Efficient implementation of lattice operations
ACM Transactions on Programming Languages and Systems (TOPLAS)
The jump number and the lattice of maximal antichains
Discrete Mathematics
Efficient handling of multiple inheritance hierarchies
OOPSLA '93 Proceedings of the eighth annual conference on Object-oriented programming systems, languages, and applications
On edge perfectness and classes of bipartite graphs
Discrete Mathematics
On the order dimension of 1-sets versus k-sets
Journal of Combinatorial Theory Series A
Sperner theory
Efficient type inclusion tests
Proceedings of the 12th ACM SIGPLAN conference on Object-oriented programming, systems, languages, and applications
Free Bits, PCPs, and Nonapproximability---Towards Tight Results
SIAM Journal on Computing
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Efficient Retrieval from Hierarchies of Objects using Lattice Operations
ICCS '93 Proceedings on Conceptual Graphs for Knowledge Representation
ECOOP '01 Proceedings of the 15th European Conference on Object-Oriented Programming
Polychotomic Encoding: A Better Quasi-Optimal Bit-Vector Encoding of Tree Hierarchies
ECOOP '02 Proceedings of the 16th European Conference on Object-Oriented Programming
Bit-Vector Encoding for Partially Ordered Sets
ORDAL '94 Proceedings of the International Workshop on Orders, Algorithms, and Applications
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A well-known method to represent a partially ordered set P (order for short) consists in associating to each element of P a subset of a fixed set S = { 1,..., k} such that the order relation coincides with subset inclusion. Such an embedding of P into 2s (the lattice of all subsets of S) is called a bit-vector encoding of P. These encodings provide an interesting way to store an order. They are economical with space and comparisons between elements can be performed efficiently via subset inclusion tests.Given an order P, minimizing the size of the encoding, i.e. the cardinal of S, is however a difficult problem. The smallest size is called the 2-dimension of P and denoted by Dim2(P). In the literature, the decision problem for the 2-dimension has been classified as NP-complete and generating small bit-vector encodings is a challenging issue.Several works deal with bit-vector encodings from a theoretical point of view In this article, we focus on computational complexity results. After a synthesis of known results, we come back on the NP-completeness by detailing a proof and enforcing the conclusion with non-approximability ratios. Besides this general result, we investigate the complexity of the 2-dimension for the class of trees. We describe a 4-approximation algorithm for this class. It uses an optimal balancing strategy which solves a conjecture of Krall, Vitek and Horspool. Several interesting open problems are listed.