Complexity issues in VLSI: optimal layouts for the shuffle-exchange graph and other networks
Complexity issues in VLSI: optimal layouts for the shuffle-exchange graph and other networks
Journal of the ACM (JACM)
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
Lectures on Discrete Geometry
Improved Approximations of Crossings in Graph Drawings and VLSI Layout Areas
SIAM Journal on Computing
Minimizing Congestion in General Networks
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Computing crossing numbers in quadratic time
Journal of Computer and System Sciences - STOC 2001
Multicommodity flow, well-linked terminals, and routing problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Crossing number is hard for cubic graphs
Journal of Combinatorial Theory Series B
Computing crossing number in linear time
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
The crossing number of K11 is 100
Journal of Graph Theory
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Expander flows, geometric embeddings and graph partitioning
Journal of the ACM (JACM)
Approximating the crossing number of toroidal graphs
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Adding one edge to planar graphs makes crossing number hard
Proceedings of the twenty-sixth annual symposium on Computational geometry
Approximating the crossing number of graphs embeddable in any orientable surface
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Progress on crossing number problems
SOFSEM'05 Proceedings of the 31st international conference on Theory and Practice of Computer Science
On graph crossing number and edge planarization
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
A tighter insertion-based approximation of the crossing number
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
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We study the Minimum Crossing Number problem: given an n-vertex graph G, the goal is to find a drawing of G in the plane with minimum number of edge crossings. This is one of the central problems in topological graph theory, that has been studied extensively over the past three decades. The first non-trivial efficient algorithm for the problem, due to Leighton and Rao, achieved an O(n log4n)-approximation for bounded degree graphs. This algorithm has since been improved by poly-logarithmic factors, with the best current approximation ratio standing on O (n poly(d) log3/2n ) for graphs with maximum degree d. In contrast, only APX-hardness is known on the negative side. In this paper we present an efficient randomized algorithm to find a drawing of any n-vertex graph G in the plane with O(OPT10poly(d log n)) crossings, where OPT is the number of crossings in the optimal solution, and d is the maximum vertex degree in G. This result implies an ~O(n9/10poly(d))-approximation for Minimum Crossing Number, thus breaking the long-standing ~O(n)-approximation barrier for bounded-degree graphs.