We introduce a new class of functions that can be minimized in polynomial time in the value oracle model. These are functions @math satisfying @math where the domain of each variable @math corresponds to nodes of a rooted binary tree, and operations @math are defined with respect to this tree. Special cases include previously studied @math -convex and bisubmodular functions, which can be obtained with particular choices of trees. We present a polynomial-time algorithm for minimizing functions in the new class. It combines Murota's steepest descent algorithm for @math -convex functions with bisubmodular minimization algorithms.
Combining multimorphisms Finally, we mention that some constructions, namely Cartesian products and Malt'stev products , can be used for obtaining new tractable classes of binary multimoprhisms from existing ones @cite_5 . Note, Krokhin and Larose @cite_5 formulated these constructions only for lattice multimorphisms @math , but the proof in @cite_5 actually applies to arbitrary binary multimorphisms @math .
@cite_5
2963192682
A novel construction of lattices is proposed. This construction can be thought of as Construction A with codes that can be represented as the Cartesian product of L linear codes over Fp1 , . . . ,FpL , respectively; hence, is referred to as the product construction. The existence of a sequence of such lattices that are good for quantization and Poltyrev-good under multistage decoding is shown. This family of lattices is then used to generate a sequence of nested lattice codes which allows one to achieve the same computation rate of Nazer and Gastpar for computeand-forward under multistage decoding, which is referred to as lattice-based multistage compute-and-forward. Motivated by the proposed lattice codes, two families of signal constellations are then proposed for the separation-based compute-and-forward framework proposed by together with a multilevel coding multistage decoding scheme tailored specifically for these constellations. This scheme is termed separation-based multistage compute-and-forward and is shown having a complexity of the channel coding dominated by the greatest common divisor of the constellation size (may not be a prime number) instead of the constellation size itself.