5,736 research outputs found
Wide partitions, Latin tableaux, and Rota's basis conjecture
Say that mu is a ``subpartition'' of an integer partition lambda if the
multiset of parts of mu is a submultiset of the parts of lambda, and define an
integer partition lambda to be ``wide'' if for every subpartition mu of lambda,
mu >= mu' in dominance order (where mu' denotes the conjugate or transpose of
mu). Then Brian Taylor and the first author have conjectured that an integer
partition lambda is wide if and only if there exists a tableau of shape lambda
such that (1) for all i, the entries in the ith row of the tableau are
precisely the integers from 1 to lambda_i inclusive, and (2) for all j, the
entries in the jth column of the tableau are pairwise distinct. This conjecture
was originally motivated by Rota's basis conjecture and, if true, yields a new
class of integer multiflow problems that satisfy max-flow min-cut and
integrality. Wide partitions also yield a class of graphs that satisfy
``delta-conjugacy'' (in the sense of Greene and Kleitman), and the above
conjecture implies that these graphs furthermore have a completely saturated
stable set partition. We present several partial results, but the conjecture
remains very much open.Comment: Joined forces with Goemans and Vondrak---several new partial results;
28 pages, submitted to Adv. Appl. Mat
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
On a registration-based approach to sensor network localization
We consider a registration-based approach for localizing sensor networks from
range measurements. This is based on the assumption that one can find
overlapping cliques spanning the network. That is, for each sensor, one can
identify geometric neighbors for which all inter-sensor ranges are known. Such
cliques can be efficiently localized using multidimensional scaling. However,
since each clique is localized in some local coordinate system, we are required
to register them in a global coordinate system. In other words, our approach is
based on transforming the localization problem into a problem of registration.
In this context, the main contributions are as follows. First, we describe an
efficient method for partitioning the network into overlapping cliques. Second,
we study the problem of registering the localized cliques, and formulate a
necessary rigidity condition for uniquely recovering the global sensor
coordinates. In particular, we present a method for efficiently testing
rigidity, and a proposal for augmenting the partitioned network to enforce
rigidity. A recently proposed semidefinite relaxation of global registration is
used for registering the cliques. We present simulation results on random and
structured sensor networks to demonstrate that the proposed method compares
favourably with state-of-the-art methods in terms of run-time, accuracy, and
scalability
Walk entropies on graphs
Entropies based on walks on graphs and on their line-graphs are defined. They are based on the summation over diagonal and off-diagonal elements of the thermal Green’s function of a graph also known as the communicability. The walk entropies are strongly related to the walk regularity of graphs and line-graphs. They are not biased by the graph size and have significantly better correlation with the inverse participation ratio of the eigenmodes of the adjacency matrix than other graph entropies. The temperature dependence of the walk entropies is also discussed. In particular, the walk entropy of graphs is shown to be non-monotonic for regular but non-walk-regular graphs in contrast to non-regular graphs
Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
We study the commutation relations within the Pauli groups built on all
decompositions of a given Hilbert space dimension , containing a square,
into its factors. Illustrative low dimensional examples are the quartit ()
and two-qubit () systems, the octit (), qubit/quartit () and three-qubit () systems, and so on. In the single qudit case,
e.g. , one defines a bijection between the maximal
commuting sets [with the sum of divisors of ] of Pauli
observables and the maximal submodules of the modular ring ,
that arrange into the projective line and a independent set
of size [with the Dedekind psi function]. In the
multiple qudit case, e.g. , the Pauli graphs rely on
symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if
) and GQ(3,3) (if ). More precisely, in dimension ( a
prime) of the Hilbert space, the observables of the Pauli group (modulo the
center) are seen as the elements of the -dimensional vector space over the
field . In this space, one makes use of the commutator to define
a symplectic polar space of cardinality , that
encodes the maximal commuting sets of the Pauli group by its totally isotropic
subspaces. Building blocks of are punctured polar spaces (i.e. a
observable and all maximum cliques passing to it are removed) of size given by
the Dedekind psi function . For multiple qudit mixtures (e.g.
qubit/quartit, qubit/octit and so on), one finds multiple copies of polar
spaces, ponctured polar spaces, hypercube geometries and other intricate
structures. Such structures play a role in the science of quantum information.Comment: 18 pages, version submiited to J. Phys. A: Math. Theo
Nice labeling problem for event structures: a counterexample
In this note, we present a counterexample to a conjecture of Rozoy and
Thiagarajan from 1991 (called also the nice labeling problem) asserting that
any (coherent) event structure with finite degree admits a labeling with a
finite number of labels, or equivalently, that there exists a function such that an event structure with degree
admits a labeling with at most labels. Our counterexample is based on
the Burling's construction from 1965 of 3-dimensional box hypergraphs with
clique number 2 and arbitrarily large chromatic numbers and the bijection
between domains of event structures and median graphs established by
Barth\'elemy and Constantin in 1993
Algorithmic Complexity of Power Law Networks
It was experimentally observed that the majority of real-world networks
follow power law degree distribution. The aim of this paper is to study the
algorithmic complexity of such "typical" networks. The contribution of this
work is twofold.
First, we define a deterministic condition for checking whether a graph has a
power law degree distribution and experimentally validate it on real-world
networks. This definition allows us to derive interesting properties of power
law networks. We observe that for exponents of the degree distribution in the
range such networks exhibit double power law phenomenon that was
observed for several real-world networks. Our observation indicates that this
phenomenon could be explained by just pure graph theoretical properties.
The second aim of our work is to give a novel theoretical explanation why
many algorithms run faster on real-world data than what is predicted by
algorithmic worst-case analysis. We show how to exploit the power law degree
distribution to design faster algorithms for a number of classical P-time
problems including transitive closure, maximum matching, determinant, PageRank
and matrix inverse. Moreover, we deal with the problems of counting triangles
and finding maximum clique. Previously, it has been only shown that these
problems can be solved very efficiently on power law graphs when these graphs
are random, e.g., drawn at random from some distribution. However, it is
unclear how to relate such a theoretical analysis to real-world graphs, which
are fixed. Instead of that, we show that the randomness assumption can be
replaced with a simple condition on the degrees of adjacent vertices, which can
be used to obtain similar results. As a result, in some range of power law
exponents, we are able to solve the maximum clique problem in polynomial time,
although in general power law networks the problem is NP-complete
Graph Isomorphism for unit square graphs
In the past decades for more and more graph classes the Graph Isomorphism
Problem was shown to be solvable in polynomial time. An interesting family of
graph classes arises from intersection graphs of geometric objects. In this
work we show that the Graph Isomorphism Problem for unit square graphs,
intersection graphs of axis-parallel unit squares in the plane, can be solved
in polynomial time. Since the recognition problem for this class of graphs is
NP-hard we can not rely on standard techniques for geometric graphs based on
constructing a canonical realization. Instead, we develop new techniques which
combine structural insights into the class of unit square graphs with
understanding of the automorphism group of such graphs. For the latter we
introduce a generalization of bounded degree graphs which is used to capture
the main structure of unit square graphs. Using group theoretic algorithms we
obtain sufficient information to solve the isomorphism problem for unit square
graphs.Comment: 31 pages, 6 figure
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