Generalized minimum 0-extension problem and discrete convexity

TL;DR

This work provides a complete tractability dichotomy for generalized minimum $0$-extension problems on finite metrics by introducing extended modular complexes and an extended notion of $L$-convexity. It extends Hirai's framework to include $F$-extensions with 2-element subsets, yielding explicit polynomial-time criteria via $F$-orientability modularity and enabling a direct $\diamond$-Steepest Descent Algorithm that avoids graph blow-ups. The authors also connect these problems to VCSP theory, showing that Basic LP relaxations suffice with binary symmetric fractional polymorphisms, and they improve the complexity bounds for Hirai’s original algorithm. Collectively, the results unify submodular, $L$-natural, and related discrete-convex function classes under a common tractable framework with practical implications for facility location, clustering, and metric labeling problems.

Abstract

Given a fixed finite metric space $(V,μ)$, the {\em minimum $0$-extension problem}, denoted as ${\tt 0\mbox{-}Ext}[μ]$, is equivalent to the following optimization problem: minimize function of the form $\min\limits_{x\in V^n} \sum_i f_i(x_i) + \sum_{ij}c_{ij}μ(x_i,x_j)$ where $c_{ij},c_{vi}$ are given nonnegative costs and $f_i:V\rightarrow \mathbb R$ are functions given by $f_i(x_i)=\sum_{v\in V}c_{vi}μ(x_i,v)$. The computational complexity of ${\tt 0\mbox{-}Ext}[μ]$ has been recently established by Karzanov and by Hirai: if metric $μ$ is {\em orientable modular} then ${\tt 0\mbox{-}Ext}[μ]$ can be solved in polynomial time, otherwise ${\tt 0\mbox{-}Ext}[μ]$ is NP-hard. To prove the tractability part, Hirai developed a theory of discrete convex functions on orientable modular graphs generalizing several known classes of functions in discrete convex analysis, such as $L^\natural$-convex functions. We consider a more general version of the problem in which unary functions $f_i(x_i)$ can additionally have terms of the form $c_{uv;i}μ(x_i,\{u,v\})$ for $\{u,v\}\in F$, where set $F\subseteq\binom{V}{2}$ is fixed. We extend the complexity classification above by providing an explicit condition on $(μ,F)$ for the problem to be tractable. In order to prove the tractability part, we generalize Hirai's theory and define a larger class of discrete convex functions. It covers, in particular, another well-known class of functions, namely submodular functions on an integer lattice. Finally, we improve the complexity of Hirai's algorithm for solving ${\tt 0\mbox{-}Ext}[μ]$ on orientable modular graphs.

Figures (2)

• Figure 1: Different 2-subdivisions for the chain $a\prec b\prec c \prec d$.
• Figure 2: (a) Each point in $I(p,q)$ is assigned a coordinate in $\mathbb R^2$. The convex hull of these coordinates ($\mathop{\rm Conv} I(p,q)$) is in gray. Distinct points may have the same coordinates (as some of the points shown in the interior of the gray region), but the coordinates of points in ${\cal E}(p,q)=\{u_0,u_1,\ldots,u_k\}$ are guaranteed to be unique. (b) Definition of $\{\theta_i\}_i$ and $p\vee_\theta q$. First, define points $\alpha_{-1},\alpha_0,\ldots,\alpha_k$ in $\mathbb R^2$ as follows: set $\alpha_{-1}=(\sqrt{2}/2,0)$, $\alpha_{k}=(0,\sqrt{2}/2)$ (so that $||\alpha_k-\alpha_{-1}||=1$), and for $i\in[k-1]$ let $\alpha_i$ be the intersection of segment $[\alpha_{-1},\alpha_k]$ and the line that goes through the origin and is perpendicular to the line passing through points $v_{pq}(u_{k})$ and $v_{pq}(u_{k+1})$. Then $\theta_i=||\alpha_i-\alpha_{-1}||$ and $p\wedge_\theta q=u_i$ for each $i\in[0,k]$ and $\theta\in(\theta_{i-1},\theta_i)$. (c) Bounded pair $(p,q)$. (d) Antipodal pair $(p,q)$.

Theorems & Definitions (133)

• Theorem 1: Karzanov:98
• Theorem 2: Hirai:0ext
• Theorem 3
• Theorem 4: kolmogorov15:power
• Theorem 5: tz16:jacm
• Theorem 6: Bandelt:85Karzanov:04
• Lemma 7: Hirai:0ext
• Lemma 8: Hirai:0ext
• Theorem 9: Hirai:0ext;Hirai:Lconvexity
• Theorem 10: Hirai:0ext