Convex Integer Optimization by Constantly Many Linear Counterparts

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

Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions

We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAC-like setting [53]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the 'curvature' of the submodular function, and provide lower and upper bounds that refine and improve previous results [3, 16, 18, 52]. Our proof techniques are fairly generic. We either use a black-box transformation of the function (for approximation and learning), or a transformation of algorithms to use an appropriate surrogate function (for minimization). Curiously, curvature has been known to influence approximations for submodular maximization [7, 55], but its effect on minimization, approximation and learning has hitherto been open. We complete this picture, and also support our theoretical claims by empirical results.Comment: 21 pages. A shorter version appeared in Advances of NIPS-201

Combinatorial Optimization

Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry

Geometric and Algebraic Combinatorics

The 2015 Oberwolfach meeting “Geometric and Algebraic Combinatorics” was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) counterexamples to the topological Tverberg conjecture, and (2) the latest results around the Heron-Rota-Welsh conjecture