317 research outputs found
All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs
We describe algorithms, based on Avis and Fukuda's reverse search paradigm,
for listing all maximal independent sets in a sparse graph in polynomial time
and delay per output. For bounded degree graphs, our algorithms take constant
time per set generated; for minor-closed graph families, the time is O(n) per
set, and for more general sparse graph families we achieve subquadratic time
per set. We also describe new data structures for maintaining a dynamic vertex
set S in a sparse or minor-closed graph family, and querying the number of
vertices not dominated by S; for minor-closed graph families the time per
update is constant, while it is sublinear for any sparse graph family. We can
also maintain a dynamic vertex set in an arbitrary m-edge graph and test the
independence of the maintained set in time O(sqrt m) per update. We use the
domination data structures as part of our enumeration algorithms.Comment: 10 page
Maximal induced matchings in triangle-free graphs
An induced matching in a graph is a set of edges whose endpoints induce a
-regular subgraph. It is known that any -vertex graph has at most
maximal induced matchings, and this bound is best
possible. We prove that any -vertex triangle-free graph has at most maximal induced matchings, and this bound is attained by any
disjoint union of copies of the complete bipartite graph . Our result
implies that all maximal induced matchings in an -vertex triangle-free graph
can be listed in time , yielding the fastest known algorithm for
finding a maximum induced matching in a triangle-free graph.Comment: 17 page
Fast algorithms for min independent dominating set
We first devise a branching algorithm that computes a minimum independent
dominating set on any graph with running time O*(2^0.424n) and polynomial
space. This improves the O*(2^0.441n) result by (S. Gaspers and M. Liedloff, A
branch-and-reduce algorithm for finding a minimum independent dominating set in
graphs, Proc. WG'06). We then show that, for every r>3, it is possible to
compute an r-((r-1)/r)log_2(r)-approximate solution for min independent
dominating set within time O*(2^(nlog_2(r)/r))
An Improved Exact Algorithm for the Exact Satisfiability Problem
The Exact Satisfiability problem, XSAT, is defined as the problem of finding
a satisfying assignment to a formula in CNF such that exactly one
literal in each clause is assigned to be "1" and the other literals in the same
clause are set to "0". Since it is an important variant of the satisfiability
problem, XSAT has also been studied heavily and has seen numerous improvements
to the development of its exact algorithms over the years.
The fastest known exact algorithm to solve XSAT runs in time,
where is the number of variables in the formula. In this paper, we propose
a faster exact algorithm that solves the problem in time. Like
many of the authors working on this problem, we give a DPLL algorithm to solve
it. The novelty of this paper lies on the design of the nonstandard measure, to
help us to tighten the analysis of the algorithm further
Immature oocytes grow during in vitro maturation culture
BACKGROUND. Oocyte competence for maturation and embryogenesis is associated with oocyte diameter in many mammals. This study aimed to test whether such a relationship exists in humans and to quantify its impact upon in vitro maturation (IVM). METHODS. We used computer-assisted image analysis daily to measure average diameter, zona thickness and other parameters in oocytes. Immature oocytes originated from unstimulated patients with polycystic ovaries, and from stimulated patients undergoing ICSI. They were cultured with or without meiosis activating sterol (FF-MAS). Oocytes maturing in vitro were inseminated using ICSI and embryo development was monitored. A sample of freshly collected in vivo matured oocytes from ICSI patients were also measured. RESULTS. Immature oocytes were usually smaller at collection than in vivo matured oocytes. Capacity for maturation was related to oocyte diameter and many oocytes grew in culture. FF-MAS stimulated growth in ICSI derived oocytes, but only stimulated growth in PCO derived oocytes if they eventually matured in vitro. Oocytes degenerating showed cytoplasmic shrinkage. Neither zona thickness, perivitelline space, nor the total diameter of the oocyte including the zona were informative regarding oocyte maturation capacity. CONCLUSIONS. Immature oocytes continue growing during maturation culture. FF-MAS promotes oocyte growth in vitro. Oocytes from different sources have different growth profiles in vitro. Measuring diameters of oocytes used in clinical IVM may provide additional non-invasive information that could potentially identify and avoid the use of oocytes that remain in the growth phase
Statistical Mechanics of maximal independent sets
The graph theoretic concept of maximal independent set arises in several
practical problems in computer science as well as in game theory. A maximal
independent set is defined by the set of occupied nodes that satisfy some
packing and covering constraints. It is known that finding minimum and
maximum-density maximal independent sets are hard optimization problems. In
this paper, we use cavity method of statistical physics and Monte Carlo
simulations to study the corresponding constraint satisfaction problem on
random graphs. We obtain the entropy of maximal independent sets within the
replica symmetric and one-step replica symmetry breaking frameworks, shedding
light on the metric structure of the landscape of solutions and suggesting a
class of possible algorithms. This is of particular relevance for the
application to the study of strategic interactions in social and economic
networks, where maximal independent sets correspond to pure Nash equilibria of
a graphical game of public goods allocation
Tur\'an Graphs, Stability Number, and Fibonacci Index
The Fibonacci index of a graph is the number of its stable sets. This
parameter is widely studied and has applications in chemical graph theory. In
this paper, we establish tight upper bounds for the Fibonacci index in terms of
the stability number and the order of general graphs and connected graphs.
Tur\'an graphs frequently appear in extremal graph theory. We show that Tur\'an
graphs and a connected variant of them are also extremal for these particular
problems.Comment: 11 pages, 3 figure
Feedback Vertex Sets in Tournaments
We study combinatorial and algorithmic questions around minimal feedback
vertex sets in tournament graphs.
On the combinatorial side, we derive strong upper and lower bounds on the
maximum number of minimal feedback vertex sets in an n-vertex tournament. We
prove that every tournament on n vertices has at most 1.6740^n minimal feedback
vertex sets, and that there is an infinite family of tournaments, all having at
least 1.5448^n minimal feedback vertex sets. This improves and extends the
bounds of Moon (1971).
On the algorithmic side, we design the first polynomial space algorithm that
enumerates the minimal feedback vertex sets of a tournament with polynomial
delay. The combination of our results yields the fastest known algorithm for
finding a minimum size feedback vertex set in a tournament
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