L-convexity on graph structures

Abstract

In this paper, we study classes of discrete convex functions: submodular functions on modular semilattices and L-convex functions on oriented modular graphs. They were introduced by the author in complexity classification of minimum 0-extension problems. We clarify the relationship to other discrete convex functions, such as $k$-submodular functions, skew-bisubmodular functions, L$^\natural$-convex functions, tree submodular functions, and UJ-convex functions. We show that they are actually viewed as submodular/L-convex functions in our sense. We also prove a sharp iteration bound of the steepest descent algorithm for minimizing our L-convex functions. The underlying structures, modular semilattices and oriented modular graphs, have rich connections to projective and polar spaces, Euclidean building, and metric spaces of global nonpositive curvature (CAT(0) spaces). By utilizing these connections, we formulate an analogue of the Lov\'asz extension, introduce well-behaved subclasses of submodular/L-convex functions, and show that these classes can be characterized by the convexity of the Lov\'asz extension. We demonstrate applications of our theory to combinatorial optimization problems that include multicommodity flow, multiway cut, and related labeling problems: these problems have been outside the scope of discrete convex analysis so far.