94 research outputs found
A lower bound on CNF encodings of the at-most-one constraint
Constraint "at most one" is a basic cardinality constraint which requires
that at most one of its boolean inputs is set to . This constraint is
widely used when translating a problem into a conjunctive normal form (CNF) and
we investigate its CNF encodings suitable for this purpose. An encoding differs
from a CNF representation of a function in that it can use auxiliary variables.
We are especially interested in propagation complete encodings which have the
property that unit propagation is strong enough to enforce consistency on input
variables. We show a lower bound on the number of clauses in any propagation
complete encoding of the "at most one" constraint. The lower bound almost
matches the size of the best known encodings. We also study an important case
of 2-CNF encodings where we show a slightly better lower bound. The lower bound
holds also for a related "exactly one" constraint.Comment: 38 pages, version 3 is significantly reorganized in order to improve
readabilit
Variations on Instant Insanity
In one of the first papers about the complexity of puzzles, Robertson and Munro [14] proved that a generalized form of the then-popular Instant Insanity puzzle is NP-complete. Here we study several variations of this puzzle, exploring how the complexity depends on the piece shapes and the allowable orientations of those shapes
On Structural Parameterizations of Hitting Set: Hitting Paths in Graphs Using 2-SAT
Hitting Set is a classic problem in combinatorial optimization. Its input
consists of a set system F over a finite universe U and an integer t; the
question is whether there is a set of t elements that intersects every set in
F. The Hitting Set problem parameterized by the size of the solution is a
well-known W[2]-complete problem in parameterized complexity theory. In this
paper we investigate the complexity of Hitting Set under various structural
parameterizations of the input. Our starting point is the folklore result that
Hitting Set is polynomial-time solvable if there is a tree T on vertex set U
such that the sets in F induce connected subtrees of T. We consider the case
that there is a treelike graph with vertex set U such that the sets in F induce
connected subgraphs; the parameter of the problem is a measure of how treelike
the graph is. Our main positive result is an algorithm that, given a graph G
with cyclomatic number k, a collection P of simple paths in G, and an integer
t, determines in time 2^{5k} (|G| +|P|)^O(1) whether there is a vertex set of
size t that hits all paths in P. It is based on a connection to the 2-SAT
problem in multiple valued logic. For other parameterizations we derive
W[1]-hardness and para-NP-completeness results.Comment: Presented at the 41st International Workshop on Graph-Theoretic
Concepts in Computer Science, WG 2015. (The statement of Lemma 4 was
corrected in this update.
Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials
This paper is our third step towards developing a theory of testing monomials
in multivariate polynomials and concentrates on two problems: (1) How to
compute the coefficients of multilinear monomials; and (2) how to find a
maximum multilinear monomial when the input is a polynomial. We
first prove that the first problem is \#P-hard and then devise a
upper bound for this problem for any polynomial represented by an arithmetic
circuit of size . Later, this upper bound is improved to for
polynomials. We then design fully polynomial-time randomized
approximation schemes for this problem for polynomials. On the
negative side, we prove that, even for polynomials with terms of
degree , the first problem cannot be approximated at all for any
approximation factor , nor {\em "weakly approximated"} in a much relaxed
setting, unless P=NP. For the second problem, we first give a polynomial time
-approximation algorithm for polynomials with terms of
degrees no more a constant . On the inapproximability side, we
give a lower bound, for any on the
approximation factor for polynomials. When terms in these
polynomials are constrained to degrees , we prove a lower
bound, assuming ; and a higher lower bound, assuming the
Unique Games Conjecture
On Embeddability of Buses in Point Sets
Set membership of points in the plane can be visualized by connecting
corresponding points via graphical features, like paths, trees, polygons,
ellipses. In this paper we study the \emph{bus embeddability problem} (BEP):
given a set of colored points we ask whether there exists a planar realization
with one horizontal straight-line segment per color, called bus, such that all
points with the same color are connected with vertical line segments to their
