12,920 research outputs found
Matching Is as Easy as the Decision Problem, in the NC Model
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for
it? This has been an outstanding open question in TCS for over three decades,
ever since the discovery of randomized NC matching algorithms [KUW85, MVV87].
Over the last five years, the theoretical computer science community has
launched a relentless attack on this question, leading to the discovery of
several powerful ideas. We give what appears to be the culmination of this line
of work: An NC algorithm for finding a minimum-weight perfect matching in a
general graph with polynomially bounded edge weights, provided it is given an
oracle for the decision problem. Consequently, for settling the main open
problem, it suffices to obtain an NC algorithm for the decision problem. We
believe this new fact has qualitatively changed the nature of this open
problem.
All known efficient matching algorithms for general graphs follow one of two
approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based
algorithm follows a new approach and uses many of the ideas discovered in the
last five years.
The difficulty of obtaining an NC perfect matching algorithm led researchers
to study matching vis-a-vis clever relaxations of the class NC. In this vein,
recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC
algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC
algorithm with the additional requirement that on the same graph, it should
return the same (i.e., unique) perfect matching for almost all choices of
random bits. A corollary of our reduction is an analogous algorithm for general
graphs.Comment: Appeared in ITCS 202
Recognizing sparse perfect elimination bipartite graphs
When applying Gaussian elimination to a sparse matrix, it is desirable to avoid turning zeros into non-zeros to preserve the sparsity. The class of perfect elimination bipartite graphs is closely related to square matrices that Gaussian elimination can be applied to without turning any zero into a non-zero. Existing literature on the recognition of this class and finding suitable pivots mainly focusses on time complexity. For matrices with m non-zero elements, the currently best known algorithm has a time complexity of . However, when viewed from a practical perspective, the space complexity also deserves attention: it may not be worthwhile to look for a suitable set of pivots for a sparse matrix if this requires space. We present two new algorithms for the recognition of sparse instances: one with a time complexity in space and one with a time complexity in space. Furthermore, if we allow only pivots on the diagonal, our second algorithm can easily be adapted to run in time
Hamilton cycles in dense vertex-transitive graphs
A famous conjecture of Lov\'asz states that every connected vertex-transitive
graph contains a Hamilton path. In this article we confirm the conjecture in
the case that the graph is dense and sufficiently large. In fact, we show that
such graphs contain a Hamilton cycle and moreover we provide a polynomial time
algorithm for finding such a cycle.Comment: 26 pages, 3 figures; referees' comments incorporated; accepted for
publication in Journal of Combinatorial Theory, series
Fully Dynamic Matching in Bipartite Graphs
Maximum cardinality matching in bipartite graphs is an important and
well-studied problem. The fully dynamic version, in which edges are inserted
and deleted over time has also been the subject of much attention. Existing
algorithms for dynamic matching (in general graphs) seem to fall into two
groups: there are fast (mostly randomized) algorithms that do not achieve a
better than 2-approximation, and there slow algorithms with \O(\sqrt{m})
update time that achieve a better-than-2 approximation. Thus the obvious
question is whether we can design an algorithm -- deterministic or randomized
-- that achieves a tradeoff between these two: a approximation
and a better-than-2 approximation simultaneously. We answer this question in
the affirmative for bipartite graphs.
Our main result is a fully dynamic algorithm that maintains a 3/2 + \eps
approximation in worst-case update time O(m^{1/4}\eps^{/2.5}). We also give
stronger results for graphs whose arboricity is at most \al, achieving a (1+
\eps) approximation in worst-case time O(\al (\al + \log n)) for constant
\eps. When the arboricity is constant, this bound is and when the
arboricity is polylogarithmic the update time is also polylogarithmic.
The most important technical developement is the use of an intermediate graph
we call an edge degree constrained subgraph (EDCS). This graph places
constraints on the sum of the degrees of the endpoints of each edge: upper
bounds for matched edges and lower bounds for unmatched edges. The main
technical content of our paper involves showing both how to maintain an EDCS
dynamically and that and EDCS always contains a sufficiently large matching. We
also make use of graph orientations to help bound the amount of work done
during each update.Comment: Longer version of paper that appears in ICALP 201
Parity balance of the -th dimension edges in Hamiltonian cycles of the hypercube
Let be an integer, and let . An -th dimension
edge in the -dimensional hypercube is an edge such that
differ just at their -th entries. The parity of an -th
dimension edge \edg{v_1}{v_2} is the number of 1's modulus 2 of any of its
vertex ignoring the -th entry. We prove that the number of -th dimension
edges appearing in a given Hamiltonian cycle of with parity zero
coincides with the number of edges with parity one. As an application of this
result it is introduced and explored the conjecture of the inscribed squares in
Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in contains
two opposite edges in a 4-cycle. We prove this conjecture for , and
for any Hamiltonian cycle containing more than edges in the same
dimension. This bound is finally improved considering the equi-independence
number of , which is a concept introduced in this paper for bipartite
graphs
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