On k-Submodular Relaxation

k-submodular functions, introduced by Huber and Kolmogorov, are functions defined on {0, 1, 2, . . . , k}n satisfying certain submodular-type inequalities. k-submodular functions typically arise as relaxations of NP-hard problems, and the relaxations by k-submodular functions play key roles in design of efficient, approximation, or FPT algorithms. Motivated by this, we consider the following problem: Given a function f : {1, 2, . . . , k}n → R ∪ {∞}, determine whether f is extended to a k-submodular function g : {0, 1, 2, . . . , k}n → R∪{∞}, where g is called a k-submodular relaxation of f . We give a polymorphic characterization of those functions which admit a k-submodular relaxation, and also give a combinatorial O((k)) time algorithm to find a k-submodular relaxation or establish that a k-submodular relaxation does not exist. Our algorithm has interesting properties: (1) if the input function is integer-valued, then our algorithm outputs a half-integral relaxation, and (2) if the input function is binary, then our algorithm outputs the unique optimal relaxation. We present applications of our algorithm to valued constraint satisfaction problems.