7 research outputs found
The complexity of general-valued CSPs seen from the other side
The constraint satisfaction problem (CSP) is concerned with homomorphisms
between two structures. For CSPs with restricted left-hand side structures, the
results of Dalmau, Kolaitis, and Vardi [CP'02], Grohe [FOCS'03/JACM'07], and
Atserias, Bulatov, and Dalmau [ICALP'07] establish the precise borderline of
polynomial-time solvability (subject to complexity-theoretic assumptions) and
of solvability by bounded-consistency algorithms (unconditionally) as bounded
treewidth modulo homomorphic equivalence.
The general-valued constraint satisfaction problem (VCSP) is a generalisation
of the CSP concerned with homomorphisms between two valued structures. For
VCSPs with restricted left-hand side valued structures, we establish the
precise borderline of polynomial-time solvability (subject to
complexity-theoretic assumptions) and of solvability by the -th level of the
Sherali-Adams LP hierarchy (unconditionally). We also obtain results on related
problems concerned with finding a solution and recognising the tractable cases;
the latter has an application in database theory.Comment: v2: Full version of a FOCS'18 paper; improved presentation and small
correction
Point-width and Max-CSPs
International audienceThe complexity of (unbounded-arity) Max-CSPs under structural restrictions is poorly understood. The two most general hypergraph properties known to ensure tractability of Max-CSPs, β-acyclicity and bounded (incidence) MIM-width, are incomparable and lead to very different algorithms. We introduce the framework of point decompositions for hypergraphs and use it to derive a new sufficient condition for the tractability of (structurally restricted) Max-CSPs, which generalises both bounded MIM-width and β-acyclicity. On the way, we give a new characterisation of bounded MIM-width and discuss other hypergraph properties which are relevant to the complexity of Max-CSPs, such as β-hypertreewidth
PTAS for Sparse General-Valued CSPs
We study polynomial-time approximation schemes (PTASes) for constraint
satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex
Cover on sparse graph classes. Baker's approach gives a PTAS on planar graphs,
excluded-minor classes, and beyond. For Max-CSPs, and even more generally,
maximisation finite-valued CSPs (where constraints are arbitrary non-negative
functions), Romero, Wrochna, and \v{Z}ivn\'y [SODA'21] showed that the
Sherali-Adams LP relaxation gives a simple PTAS for all
fractionally-treewidth-fragile classes, which is the most general "sparsity"
condition for which a PTAS is known. We extend these results to general-valued
CSPs, which include "crisp" (or "strict") constraints that have to be satisfied
by every feasible assignment. The only condition on the crisp constraints is
that their domain contains an element which is at least as feasible as all the
others (but possibly less valuable). For minimisation general-valued CSPs with
crisp constraints, we present a PTAS for all Baker graph classes -- a
definition by Dvo\v{r}\'ak [SODA'20] which encompasses all classes where
Baker's technique is known to work, except possibly for
fractionally-treewidth-fragile classes. While this is standard for problems
satisfying a certain monotonicity condition on crisp constraints, we show this
can be relaxed to diagonalisability -- a property of relational structures
connected to logics, statistical physics, and random CSPs
Treewidth-Pliability and PTAS for Max-CSPs
We identify a sufficient condition, treewidth-pliability, that gives a
polynomial-time approximation scheme (PTAS) for a large class of Max-2-CSPs
parametrised by the class of allowed constraint graphs (with arbitrary
constraints on an unbounded alphabet). Our result applies more generally to the
maximum homomorphism problem between two rational-valued structures.
The condition unifies the two main approaches for designing PTASes. One is
Baker's layering technique, which applies to sparse graphs such as planar or
excluded-minor graphs. The other is based on Szemer\'{e}di's regularity lemma
and applies to dense graphs. We extend the applicability of both techniques to
new classes of Max-CSPs.
Treewidth-pliability turns out to be a robust notion that can be defined in
several equivalent ways, including characterisations via size, treedepth, or
the Hadwiger number. We show connections to the notions of
fractional-treewidth-fragility from structural graph theory, hyperfiniteness
from the area of property testing, and regularity partitions from the theory of
dense graph limits. These may be of independent interest. In particular we show
that a monotone class of graphs is hyperfinite if and only if it is
fractionally-treewidth-fragile and has bounded degree
The complexity of general-valued CSPs seen from the other side
The constraint satisfaction problem (CSP) is concerned with homomorphisms between two structures. For CSPs with restricted left-hand side structures, the results of Dalmau, Kolaitis, and Vardi [CP'02], Grohe [FOCS'03/JACM'07], and Atserias, Bulatov, and Dalmau [ICALP'07] establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by bounded-consistency algorithms (unconditionally) as bounded treewidth modulo homomorphic equivalence. The general-valued constraint satisfaction problem (VCSP) is a generalisation of the CSP concerned with homomorphisms between two valued structures. For VCSPs with restricted left-hand side valued structures, we establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by the k-th level of the Sherali-Adams LP hierarchy (unconditionally). We also obtain results on related problems concerned with finding a solution and recognising the tractable cases; the latter has an application in database theory
The complexity of general-valued CSPs seen from the other side
The constraint satisfaction problem (CSP) is concerned with homomorphisms between two structures. For CSPs with restricted left-hand side structures, the results of Dalmau, Kolaitis, and Vardi [CP'02], Grohe [FOCS'03/JACM'07], and Atserias, Bulatov, and Dalmau [ICALP'07] establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by bounded-consistency algorithms (unconditionally) as bounded treewidth modulo homomorphic equivalence. The general-valued constraint satisfaction problem (VCSP) is a generalisation of the CSP concerned with homomorphisms between two valued structures. For VCSPs with restricted left-hand side valued structures, we establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by the k-th level of the Sherali-Adams LP hierarchy (unconditionally). We also obtain results on related problems concerned with finding a solution and recognising the tractable cases; the latter has an application in database theory