1,713 research outputs found
Algorithms and complexity for approximately counting hypergraph colourings and related problems
The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Δ, this problem falls into the regime of Lovász local lemma when Δ ≲ qᵏ. In prior, however, fast approximate counting algorithms exist when Δ ≲ qᵏ/³, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows.
• When q, k ≥ 4 are evens and Δ ≥ 5·qᵏ/², approximating the number of hypergraph colourings is NP-hard.
• When the input hypergraph is linear and Δ ≲ qᵏ/², a fast approximate counting algorithm does exist
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
The Diophantine problem in Chevalley groups
In this paper we study the Diophantine problem in Chevalley groups , where is an indecomposable root system of rank , is
an arbitrary commutative ring with . We establish a variant of double
centralizer theorem for elementary unipotents . This theorem is
valid for arbitrary commutative rings with . The result is principle to show
that any one-parametric subgroup , , is Diophantine
in . Then we prove that the Diophantine problem in is
polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine
problem in . This fact gives rise to a number of model-theoretic corollaries
for specific types of rings.Comment: 44 page
Bounded Relativization
Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization. For a complexity class ?, we say that a statement is ?-relativizing if the statement holds relative to every oracle ? ? ?. It is easy to see that every result that relativizes also ?-relativizes for every complexity class ?. On the other hand, we observe that many non-relativizing results, such as IP = PSPACE, are in fact PSPACE-relativizing.
First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ? > 0, BPE^{MCSP}/2^{?n} ? SIZE[2?/n].
We prove this by PSPACE-relativizing the recent pseudodeterministic pseudorandom generator by Lu, Oliveira, and Santhanam (STOC 2021).
Next, we study the limitations of PSPACE-relativizing proof techniques, and show that a seemingly minor improvement over the known results using PSPACE-relativizing techniques would imply a breakthrough separation NP ? L. For example:
- Impagliazzo and Wigderson (JCSS 2001) proved that if EXP ? BPP, then BPP admits infinitely-often subexponential-time heuristic derandomization. We show that their result is PSPACE-relativizing, and that improving it to worst-case derandomization using PSPACE-relativizing techniques implies NP ? L.
- Oliveira and Santhanam (STOC 2017) recently proved that every dense subset in P admits an infinitely-often subexponential-time pseudodeterministic construction, which we observe is PSPACE-relativizing. Improving this to almost-everywhere (pseudodeterministic) or (infinitely-often) deterministic constructions by PSPACE-relativizing techniques implies NP ? L.
- Santhanam (SICOMP 2009) proved that pr-MA does not have fixed polynomial-size circuits. This lower bound can be shown PSPACE-relativizing, and we show that improving it to an almost-everywhere lower bound using PSPACE-relativizing techniques implies NP ? L.
In fact, we show that if we can use PSPACE-relativizing techniques to obtain the above-mentioned improvements, then PSPACE ? EXPH. We obtain our barrier results by constructing suitable oracles computable in EXPH relative to which these improvements are impossible
Complexity of Chess Domination Problems
We study different domination problems of attacking and non-attacking rooks
and queens on polyominoes and polycubes of all dimensions. Our main result
proves that maximal domination is NP-complete for non-attacking queens and for
non-attacking rooks on polycubes of dimension three and higher. We also analyse
these problems for polyominoes and convex polyominoes, conjecture the
complexity classes and provide a computer tool for investigation. We have also
computed new values for classical queen domination problems on chessboards
(square polyominoes). For our computations, we have translated the problem into
an integer linear programming instance. Finally, using this computational
implementation and the game engine Godot, we have developed a video game of
minimal domination of queens and rooks on randomly generated polyominoes.Comment: 19 pages, 20 figures, 4 tables. Theorem 1 now for d>2, added results
on approximation, fixed typos, reorganised some proof
A computational approach for finding 6-List-critical graphs on the Torus
