602 research outputs found
The Connectivities of Leaf Graphs of 2-Connected Graphs
AbstractGiven a connected graph G, denote by V the family of all the spanning trees of G. Define an adjacency relation in V as follows: the spanning trees t and tâČ are said to be adjacent if for some vertex uâV, tâu is connected and coincides with tâČâu. The resultant graph G is called the leaf graph of G. The purpose of this paper is to show that if G is 2-connected with minimal degree ÎŽ, then G is (2ÎŽâ2)-connected
Statistical mechanics of the vertex-cover problem
We review recent progress in the study of the vertex-cover problem (VC). VC
belongs to the class of NP-complete graph theoretical problems, which plays a
central role in theoretical computer science. On ensembles of random graphs, VC
exhibits an coverable-uncoverable phase transition. Very close to this
transition, depending on the solution algorithm, easy-hard transitions in the
typical running time of the algorithms occur.
We explain a statistical mechanics approach, which works by mapping VC to a
hard-core lattice gas, and then applying techniques like the replica trick or
the cavity approach. Using these methods, the phase diagram of VC could be
obtained exactly for connectivities , where VC is replica symmetric.
Recently, this result could be confirmed using traditional mathematical
techniques. For , the solution of VC exhibits full replica symmetry
breaking.
The statistical mechanics approach can also be used to study analytically the
typical running time of simple complete and incomplete algorithms for VC.
Finally, we describe recent results for VC when studied on other ensembles of
finite- and infinite-dimensional graphs.Comment: review article, 26 pages, 9 figures, to appear in J. Phys. A: Math.
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Dynamics of heuristic optimization algorithms on random graphs
In this paper, the dynamics of heuristic algorithms for constructing small
vertex covers (or independent sets) of finite-connectivity random graphs is
analysed. In every algorithmic step, a vertex is chosen with respect to its
vertex degree. This vertex, and some environment of it, is covered and removed
from the graph. This graph reduction process can be described as a Markovian
dynamics in the space of random graphs of arbitrary degree distribution. We
discuss some solvable cases, including algorithms already analysed using
different techniques, and develop approximation schemes for more complicated
cases. The approximations are corroborated by numerical simulations.Comment: 19 pages, 3 figures, version to app. in EPJ
Phase transition for cutting-plane approach to vertex-cover problem
We study the vertex-cover problem which is an NP-hard optimization problem
and a prototypical model exhibiting phase transitions on random graphs, e.g.,
Erdoes-Renyi (ER) random graphs. These phase transitions coincide with changes
of the solution space structure, e.g, for the ER ensemble at connectivity
c=e=2.7183 from replica symmetric to replica-symmetry broken. For the
vertex-cover problem, also the typical complexity of exact branch-and-bound
algorithms, which proceed by exploring the landscape of feasible
configurations, change close to this phase transition from "easy" to "hard". In
this work, we consider an algorithm which has a completely different strategy:
The problem is mapped onto a linear programming problem augmented by a
cutting-plane approach, hence the algorithm operates in a space OUTSIDE the
space of feasible configurations until the final step, where a solution is
found. Here we show that this type of algorithm also exhibits an "easy-hard"
transition around c=e, which strongly indicates that the typical hardness of a
problem is fundamental to the problem and not due to a specific representation
of the problem.Comment: 4 pages, 3 figure
Efficient algorithm for computing all low s-t edge connectivities in directed graphs
LNCS v. 9235 entitled: Mathematical Foundations of Computer Science 2015: 40th International Symposium, MFCS 2015, Milan, Italy, August 24-28, 2015, Proceedings, Part 2Given a directed graph with n nodes and m edges, the (strong) edge connectivity λ (u; v) between two nodes u and v is the minimum number of edges whose deletion makes u and v not strongly connected. The problem of computing the edge connectivities between all pairs of nodes of a directed graph can be done in O(m Ï) time by Cheung, Lau and Leung (FOCS 2011), where Ï is the matrix multiplication factor (â 2:373), or in Ă (mn1:5) time using O(n) computations of max-flows by Cheng and Hu (IPCO 1990).
We consider in this paper the âlow edge connectivityâ problem, which aims at computing the edge connectivities for the pairs of nodes (u; v) such that λ (u; v) †k. While the undirected version of this problem was considered by Hariharan, Kavitha and Panigrahi (SODA 2007), who presented an algorithm with expected running time Ă (m+nk3), no algorithm better than computing all-pairs edge connectivities was proposed for directed graphs. We provide an algorithm that computes all low edge connectivities in O(kmn) time, improving the previous best result of O (min(m Ï, mn1:5)) when k †â n. Our algorithm also computes a minimum u-v cut for each pair of nodes (u; v) with λ (u; v) †k.postprin
Minimum Cycle Base of Graphs Identified by Two Planar Graphs
In this paper, we study the minimum cycle base of the planar graphs obtained
from two 2-connected planar graphs by identifying an edge (or a cycle) of one graph with the corresponding edge (or cycle) of another, related with map geometries, i.e., Smarandache 2-dimensional manifolds
The Phase Diagram of 1-in-3 Satisfiability Problem
We study the typical case properties of the 1-in-3 satisfiability problem,
the boolean satisfaction problem where a clause is satisfied by exactly one
literal, in an enlarged random ensemble parametrized by average connectivity
and probability of negation of a variable in a clause. Random 1-in-3
Satisfiability and Exact 3-Cover are special cases of this ensemble. We
interpolate between these cases from a region where satisfiability can be
typically decided for all connectivities in polynomial time to a region where
deciding satisfiability is hard, in some interval of connectivities. We derive
several rigorous results in the first region, and develop the
one-step--replica-symmetry-breaking cavity analysis in the second one. We
discuss the prediction for the transition between the almost surely satisfiable
and the almost surely unsatisfiable phase, and other structural properties of
the phase diagram, in light of cavity method results.Comment: 30 pages, 12 figure
The dynamics of proving uncolourability of large random graphs I. Symmetric Colouring Heuristic
We study the dynamics of a backtracking procedure capable of proving
uncolourability of graphs, and calculate its average running time T for sparse
random graphs, as a function of the average degree c and the number of vertices
N. The analysis is carried out by mapping the history of the search process
onto an out-of-equilibrium (multi-dimensional) surface growth problem. The
growth exponent of the average running time is quantitatively predicted, in
agreement with simulations.Comment: 5 figure
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