar 2024, P. Marimon, seminar talk When measures don’t care about structures (and when they do) at Model theory seminar, Yonsei University, Korea [Slides]
5 Mar 2024, M. Pinsker, seminar talk CSPs: distinguishing the easy from the hard at Algebra Colloquium, Charles University, Prague, Czechia [Slides]
11 Feb 2024, P. Kawałek, seminar talk Proving lower bounds for symmetric circuits at Institute for Formal Methods of Computer Science seminar, Univeristy of Stuttgart, Germany
8 Feb 2024, Ž. Semanišinová, seminar talk Constraint satisfaction problems: an algebraic approach to classifying computational complexity at Algebra seminar, P. J. Šafárik University, Košice, Slovakia [Slides]
29 Jan 2024, M. Bodirsky, seminar talk (Concrete and Abstract) Minions and Applications at Seminar Geometric Methods in Mathematics, TU Dresden, Germany
16 Jan 2024, L. Barto, talk Symetrie ve výpočetní složitosti (in Czech) for Learned Society of the Czech Republic, Prague, Czechia [Slides] [YouTube video]
4 Jan 2024, M. Konečný, seminar talk Extending partial automorphisms at AGK Seminar, TU Dresden, Germany [Slides]
18 Dec 2023, M. Pinsker, talk Maximal clones on omega at Colloquium for Martin Goldstern’s 60th birthday, TU Wien, Austria
15 Dec 2023, P. Kawałek, seminar talk Satisfiability of circuits and equations in algebras at Forschungsseminarder Algebra-Gruppe, TU Wien, Austria [Slides]
7 Dec 2023, P. Kawałek, seminar talk Notes on depth 2 CC-circuits at Institute of Algebra seminar, JKU, Linz, Austria
6 Dec 2023, S. G. Pro, online seminar talk Forbidden tournaments and the orientation (completion) problem at Atlantic Graph Theory Seminar, AARMS [Slides]
28 Nov 2023, P. Kawałek, seminar talk An etude on polynomials over finite rings in the Computational Complexity Theory at Algebra Colloquium, Charles University, Prague, Czechia [Slides]
27 Nov 2023, M. Konečný, seminar talk EPPA numbers of graphs at G^2OAT Monday Seminar, Czech Technical University, Prague, Czechia [Slides]
6-8 Nov 2023, P. Marimon, M. Pinsker, Ž. Semanišinová attending workshop Combinatorial Problems in Model Theory and Computer Science, Leeds, UK
1. M. Pinsker: invited talk Constraint Satisfaction Problems: algebraic and model-theoretic challenges to distinguish the easy from the hard (7 Nov) [Slides]
2. Ž. Semanišinová: invited talk Valued Constraint Satisfaction Problem and Resilience in Database Theory (7 Nov) [Slides]
23-28 Oct 2023, M. Bodirsky, M. Pinsker attending Conference on Generic Structures, Będlewo, Poland
1. M. Bodirsky: invited talk Generic Structures for Monotone Sentences in Monadic Second-Order Logic (27 Oct) [Slides]
2. M. Pinsker: invited talk Topologies on endomorphism monoids of generic structures (27 Oct) [Slides]
19 Oct 2023, M. Konečný, invited talk EPPA numbers of graphs at Annual DYNASNET meeting (16-20 Oct 2023), Lednice, Czechia [Slides]
