Four pages are necessary and sufficient for planar graphs
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Linear and book embeddings of graphs
Proc. of the Aegean workshop on computing on VLSI algorithms and architectures
Embedding graphs in books: a layout problem with applications to VLSI design
SIAM Journal on Algebraic and Discrete Methods
Crossing Minimization in Linear Embeddings of Graphs
IEEE Transactions on Computers
Introduction to parallel algorithms and architectures: array, trees, hypercubes
Introduction to parallel algorithms and architectures: array, trees, hypercubes
Graphs with E edges have pagenumber E O
Journal of Algorithms
Randomized algorithms
Parallel and distributed computing handbook
Parallel and distributed computing handbook
The book crossing number of a graph
Journal of Graph Theory
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
CNMGRAF—graphic presentation services for network management
SIGCOMM '85 Proceedings of the ninth symposium on Data communications
Sorting Using Networks of Queues and Stacks
Journal of the ACM (JACM)
Discrete Applied Mathematics
Embedding Graphs into a Three Page Book with O(m log n) Crossings of Edges over the Spine
SIAM Journal on Discrete Mathematics
Drawing graphs: methods and models
Drawing graphs: methods and models
Graph Drawing: Algorithms for the Visualization of Graphs
Graph Drawing: Algorithms for the Visualization of Graphs
Algorithms for the fixed linear crossing number problem
Discrete Applied Mathematics
Improved Approximation Algorithms for MAX k-CUT and MAX BISECTION
Proceedings of the 4th International IPCO Conference on Integer Programming and Combinatorial Optimization
Book Embeddings and Crossing Numbers
WG '94 Proceedings of the 20th International Workshop on Graph-Theoretic Concepts in Computer Science
Derandomizing semidefinite programming based approximation algorithms
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Computational Aspects of VLSI
Genetic algorithms for the 2-page book drawing problem of graphs
Journal of Heuristics
IEEE Transactions on Computers
The Diogenes Approach to Testable Fault-Tolerant Arrays of Processors
IEEE Transactions on Computers
A neural-network algorithm for a graph layout problem
IEEE Transactions on Neural Networks
The 2-page crossing number of Kn
Proceedings of the twenty-eighth annual symposium on Computational geometry
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We present a randomized polynomial-time approximation algorithm for the fixed linear crossing number problem (FLCNP). In this problem, the vertices of a graph are placed in a fixed order along a horizontal ''node line'' in the plane, each edge is drawn as an arc in one of the two half-planes (pages), and the objective is to minimize the number of edge crossings. FLCNP is NP-hard, and no previous polynomial-time approximation algorithms are known. We show that the problem can be generalized to k pages and transformed to the maximum k-cut problem which admits a randomized polynomial-time approximation. For the 2-page case, our approach leads to a randomized polynomial time 0.878+0.122@r approximation algorithm for FLCNP, where @r is the ratio of the number of conflicting pairs (pairs of edges that cross if drawn in the same page) to the crossing number. We further investigate this performance ratio on the random graph family G"n","1"/"2, where each edge of the complete graph K"n occurs with probability 12. We show that a longstanding conjecture for the crossing number of K"n implies that with probability at least 1-4e^-^@l^^^2, the expected performance bound of the algorithm on a random graph from G"n","1"/"2 is 1.366+O(@l/n). A series of experiments is performed to compare the algorithm against two other leading heuristics on a set of test graphs. The results indicate that the randomized algorithm yields near-optimal solutions and outperforms the other heuristics overall.