La coloració de grafs dibuixats a superfícies és un àrea antiga i molt estudiada de la teoria de grafs. Thomassen va demostrar que hi ha un nombre finit de grafs 6-crítics a qualsevol superfície fixa i va proporcionar el conjunt explícit dels grafs 6-crítics al torus. Després, Postle va demostrar que hi ha un nombre finit de grafs 6-llista-crítics a qualsevol superfície fixa. Amb l'objectiu de trobar el conjunt de grafs 6-llista-crítics al torus, desenvolupem i implementem tècniques algorítmiques per la cerca per ordinador de grafs crítics en diferents situacions de coloració per llistes.La coloración de grafos dibujados en superficies es un área antigua y muy estudiada de la teoría de grafos. Thomassen demostró que hay un número finito de grafos 6-críticos en cualquier superficie fija y proporcionó el conjunto explícito de los grafos 6-críticos en el toro. Después, Postle demostró que hay un número finito de grafos 6-lista-críticos en cualquier superficie fija. Con el objetivo de encontrar el conjunto de grafos 6-lista-críticos en el toro, desarrollamos e implementamos técnicas algorítmicas para la búsqueda por ordenador de grafos críticos en diferentes situaciones de coloración por listas.Coloring graphs embedded on surfaces is an old and well-studied area of graph theory. Thomassen proved that there are finitely many 6-critical graphs on any fixed surface and provided the explicit set of 6-critical graphs on the torus. Later, Postle proved that there are finitely many 6-list-critical graphs on any fixed surface. With the goal of finding the set of 6-list-critical graphs on the torus, we develop and implement algorithmic techniques for computer search of critical graphs in different list-coloring settings.Outgoin
Clones over Finite Sets and Minor Conditions
Achieving a classification of all clones of operations over a finite set is one of the goals at the heart of universal algebra. In 1921 Post provided a full description of the lattice of all clones over a two-element set. However, over the following years, it has been shown that a similar classification seems arduously reachable even if we only focus on clones over three-element sets: in 1959 Janov and Mučnik proved that there exists a continuum of clones over a k-element set for every k > 2. Subsequent research in universal algebra therefore focused on understanding particular aspects of clone lattices over finite domains. Remarkable results in this direction are the description of maximal and minimal clones. One might still hope to classify all operation clones on finite domains up to some equivalence relation so that equivalent clones share many of the properties that are of interest in universal algebra.
In a recent turn of events, a weakening of the notion of clone homomorphism was introduced: a minor-preserving map from a clone C to D is a map which preserves arities and composition with projections. The minor-equivalence relation on clones over finite sets gained importance both in universal algebra and in computer science: minor-equivalent clones satisfy the same set identities of the form f(x_1,...,x_n) = g(y_1,...,y_m), also known as minor-identities. Moreover, it was proved that the complexity of the CSP of a finite structure A only depends on the set of minor-identities satisfied by the polymorphism clone of A. Throughout this dissertation we focus on the poset that arises by considering clones over a three-element set with the following order: we write C ≤_{m} D if there exist a minor-preserving map from C to D. It has been proved that ≤_{m} is a preorder; we call the poset arising from ≤_{m} the pp-constructability poset.
We initiate a systematic study of the pp-constructability poset. To this end, we distinguish two cases that are qualitatively distinct: when considering clones over a finite set A, one can either set a boundary on the cardinality of A, or not. We denote by P_n the pp-constructability poset restricted to clones over a set A such that |A|=n and by P_{fin} we denote the whole pp-constructability poset, i.e., we only require A to be finite. First, we prove that P_{fin} is a semilattice and that it has no atoms. Moreover, we provide a complete description of P_2 and describe a significant part of P_3: we prove that P_3 has exactly three submaximal elements and present a full description of the ideal generated by one of these submaximal elements. As a byproduct, we prove that there are only countably many clones of self-dual operations over {0,1,2} up to minor-equivalence
Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius
The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete, leading to many complexity results for both problems on special graph classes. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we introduce the Maximum Matching Cut problem. This problem is to determine a largest matching cut in a graph. We generalize and unify known polynomial-time algorithms for Matching Cut and Perfect Matching Cut restricted to graphs of diameter at most 2 and to (P?+sP?)-free graphs. We also show that the complexity of Maximum Matching Cut differs from the complexities of Matching Cut and Perfect Matching Cut by proving NP-hardness of Maximum Matching Cut for 2P?-free quadrangulated graphs of diameter 3 and radius 2 and for subcubic line graphs of triangle-free graphs. In this way, we obtain full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs
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