Convex Integer Optimization by Constantly Many Linear Counterparts

TL;DR

This paper introduces the concept of edge complexity to establish bounds on the number of vertices and linear counterparts needed to solve convex integer maximization problems with composite functions, leading to efficient solutions for specific combinatorial structures.

Contribution

It defines edge complexity of finite sets and uses it to derive polynomial bounds on the number of vertices and linear solutions for convex integer problems, extending previous research.

Findings

Bounded the number of vertices of projected sets for matroids and transportation polytopes.

Reduced convex problems to a fixed number of linear optimization problems.

Provided polynomial-time solutions for specific convex integer maximization problems.

Abstract

In this article we study convex integer maximization problems with composite objective functions of the form $f(Wx)$, where $f$ is a convex function on $\R^d$ and $W$ is a $d\times n$ matrix with small or binary entries, over finite sets $S\subset \Z^n$ of integer points presented by an oracle or by linear inequalities. Continuing the line of research advanced by Uri Rothblum and his colleagues on edge-directions, we introduce here the notion of {\em edge complexity} of $S$, and use it to establish polynomial and constant upper bounds on the number of vertices of the projection $\conv(WS)$ and on the number of linear optimization counterparts needed to solve the above convex problem. Two typical consequences are the following. First, for any $d$, there is a constant $m(d)$ such that the maximum number of vertices of the projection of any matroid $S\subset\{0,1\}^n$ by any binary $d\times n$ matrix $W$ is $m(d)$ regardless of $n$ and $S$; and the convex matroid problem reduces to $m(d)$ greedily solvable linear counterparts. In particular, $m(2)=8$. Second, for any $d,l,m$, there is a constant $t(d;l,m)$ such that the maximum number of vertices of the projection of any three-index $l\times m\times n$ transportation polytope for any $n$ by any binary $d\times(l\times m\times n)$ matrix $W$ is $t(d;l,m)$; and the convex three-index transportation problem reduces to $t(d;l,m)$ linear counterparts solvable in polynomial time.