jakub's notes on jakub opršal

Is all NP-hardness in the promise CSP explained by Datalog reductions?

Sat, 19 Sep 2026 14:00:00 +0000

There is an old tradition of putting prices on problems in mathematics, and in universal algebra, we used to offer alcohol. I remember one such price as a student, a few people contributed, and the final amount was four beers and a shot of vodka (or something like that). Building on this tradition, I decided to offer my own price of a bottle of whisky to a negative counterexample to one of my conjectures. This price was claimed sooner than expected, but I am happy with the attention it created, so I decided to announce a second Whisky problem at Dagstuhl in 2025.

Homomorphism complexes on three levels of abstraction

Fri, 03 Jan 2025 14:00:00 +0000

Throughout this note, I assume that $G$ and $H$ are simple undirected graphs, and loops (i.e., edges of the form $uu$) are allowed. I will also not pay attention to the distinction between simplicial complexes and their geometric realisations, using topological notions freely for simplicial complexes.

My goal with this note is to give a brief overview of several constructions of topological spaces from graphs. I will pick a few of these constructions on various levels of abstraction — from purely combinatorial to deeply categorical. Each one of these will suit different audience, and you may pick your favourite one.

On the hardness of 3-colouring H-colourable graphs

Sat, 15 Jun 2024 11:36:25 +0100

In 2019, me and Andrei Krokhin proved that colouring graphs that are promised to map homomorphically to a fixed (but arbitrary long) odd cycle with 3 colours is NP-hard [1]. The original proof was a combinatorial argument based on some ideas of homotopy and a degree of a continuous endomap of a cycle. For that reason, we have included an overview of a topological intuition behind our combinatorial proof. Time has shown that this was a good decision since that intuition was immediately formalised by Marcin Wrochna and Standa Živný [2].

Yoneda Lemma in promise CSP

Mon, 03 Jul 2023 06:00:00 +0200

As I am working on write-ups on our recent work on the complexity of CSPs [1, 2], it become more and more obvious that the categorical language might be useful in formulating our results and providing formal proofs thereof. This is why me and Tomáš Jakl started working on categorical understanding of the theory of promise CSPs—this will take us a while as we are slowly learning that everything has been proved in 70’s.

The promised open problems

Sun, 03 Jan 2021 17:21:20 +0000

This note is a list of open problems and ideas for research directions in and around promise constraint satisfaction problems. It is certainly not an exhaustive list of open problems that I personally find interesting, or I believe they will open interesting further directions. I hope it will be useful nevertheless. Let me first start with the definition of a promise constraint satisfaction problem with a given template. Definition A promise template is a pair of relational structures A, B such that A maps homomorphically to B (the two structures have the same signature).

jakub's notes on jakub opršal

Is all NP-hardness in the promise CSP explained by Datalog reductions?

Homomorphism complexes on three levels of abstraction

On the hardness of 3-colouring *H*-colourable graphs

Yoneda Lemma in promise CSP

The promised open problems

On the hardness of 3-colouring H-colourable graphs