On the Convex Hull of Huffman Trees

A well-known kind of binary tree in information theory is Huffman tree. Given any linear objective function f, Huffman has given an algorithm allowing to find an optimal Huffman tree minimizing f. In this paper, we associate to each Huffman point of n nodes, a point in Q n called Huffman point whose components are the length of the path from the root of the tree to each leaf. In this paper, we study the Huffmanhedron, PH n , which is the convex hull of the Huffman points in Q n . In particular, we present a family of facet-defining inequalities for PH n whose coefficients form a Fibonacci sequence. We describe several lifting and composition methods which allow to derive new facet-defining inequalities from existing ones. Finally, we show that these methods together with the Fibonacci family of facet-defining inequalities characterize all facets of non-negative coefficients for PH n containing a deepest Huffman point, i.e. a permutation of the Huffman point ( n − 1 , n − 1 , n − 2 , … , 3 , 2 , 1 ) .