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The Planar k-Means Problem is NP-Hard
In the k-means problem, we are given a finite set S of points in $\Re^m$, and integer k ≥ 1, and we want to find k points (centers) so as to minimize the sum of the square of the Euclidean distanceExpand
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  • Open Access
The planar k-means problem is NP-hard
In the k-means problem, we are given a finite set S of points in @?^m, and integer k>=1, and we want to find k points (centers) so as to minimize the sum of the square of the Euclidean distance ofExpand
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Planar Graph Isomorphism is in Log-Space
Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. There is a significant gap between extantExpand
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Pseudorandom generators for group products: extended abstract
We prove that the pseudorandom generator introduced by Impagliazzo et al. (1994) with proper choice of parameters fools group products of a given finite group G. The seed length is O((|G|O(1) + logExpand
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Longest Paths in Planar DAGs in Unambiguous Log-Space
Reachability and distance computation are known to be NL-complete in general graphs, but within UL ∩ co-UL if the graphs are planar. However, finding longest paths is known to be NP-complete, evenExpand
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Pseudorandom Generators for Group Products
We prove that the pseudorandom generator introduced in [INW94] fools group products of a given finite group. The seed length is O(log n log 1e ), where n the length of the word and e is theExpand
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A Log-space Algorithm for Canonization of Planar Graphs
Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. We bridge this gap for a natural andExpand
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Popularity at Minimum Cost
We consider an extension of the popular matching problem in this paper. The input to the popular matching problem is a bipartite graph \(G = (\mathcal{A} \cup \mathcal{B},E)\), where \(\mathcal{A}\)Expand
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Dynamic Rank-Maximal Matchings
We consider the problem of matching applicants to posts where applicants have preferences over posts. Thus the input to our problem is a bipartite graph \(G=(\mathcal {A}\cup \mathcal {P},E)\), whereExpand
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Graph Isomorphism for K_{3, 3}-free and K_5-free graphs is in Log-space
Graph isomorphism is an important and widely studied computational problem with a yet unsettled complexity. However, the exact complexity is known for isomorphism of various classes of graphs.Expand
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