13 research outputs found
Constraint satisfaction problems for reducts of homogeneous graphs
For n >= 3, let (Hn, E) denote the n-th Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on n vertices. We show that for all structures Gamma with domain Hn whose relations are first-order definable in (Hn, E) the constraint satisfaction problem for Gamma is either in P or is NP-complete. We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation. Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete
Constraint Satisfaction Problems for Reducts of Homogeneous Graphs
For n >= 3, let (Hn, E) denote the n-th Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on n vertices. We show that for all structures Gamma with domain Hn whose relations are first-order definable in (Hn, E) the constraint satisfaction problem for Gamma is either in P or is NP-complete. We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation. Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete
Constraint satisfaction problems for reducts of homogeneous graphs
For n >= 3, let (H-n, E) denote the nth Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on n vertices. We show that for all structures Gamma with domain H-n whose relations are first-order definable in (H-n, E) the constraint satisfaction problem for F either is in P or is NP-complete. We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation. Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete
The Complexity of Combinations of Qualitative Constraint Satisfaction Problems
The CSP of a first-order theory is the problem of deciding for a given
finite set of atomic formulas whether is satisfiable. Let
and be two theories with countably infinite models and disjoint
signatures. Nelson and Oppen presented conditions that imply decidability (or
polynomial-time decidability) of under the
assumption that and are decidable (or
polynomial-time decidable). We show that for a large class of
-categorical theories the Nelson-Oppen conditions are not
only sufficient, but also necessary for polynomial-time tractability of
(unless P=NP)
On The Relational Width of First-Order Expansions of Finitely Bounded Homogeneous Binary Cores with Bounded Strict Width
The relational width of a finite structure, if bounded, is always (1,1) or
(2,3). In this paper we study the relational width of first-order expansions of
finitely bounded homogeneous binary cores where binary cores are structures
with equality and some anti-reflexive binary relations such that for any two
different elements a, b in the domain there is exactly one binary relation R
with (a, b) in R.
Our main result is that first-order expansions of liberal finitely bounded
homogeneous binary cores with bounded strict width have relational width (2,
MaxBound) where MaxBound is the size of the largest forbidden substructure, but
is not less than 3, and liberal stands for structures that do not forbid
certain finite structures of small size. This result is built on a new approach
and concerns a broad class of structures including reducts of homogeneous
digraphs for which the CSP complexity classification has not yet been obtained.Comment: A long version of an extended abstract that appeared in LICS 202
The Complexity of Combinations of Qualitative Constraint Satisfaction Problems
The CSP of a first-order theory is the problem of deciding for a given
finite set of atomic formulas whether is satisfiable. Let
and be two theories with countably infinite models and disjoint
signatures. Nelson and Oppen presented conditions that imply decidability (or
polynomial-time decidability) of under the
assumption that and are decidable (or
polynomial-time decidable). We show that for a large class of
-categorical theories the Nelson-Oppen conditions are not
only sufficient, but also necessary for polynomial-time tractability of
(unless P=NP).Comment: Version 2: stronger main result with better presentation of the
proof; multiple improvements in other proofs; new section structure; new
example
The combined basic LP and affine IP relaxation for promise VCSPs on infinite domains
Convex relaxations have been instrumental in solvability of constraint
satisfaction problems (CSPs), as well as in the three different generalisations
of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In
this work, we extend an existing tractability result to the three
generalisations of CSPs combined: We give a sufficient condition for the
combined basic linear programming and affine integer programming relaxation for
exact solvability of promise valued CSPs over infinite-domains. This extends a
result of Brakensiek and Guruswami [SODA'20] for promise (non-valued) CSPs (on
finite domains).Comment: Full version of an MFCS'20 pape
Canonical Polymorphisms of Ramsey Structures and the Unique Interpolation Property
Constraint satisfaction problems for first-order reducts of finitely bounded
homogeneous structures form a large class of computational problems that might
exhibit a complexity dichotomy, P versus NP-complete. A powerful method to
obtain polynomial-time tractability results for such CSPs is a certain
reduction to polynomial-time tractable finite-domain CSPs defined over k-types,
for a sufficiently large k. We give sufficient conditions when this method can
be applied and illustrate how to use the general results to prove a new
complexity dichotomy for first-order expansions of the basic relations of the
spatial reasoning formalism RCC5