434 research outputs found
Hybrid VCSPs with crisp and conservative valued templates
A constraint satisfaction problem (CSP) is a problem of computing a
homomorphism between two relational
structures. Analyzing its complexity has been a very fruitful research
direction, especially for fixed template CSPs, denoted , in
which the right side structure is fixed and the left side
structure is unconstrained.
Recently, the hybrid setting, written ,
where both sides are restricted simultaneously, attracted some attention. It
assumes that is taken from a class of relational structures
that additionally is closed under inverse homomorphisms. The last
property allows to exploit algebraic tools that have been developed for fixed
template CSPs. The key concept that connects hybrid CSPs with fixed-template
CSPs is the so called "lifted language". Namely, this is a constraint language
that can be constructed from an input . The
tractability of that language for any input is a
necessary condition for the tractability of the hybrid problem.
In the first part we investigate templates for which the
latter condition is not only necessary, but also is sufficient. We call such
templates widely tractable. For this purpose, we construct from
a new finite relational structure and define
as a class of structures homomorphic to . We
prove that wide tractability is equivalent to the tractability of
. Our proof is based on the key observation
that is homomorphic to if and only if the core of
is preserved by a Siggers polymorphism. Analogous
result is shown for valued conservative CSPs.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1504.0706
Minimum Cost Homomorphisms to Reflexive Digraphs
For digraphs and , a homomorphism of to is a mapping $f:\
V(G)\dom V(H)uv\in A(G)f(u)f(v)\in A(H)u \in V(G)c_i(u), i \in V(H)f\sum_{u\in V(G)}c_{f(u)}(u)H, the {\em minimum cost homomorphism problem} for HHGc_i(u)u\in V(G)i\in V(H)kGHk. We focus on the
minimum cost homomorphism problem for {\em reflexive} digraphs HHHH has a {\em Min-Max ordering}, i.e.,
if its vertices can be linearly ordered by <i<j, s<rir, js
\in A(H)is \in A(H)jr \in A(H)H$ which does not admit a Min-Max ordering, the minimum cost
homomorphism problem is NP-complete. Thus we obtain a full dichotomy
classification of the complexity of minimum cost homomorphism problems for
reflexive digraphs
Approximation for Maximum Surjective Constraint Satisfaction Problems
Maximum surjective constraint satisfaction problems (Max-Sur-CSPs) are
computational problems where we are given a set of variables denoting values
from a finite domain B and a set of constraints on the variables. A solution to
such a problem is a surjective mapping from the set of variables to B such that
the number of satisfied constraints is maximized. We study the approximation
performance that can be acccchieved by algorithms for these problems, mainly by
investigating their relation with Max-CSPs (which are the corresponding
problems without the surjectivity requirement). Our work gives a complexity
dichotomy for Max-Sur-CSP(B) between PTAS and APX-complete, under the
assumption that there is a complexity dichotomy for Max-CSP(B) between PO and
APX-complete, which has already been proved on the Boolean domain and 3-element
domains
Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs
For digraphs and , a homomorphism of to is a mapping $f:\
V(G)\dom V(H)uv\in A(G)f(u)f(v)\in A(H)u \in V(G)c_i(u), i \in V(H)f\sum_{u\in V(G)}c_{f(u)}(u)HHHGc_i(u)u\in V(G)i\in V(H)GH$ and, if one exists, to find one of minimum cost.
Minimum cost homomorphism problems encompass (or are related to) many well
studied optimization problems such as the minimum cost chromatic partition and
repair analysis problems. We focus on the minimum cost homomorphism problem for
locally semicomplete digraphs and quasi-transitive digraphs which are two
well-known generalizations of tournaments. Using graph-theoretic
characterization results for the two digraph classes, we obtain a full
dichotomy classification of the complexity of minimum cost homomorphism
problems for both classes
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