bus. We present an ILP and an FPT algorithm for the general problem. For
restricted versions of this problem, such as when the relative order of buses
is predefined, or when a bus must be placed above all its points, we provide
efficient algorithms. We show that another restricted version of the problem
can be solved using 2-stack pushall sorting. On the negative side we prove the
NP-completeness of a special case of BEP.Comment: 19 pages, 9 figures, conference version at GD 201
Computing NodeTrix Representations of Clustered Graphs
NodeTrix representations are a popular way to visualize clustered graphs;
they represent clusters as adjacency matrices and inter-cluster edges as curves
connecting the matrix boundaries. We study the complexity of constructing
NodeTrix representations focusing on planarity testing problems, and we show
several NP-completeness results and some polynomial-time algorithms. Building
on such algorithms we develop a JavaScript library for NodeTrix representations
aimed at reducing the crossings between edges incident to the same matrix.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Tree decompositions with small cost
The f-cost of a tree decomposition ({Xi | i e I}, T = (I;F))
for a function f : N -> R+ is defined as EieI f(|Xi|). This measure
associates with the running time or memory use of some algorithms
that use the tree decomposition. In this paper we investigate the
problem to find tree decompositions of minimum f-cost.
A function f : N -> R+ is fast, if for every i e N: f(i+1) => 2*f(i).
We show that for fast functions f, every graph G has a tree decomposition
of minimum f-cost that corresponds to a minimal triangulation
of G; if f is not fast, this does not hold. We give polynomial time
algorithms for the problem, assuming f is a fast function, for graphs
that has a polynomial number of minimal separators, for graphs of
treewidth at most two, and for cographs, and show that the problem
is NP-hard for bipartite graphs and for cobipartite graphs.
We also discuss results for a weighted variant of the problem derived
of an application from probabilistic networks
Simplest random K-satisfiability problem
We study a simple and exactly solvable model for the generation of random
satisfiability problems. These consist of random boolean constraints
which are to be satisfied simultaneously by logical variables. In
statistical-mechanics language, the considered model can be seen as a diluted
p-spin model at zero temperature. While such problems become extraordinarily
hard to solve by local search methods in a large region of the parameter space,
still at least one solution may be superimposed by construction. The
statistical properties of the model can be studied exactly by the replica
method and each single instance can be analyzed in polynomial time by a simple
global solution method. The geometrical/topological structures responsible for
dynamic and static phase transitions as well as for the onset of computational
complexity in local search method are thoroughly analyzed. Numerical analysis
on very large samples allows for a precise characterization of the critical
scaling behaviour.Comment: 14 pages, 5 figures, to appear in Phys. Rev. E (Feb 2001). v2: minor
errors and references correcte
A Satisfiability-based Approach for Embedding Generalized Tanglegrams on Level Graphs
A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in computational biology to compare evolutionary histories of species. In this paper we present a formulation of two related combinatorial embedding problems concerning tanglegrams in terms of CNF-formulas. The first problem is known as planar embedding and the second as crossing minimization problem. We show that our satisfiability formulation of these problems can handle a much more general case with more than two, not necessarily binary or complete, trees defined on arbitrary sets of leaves and allowed to vary their layouts
First-Digit Law in Nonextensive Statistics
Nonextensive statistics, characterized by a nonextensive parameter , is a
promising and practically useful generalization of the Boltzmann statistics to
describe power-law behaviors from physical and social observations. We here
explore the unevenness of the first digit distribution of nonextensive
statistics analytically and numerically. We find that the first-digit
distribution follows Benford's law and fluctuates slightly in a periodical
manner with respect to the logarithm of the temperature. The fluctuation
decreases when increases, and the result converges to Benford's law exactly
as approaches 2. The relevant regularities between nonextensive statistics
and Benford's law are also presented and discussed.Comment: 11 pages, 3 figures, published in Phys. Rev.
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