MATHEMATICAL ENGINEERING TECHNICAL REPORTS A Proof of the M-Convex Intersection Theorem

This short note gives an alternative proof of the M-convex intersection theorem, which is one of the central results in discrete convex analysis. This note is intended to provide a direct simpler proof accessible to nonexperts. 1 M-Convex Intersection Theorem The M-convex intersection theorem [3, Theorem 8.17] reads as follows, where V is a nonempty finite set, and Z and R are the sets of integers and reals, respectively; see §3 for the definitions of M\-convex functions and notation arg min. This theorem is equivalent to the M-separation theorem, to the Fenchel-type min-max duality theorem, and to an optimality criterion of the M-convex submodular flow problem. Theorem 1 (M-convex intersection theorem). For M\-convex functions f1, f2 and a point x∗ ∈ domf1 ∩ domf2 we have f1(x∗) + f2(x∗) ≤ f1(x) + f2(x) (∀x ∈ Z ) (1) if and only if there exists p∗ ∈ RV such that1 f1[−p∗](x∗) ≤ f1[−p∗](x) (∀x ∈ Z ), (2) f2[+p∗](x∗) ≤ f2[+p∗](x) (∀x ∈ Z ). (3) For such p∗ we have arg min(f1 + f2) = arg minf1[−p∗] ∩ arg minf2[+p∗]. (4) Moreover, if f1 and f2 are integer-valued, we can choose integer-valued p∗ ∈ ZV . We shall give a constructive proof of Theorem 1 based on the successive shortest path algorithm. Different proofs available in [3] are: ∗Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan E-mail: [email protected] Notation: f1[−p](x) = f1(x)− X v∈V p∗(v)x(v), f2[+p ∗](x) = f2(x) + X