2,115 research outputs found
Finite symmetric functions with non-trivial arity gap
Given an -ary
valued function , denotes the essential arity gap of
which is the minimal number of essential variables in which become fictive
when identifying any two distinct essential variables in . In the present
paper we study the properties of the symmetric function with non-trivial arity
gap (). We prove several results concerning decomposition of the
symmetric functions with non-trivial arity gap with its minors or subfunctions.
We show that all non-empty sets of essential variables in symmetric functions
with non-trivial arity gap are separable.Comment: 12 page
The arity gap of polynomial functions over bounded distributive lattices
Let A and B be arbitrary sets with at least two elements. The arity gap of a
function f: A^n \to B is the minimum decrease in its essential arity when
essential arguments of f are identified. In this paper we study the arity gap
of polynomial functions over bounded distributive lattices and present a
complete classification of such functions in terms of their arity gap. To this
extent, we present a characterization of the essential arguments of polynomial
functions, which we then use to show that almost all lattice polynomial
functions have arity gap 1, with the exception of truncated median functions,
whose arity gap is 2.Comment: 7 page
Generalizations of Swierczkowski's lemma and the arity gap of finite functions
Swierczkowski's Lemma - as it is usually formulated - asserts that if f is an
at least quaternary operation on a finite set A and every operation obtained
from f by identifying a pair of variables is a projection, then f is a
semiprojection. We generalize this lemma in various ways. First, it is extended
to B-valued functions on A instead of operations on A and to essentially at
most unary functions instead of projections. Then we characterize the arity gap
of functions of small arities in terms of quasi-arity, which in turn provides a
further generalization of Swierczkowski's Lemma. Moreover, we explicitly
classify all pseudo-Boolean functions according to their arity gap. Finally, we
present a general characterization of the arity gaps of B-valued functions on
arbitrary finite sets A.Comment: 11 pages, proofs simplified, contents reorganize
On the effect of variable identification on the essential arity of functions
We show that every function of several variables on a finite set of k
elements with n>k essential variables has a variable identification minor with
at least n-k essential variables. This is a generalization of a theorem of
Salomaa on the essential variables of Boolean functions. We also strengthen
Salomaa's theorem by characterizing all the Boolean functions f having a
variable identification minor that has just one essential variable less than f.Comment: 10 page
Additive decomposability of functions over abelian groups
Abelian groups are classified by the existence of certain additive
decompositions of group-valued functions of several variables with arity gap 2.Comment: 17 page
The arity gap of order-preserving functions and extensions of pseudo-Boolean functions
The aim of this paper is to classify order-preserving functions according to
their arity gap. Noteworthy examples of order-preserving functions are
so-called aggregation functions. We first explicitly classify the Lov\'asz
extensions of pseudo-Boolean functions according to their arity gap. Then we
consider the class of order-preserving functions between partially ordered
sets, and establish a similar explicit classification for this function class.Comment: 11 pages, material reorganize
The power of linear programming for general-valued CSPs
Let , called the domain, be a fixed finite set and let , called
the valued constraint language, be a fixed set of functions of the form
, where different functions might have
different arity . We study the valued constraint satisfaction problem
parametrised by , denoted by VCSP. These are minimisation
problems given by variables and the objective function given by a sum of
functions from , each depending on a subset of the variables.
Finite-valued constraint languages contain functions that take on only rational
values and not infinite values.
Our main result is a precise algebraic characterisation of valued constraint
languages whose instances can be solved exactly by the basic linear programming
relaxation (BLP). For a valued constraint language , BLP is a decision
procedure for if and only if admits a symmetric fractional
polymorphism of every arity. For a finite-valued constraint language ,
BLP is a decision procedure if and only if admits a symmetric
fractional polymorphism of some arity, or equivalently, if admits a
symmetric fractional polymorphism of arity 2.
Using these results, we obtain tractability of several novel classes of
problems, including problems over valued constraint languages that are: (1)
submodular on arbitrary lattices; (2) -submodular on arbitrary finite
domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: A full version of a FOCS'12 paper by the last two authors
(arXiv:1204.1079) and an ICALP'13 paper by the first author (arXiv:1207.7213)
to appear in SIAM Journal on Computing (SICOMP
Join-irreducible Boolean functions
This paper is a contribution to the study of a quasi-order on the set
of Boolean functions, the \emph{simple minor} quasi-order. We look at
the join-irreducible members of the resulting poset . Using a
two-way correspondence between Boolean functions and hypergraphs,
join-irreducibility translates into a combinatorial property of hypergraphs. We
observe that among Steiner systems, those which yield join-irreducible members
of are the -2-monomorphic Steiner systems. We also describe
the graphs which correspond to join-irreducible members of .Comment: The current manuscript constitutes an extension to the paper
"Irreducible Boolean Functions" (arXiv:0801.2939v1
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