17 Oct 2023, A. Vucaj, seminar talk Clones over finite sets and Minor Conditions at Seminar: Algebra and Discrete Mathematics, JKU, Linz, Austria [Slides]
17 Oct 2023, A. Barsukov, seminar talk Homogeneous graphs: construction, examples, and applications at Algebra Colloquium, Charles University, Prague, Czechia [Slides]
13 Oct 2023, S. G. Pro, talk Forbidden tournaments and the orientation (completion) problem at KOLKOM 2023 (13-14 Oct 2023), Heidelberg, Germany [Slides]
28 Sep 2023, M. Bodirsky, invited talk Model theoretic challenges in constraint satisfaction at Mathematical Logic section of DMV Meeting (25-28 Sep 2023), Ilmenau, Germany [Slides]
16 Sep 2023, P. Marimon, invited talk Measures in homogeneous 3-hypergraphs at Model Theory Workshop (15-18 Sep 2023), Warsaw, Poland [Slides]
18-22 Sep 2023, POCOCOP attending CWC, Weissensee, Austria
28 Aug - 1 Sep 2023, S. G. Pro attending EUROCOMB'23, Prague, Czechia
10-14 Jul 2023, M. Bodirsky attending ICALP, Paderborn, Germany
4-7 Jul 2023, Ž. Semanišinová, M. Pinsker, A. Vucaj attending Algebra Week, Siena, Italy
1. Ž. Semanišinová: invited talk Valued Constraint Satisfaction Problem and Resilience in Database Theory (5 Jul) [Slides]
2. M. Pinsker: invited talk Loop conditions (6 Jul) [Slides]
3. A. Vucaj: invited talk Clones over finite sets up to minor-equivalence (7 Jul) [Slides]
29 Jun 2023, S. G. Pro, seminar talk Structural Graph Theory, finite bounds, and some CSPs at AGK Seminar, TU Dresden, Germany [Slides]
28 Jun 2023, M. Pinsker, talk Symmetries of graphs and structures that fail to interpret a finite thing at LICS'23 (26-29 Jun 2023), Boston, USA [Slides]
9-11 Jun 2023, S. Meyer, Ž. Semanišinová, M. Pinsker, D. Zhuk, attending AAA 103, Tartu, Estonia
1. S. Meyer: talk Infinitary pp-definabillity over the real numbers with convex relations (9 Jun) [Slides]
2. Ž. Semanišinová: talk Valued Constraint Satisfaction Problem and Resilience in Database Theory (10 Jun) [Slides]
3. M. Pinsker: talk The semigroup of increasing functions on the rationals and its unique Polish topology (9 Jun)
4. D. Zhuk: talk On symmetric term operations in finite Taylor algebras (9 Jun) [Slides]
1 Jun 2023, M. Pinsker, outreach talk POCOCOP at TU Wien Faculty meeting
18 May 2023, M. Pinsker, outreach talk Universal algebra, the Sudoku revolution, and POCOCOP at Austrian Mathematical Society Early Student Awards
30 Mar 2023, M. Pinsker, seminar talk Constraint Satisfaction Problems: algebraic and model-theoretic challenges to distinguish the easy from the hard at Logic Colloquium, University of Vienna, KGRC [Slides]
23-26 Mar 2023, L. Barto, attending Spring school, Department of Algebra (with A. Krokhin)
14 Mar 2023, D. Zhuk, online seminar talk Clones on 3 Elements: A New Hope at PALS [Slides]
14 Mar 2023, M. Bodirsky, seminar talk Valued Constraint Satisfaction Problems and Resilience in Database Theory at Algebra Colloquium, Charles University [Slides]

