On the complexity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml11" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi mathvariant="double-struck">H</mml:mi></mml:math>-coloring for special oriented trees

Bulín, Jakub

doi:10.1016/j.ejc.2017.10.001

Cited by 5 publications

(5 citation statements)

References 35 publications

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“…We will introduce several such concrete constructions in the next section. Similar constructions have appeared in [4,8,22,23]. This optimization is particularly useful when testing the condition TS(n) for all n; see Section 7.1.4.…”

Section: Level-wise Satisfiabilitymentioning

confidence: 94%

“…If T has the ability to count, then T is NP-hard. This conjecture (which is a rephrasing of Conjecture 2 in [22]) would answer an open question posed by Hell, Nešetřil, and Zhu [44] (Open Problem 1 at the end of the article): they asked whether there exists a tractable tree which does not have bounded treewidth duality.…”

Section: Conjecturementioning

confidence: 96%

“…An NP-hard triad with 39 vertices was found by Barto, Kozik, Maróti, and Niven [8,9] using an in-depth analysis of the polymorphisms of triads; they conjectured that their triad is the smallest NP-hard tree (assuming P = NP). This approach lead to a study of certain classes of trees [4,22]. Fischer [36] used a computer search and found an NP-hard tree with just 30 vertices (refuting the conjecture of Barto et al mentioned above).…”

Section: Conjecture 1 (Larose and Tessonmentioning

confidence: 99%

“…In [22] Bulín conjectured that 'easy trees cannot count' (i.e., only NP-hard trees possess the ability to count) and showed that this conjecture held for a large yet structurally limited subclass of trees.…”

Section: Descriptive Complexitymentioning

confidence: 99%

“…In fact, up to 20 vertices all trees that have Kearnes-Marković-McKenzie polymorphisms either have a majority polymorphism or totally symmetric polymorphisms of all arities. The picture is however more complex for larger trees: there exists a tree which can be solved by Datalog but does not have a nearunanimity polymorphism (of any arity) and does not have totally symmetric polymorphisms of all arities, see [4, Proposition 5.5] for an example and [22] for its solvability in Datalog.…”

Section: The Smallest Tree Not Solved By Datalogmentioning

confidence: 99%

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The smallest hard trees

et al. 2023

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We find an orientation of a tree with 20 vertices such that the corresponding fixed-template constraint satisfaction problem (CSP) is NP-complete, and prove that for every orientation of a tree with fewer vertices the corresponding CSP can be solved in polynomial time. We also compute the smallest tree that is NL-hard (assuming L≠NL), the smallest tree that cannot be solved by arc consistency, and the smallest tree that cannot be solved by Datalog. Our experimental results also support a conjecture of Bulín concerning a question of Hell, Nešetřil and Zhu, namely that ‘easy trees lack the ability to count’. Most proofs are computer-based and make use of the most recent universal-algebraic theory about the complexity of finite-domain CSPs. However, further ideas are required because of the huge number of orientations of trees. In particular, we use the well-known fact that it suffices to study orientations of trees that are cores and show how to efficiently decide whether a given orientation of a tree is a core using the arc-consistency procedure. Moreover, we present a method to generate orientations of trees that are cores that works well in practice. In this way we found interesting examples for the open research problem to classify finite-domain CSPs in NL.

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Section: Level-wise Satisfiabilitymentioning

confidence: 94%

Section: Conjecturementioning

confidence: 96%