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On k-Submodular Relaxation

$k$-submodular functions, introduced by Huber and Kolmogorov, are functions defined on $\{0, 1, 2, \dots, k\}^n$ 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 fixed-parameter tractable algorithms. Motivated by this, we consider the following problem: Given a function $f : \{1, 2, \dots, k\}^n \rightarrow \mathbb{R} \cup \{\infty\}$, determine whether $f$ is extended to a $k$-submodular function $g : \{0, 1, 2, \dots, k\}^n \rightarrow \mathbb{R} \cup \{\infty\}$, 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 $O((k^n)^2)$-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.