Visitors

3 Jul 2024, Tamio-Vesa Nakajima from the University of Oxford visiting CU
25 Jun 2024, Venkatesan Guruswami from UC Berkeley visiting CU. Gave a talk How Non-Commutativity Helps Data Centers: Maximally Recoverable Codes from Skew Polynomials
23-30 May 2024, Javier de la Nuez Gonzalez from KIAS (Korean Institute of Advanced Studies) and Zaniar Ghadernezhad from University of Buckingham visiting TU Wien. Zaniar Ghadernezhad gave a talk The small index property of some Hrushovski constructions
9-19 Apr 2024, Matthieu Joseph from Paris-Saclay University visiting TU Dresden. Gave a talk From exchangeability theory to stabilizer rigidity of ergodic actions for Polish non-archimedean groups
8-12 Apr 2024, Jakub Opršal from the University of Birmingham visiting TU Dresden. Gave a talk Local consistency as a reduction between constraint satisfaction problems
2-12 Apr 2024, Andrei Krokhin from Durham University visiting CU. Gave a talk Reasons for NP-hardness of digraph PCSPs
11-15 Mar 2024, Lionel Nguyen van Thé from Aix-Marseille University visiting TU Dresden and CU
6 Mar 2024, Lionel Nguyen van Thé from Aix-Marseille University visiting TU Wien
22-26 Jan 2024, Andy Zucker from the University of Waterloo visiting TU Dresden. Gave a talk Ultracoproducts and weak containment for flows of topological groups at AGK Seminar
9-18 Jan 2024 Armin Weiss from the University of Stuttgart visiting TU Wien. PK guest
Jacek Krzaczkowski from Maria Curie-Skłodowska University visiting TU Wien. PK guest
14 Dec 2023, Martin Otto from TU Darmstadt visiting TU Wien.
13 Dec 2023, André Nies from the University of Auckland visiting TU Wien.
13-14 Dec 2023, Jan Hubička from CU visiting TU Dresden. Gave a talk Recent progress on big Ramsey degrees
3-12 Dec 2023 Colin Jahel from the TU Dresden visiting TU Wien. Collaboration with Paolo Marimon and gave a talk Asymptotic theories and homomorphically avoided structures
3-9 Dec 2023, Édouard Bonnet from ENS Lyon visiting TU Dresden. Gave a talk Introduction to twin-width
16-18 Oct 2023, André Nies from the University of Auckland visiting TU Dresden. Gave a talk Oligomorphic groups, their interactions and generalisations
14-20 Aug 2023, Bertalan Bodor from the University of Szeged visiting TU Dresden. Gave a talk Pseudoloop lemmas
7-12 Aug 2023, Marcin Kozik, Michał Wrona from the Jagiellonian University (Kraków) visiting TU Wien
19-22 Jun 2023, Jakub Opršal from ISTA visiting CU
4-8 Jun 2023, Bertalan Bodor from the University of Szeged visiting CU
28 May - 2 Jun 2023, Colin Jahel from Carnegie Mellon University visiting TU Dresden
10-20 Apr 2023, Silvia Butti from the University of Oxford visiting CU
3-5 Apr 2023, Andrei Krokhin from Durham University visiting TU Wien
20 Mar - 1 Apr 2023, Andrei Krokhin from Durham University visiting CU

Visits

26-27 Jun 2024: J. Brunar and C. Spiess visiting Antoine Mottet at TU Hamburg
17-19 Jun 2024: J. Brunar visiting Marcin Kozik and Tomáš Nagy at Jagiellonian University (Kraków)
6-7 Jun 2024: M. Bodirsky visiting Czech Science Foundation, Project Proposal Evaluation in Prague
2-8 Jun 2024: D. Zhuk visiting Stanislav Živný at University of Oxford
8-14 May 2024: D. Zhuk visiting Petar Markovic at University of Novi Sad
18-22 Mar 2024: P. Kawałek visiting Jacek Krzaczkowski at UMCS (Lublin)
1-17 Mar 2024: P. Marimon visiting Javier de la Nuez Gonzalez at KIAS (Korean Institute of advanced Studies)
28 Jan - 2 Feb 2024: L. Barto visiting Marcin Kozik at Jagiellonian University (Kraków)
10-16 Dec 2023: L. Barto visiting Andrei Krokhin at Durham University
5-7 Dec 2023: P. Kawałek visiting Erhard Aichinger at JKU, Linz
14-16 Nov 2023: P. Kawałek visiting Jacek Krzaczkowski
8-14 Nov 2023: P. Kawałek visiting Paweł Idziak at Jagiellonian University (Kraków)
29 Oct - 5 Nov 2023: P. Kawałek visiting Armin Weiss at University of Stuttgart
10-14 Oct 2023: P. Kawałek visiting Paweł Idziak and Jacek Krzaczkowski at Jagiellonian University (Kraków)

