9 research outputs found
QCSP monsters and the demise of the chen conjecture.
We give a surprising classification for the computational complexity
of the Quantified Constraint Satisfaction Problem over a constraint
language Γ, QCSP(Γ), where Γ is a finite language over 3 elements
which contains all constants. In particular, such problems are either in P, NP-complete, co-NP-complete or PSpace-complete. Our
classification refutes the hitherto widely-believed Chen Conjecture.
Additionally, we show that already on a 4-element domain there
exists a constraint language Γ such that QCSP(Γ) is DP-complete
(from Boolean Hierarchy), and on a 10-element domain there exists
a constraint language giving the complexity class Θ
????
2
.
Meanwhile, we prove the Chen Conjecture for finite conservative languages Γ. If the polymorphism clone of such Γ has the
polynomially generated powers (PGP) property then QCSP(Γ) is in
NP. Otherwise, the polymorphism clone of Γ has the exponentially
generated powers (EGP) property and QCSP(Γ) is PSpace-complete
QCSP on reflexive tournaments
We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well known that reflexive tournaments can be split into a sequence of strongly connected components H1,…,Hn so that there exists an edge from every vertex of Hi to every vertex of Hj if and only if
Finding Small Satisfying Assignments Faster Than Brute Force: {A} Fine-grained Perspective into {B}oolean Constraint Satisfaction
To study the question under which circumstances small solutions can be found faster than by exhaustive search (and by how much), we study the fine-grained complexity of Boolean constraint satisfaction with size constraint exactly . More precisely, we aim to determine, for any finite constraint family, the optimal running time required to find satisfying assignments that set precisely of the variables to . Under central hardness assumptions on detecting cliques in graphs and 3-uniform hypergraphs, we give an almost tight characterization of into four regimes: (1) Brute force is essentially best-possible, i.e., , (2) the best algorithms are as fast as current -clique algorithms, i.e., , (3) the exponent has sublinear dependence on with , or (4) the problem is fixed-parameter tractable, i.e., . This yields a more fine-grained perspective than a previous FPT/W[1]-hardness dichotomy (Marx, Computational Complexity 2005). Our most interesting technical contribution is a -time algorithm for SubsetSum with precedence constraints parameterized by the target -- particularly the approach, based on generalizing a bound on the Frobenius coin problem to a setting with precedence constraints, might be of independent interest
Unifying the Three Algebraic Approaches to the CSP via Minimal Taylor Algebras
This paper focuses on the algebraic theory underlying the study of the
complexity and the algorithms for the Constraint Satisfaction Problem (CSP). We
unify, simplify, and extend parts of the three approaches that have been
developed to study the CSP over finite templates - absorption theory that was
used to characterize CSPs solvable by local consistency methods (JACM'14), and
Bulatov's and Zhuk's theories that were used for two independent proofs of the
CSP Dichotomy Theorem (FOCS'17, JACM'20).
As the first contribution we present an elementary theorem about primitive
positive definability and use it to obtain the starting points of Bulatov's and
Zhuk's proofs as corollaries. As the second contribution we propose and
initiate a systematic study of minimal Taylor algebras. This class of algebras
is broad enough so that it suffices to verify the CSP Dichotomy Theorem on this
class only, but still is unusually well behaved. In particular, many concepts
from the three approaches coincide in the class, which is in striking contrast
with the general setting.
We believe that the theory initiated in this paper will eventually result in
a simple and more natural proof of the Dichotomy Theorem that employs a simpler
and more efficient algorithm, and will help in attacking complexity questions
in other CSP-related problems
QCSP monsters and the demise of the chen conjecture
We give a surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language Γ, QCSP(Γ), where Γ is a finite language over 3 elements which contains all constants. In particular, such problems are either in P, NP-complete, co-NP-complete or PSpace-complete. Our classification refutes the hitherto widely-believed Chen Conjecture. Additionally, we show that already on a 4-element domain there exists a constraint language Γ such that QCSP(Γ) is DP-complete (from Boolean Hierarchy), and on a 10-element domain there exists a constraint language giving the complexity class Θ ???? 2 . Meanwhile, we prove the Chen Conjecture for finite conservative languages Γ. If the polymorphism clone of such Γ has the polynomially generated powers (PGP) property then QCSP(Γ) is in NP. Otherwise, the polymorphism clone of Γ has the exponentially generated powers (EGP) property and QCSP(Γ) is PSpace-complete