701 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
Tropical Dominating Sets in Vertex-Coloured Graphs
Given a vertex-coloured graph, a dominating set is said to be tropical if
every colour of the graph appears at least once in the set. Here, we study
minimum tropical dominating sets from structural and algorithmic points of
view. First, we prove that the tropical dominating set problem is NP-complete
even when restricted to a simple path. Then, we establish upper bounds related
to various parameters of the graph such as minimum degree and number of edges.
We also give upper bounds for random graphs. Last, we give approximability and
inapproximability results for general and restricted classes of graphs, and
establish a FPT algorithm for interval graphs.Comment: 19 pages, 4 figure
Finiteness conditions for graph algebras over tropical semirings
Connection matrices for graph parameters with values in a field have been
introduced by M. Freedman, L. Lov{\'a}sz and A. Schrijver (2007). Graph
parameters with connection matrices of finite rank can be computed in
polynomial time on graph classes of bounded tree-width. We introduce join
matrices, a generalization of connection matrices, and allow graph parameters
to take values in the tropical rings (max-plus algebras) over the real numbers.
We show that rank-finiteness of join matrices implies that these graph
parameters can be computed in polynomial time on graph classes of bounded
clique-width. In the case of graph parameters with values in arbitrary
commutative semirings, this remains true for graph classes of bounded linear
clique-width. B. Godlin, T. Kotek and J.A. Makowsky (2008) showed that
definability of a graph parameter in Monadic Second Order Logic implies rank
finiteness. We also show that there are uncountably many integer valued graph
parameters with connection matrices or join matrices of fixed finite rank. This
shows that rank finiteness is a much weaker assumption than any definability
assumption.Comment: 12 pages, accepted for presentation at FPSAC 2014 (Chicago, June 29
-July 3, 2014), to appear in Discrete Mathematics and Theoretical Computer
Scienc
Entropy algebras and Birkhoff factorization
We develop notions of Rota-Baxter structures and associated Birkhoff
factorizations, in the context of min-plus semirings and their thermodynamic
deformations, including deformations arising from quantum information measures
such as the von Neumann entropy. We consider examples related to Manin's
renormalization and computation program, to Markov random fields and to
counting functions and zeta functions of algebraic varieties.Comment: 28 pages, LaTe
Dynamic Programming on Nominal Graphs
Many optimization problems can be naturally represented as (hyper) graphs,
where vertices correspond to variables and edges to tasks, whose cost depends
on the values of the adjacent variables. Capitalizing on the structure of the
graph, suitable dynamic programming strategies can select certain orders of
evaluation of the variables which guarantee to reach both an optimal solution
and a minimal size of the tables computed in the optimization process. In this
paper we introduce a simple algebraic specification with parallel composition
and restriction whose terms up to structural axioms are the graphs mentioned
above. In addition, free (unrestricted) vertices are labelled with variables,
and the specification includes operations of name permutation with finite
support. We show a correspondence between the well-known tree decompositions of
graphs and our terms. If an axiom of scope extension is dropped, several
(hierarchical) terms actually correspond to the same graph. A suitable
graphical structure can be found, corresponding to every hierarchical term.
Evaluating such a graphical structure in some target algebra yields a dynamic
programming strategy. If the target algebra satisfies the scope extension
axiom, then the result does not depend on the particular structure, but only on
the original graph. We apply our approach to the parking optimization problem
developed in the ASCENS e-mobility case study, in collaboration with
Volkswagen. Dynamic programming evaluations are particularly interesting for
autonomic systems, where actual behavior often consists of propagating local
knowledge to obtain global knowledge and getting it back for local decisions.Comment: In Proceedings GaM 2015, arXiv:1504.0244
Computing the GIT-fan
We present an algorithm to compute the GIT-fan of algebraic torus actions on
affine varieties.Comment: minor changes; to appear in Internat. J. Algebra Compu
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