Team visits

28 Jun 2024: P. Grzywaczyk visiting TU Wien
24-29 Jun 2024: S. Guzmán Pro visiting CU
10-14 Jun 2024: M. Hadek visiting TU Wien
6 Jun 2024: M. Bodirsky visiting CU
23 May 2024: M. Konečný visiting CU
21-25 May 2024: P. Grzywaczyk visiting TU Wien
21 May 2024: M. Konečný visiting CU
16 May 2024: M. Konečný visiting CU
9 May 2024: M. Konečný visiting CU
2 May 2024: M. Konečný visiting CU
19-26 Apr 2024: S. Guzmán Pro visiting TU Wien
23-28 Mar 2024 3rd POCOCOP meeting: A. Barsukov, L. Barto, D. Zhuk, M. Hadek, J. Brunar, R. Feller, P. Kawałek, P. Marimon, M. Pinsker, C. Spiess, A. Vucaj visiting TU Dresden
18-22 Mar 2024: J. Brunar visiting CU
21 Mar 2024: M. Konečný visiting CU
14 Mar 2024: A. Moorhead and M. Konečný visiting CU
7 Mar 2024: M. Konečný visiting CU
29 Feb 2024: A. Moorhead and M. Konečný visiting CU
4-10 Feb 2024: P. Kawałek visiting TU Dresden
29 Nov - 2 Dec 2023 2nd POCOCOP meeting: M. Bodirsky, S. Meyer, A. Moorhead, P. Grzywaczyk, S. Guzmán Pro, Ž. Semanišinová, M. Konečný, M. Pinsker, P. Marimon, A. Vucaj, P. Kawalek, J. Brunar, R. Feller visiting CU
28-30 May 2023 1st POCOCOP meeting: M. Pinsker, A. Vucaj, M. Bodirsky, Z. Semanišinová, S. Meyer visiting CU
4-8 May 2023: Z. Semanišinová visiting TU Wien
13-15 Mar 2023: M. Bodirsky, S. Meyer visiting CU

Other activities

21 Apr 2024: teams POCOCOP_PRAGUE (S. Guzmán Pro / A. Barsukov / A. Vucaj / M. Kompatscher) and Polynomial-time Runners (J. Brunar / R. Feller / J. Rydval / M. Pinsker) participated in Vienna City Marathon

Abstract

The class P of polynomial-time computable computational problems is the most important and robust complexity class for the study of efficient computation. Answering what problems belong to P will lead to groundbreaking applications in science and modern society where computation is omnipresent. Moreover, P is a relatively recent mathematical object and radically different from classical notions studied for centuries; thus, capturing it promises the discovery of new fundamental theorems in mathematics.

Our current understanding of P is limited; for instance, the P=NP millennium problem is wide open. There neither exists a uniform reduction technique, nor a single algorithmic scheme capturing the power of P, nor a description of P in purely logical terms. We intend to provide these in a context which is so rich and vast that it requires the unification of some of the most important techniques, and will enhance our general understanding of P.

Within the microcosm of finite-domain constraint satisfaction problems (CSPs), the recent resolution of the Feder-Vardi conjecture by Bulatov and by Zhuk provides a satisfactory picture of P. Our goal is a vast and uniform generalisation of this result in three directions: towards approximation via Promise CSPs, towards optimisation via Valued CSPs, and towards infinite domains via omega-categorical CSPs and CSPs over numeric domains. In particular, our setting includes the linear programming problem as a numeric Valued CSP, the approximate graph coloring problem as a Promise CSP, and many problems from qualitative reasoning as infinite-domain CSPs. Our methods range from universal algebra, model theory, Ramsey theory, to complexity theory. Building on cross-connections between these extensions, we will provide a uniform description of P within this diverse and applicable universe, thus making a revolutionary leap in the resolution of the general problem.

Funding statement: Funded by the European Union (ERC, POCOCOP, 101071674). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.