We consider the problem of linearizing a pseudo-Boolean function $f : \{0,1\}^n \to \mathbb{R}$ by means of $k$ Boolean functions. Such a linearization yields an integer linear programming formulation with only $k$ auxiliary variables. This motivates the definition of the linarization complexity of $f$ as the minimum such $k$. Our theoretical contributions are the proof that random polynomials almost surely have a high linearization complexity and characterizations of its value in case we do or do not restrict the set of admissible Boolean functions. The practical relevance is shown by devising and evaluating integer linear programming models of two such linearizations for the low auto-correlation binary sequences problem. Still, many problems around this new concept remain open.
Learning pseudo-Boolean k-DNF and Submodular Functions
Pennsylvania State University（宾夕法尼亚州立大学）
We prove that any submodular function f: {0,1}^n -> {0,1,...,k} can be represented as a pseudo-Boolean 2k-DNF formula. Pseudo-Boolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in such a formula has an associated integral constant. We show that an analog of Hastad's switching lemma holds for pseudo-Boolean k-DNFs if all constants associated with the terms of the formula are bounded. This allows us to generalize Mansour's PAC- learning algorithm for k-DNFs to pseudo-Boolean k-DNFs, and hence gives a PAC- learning algorithm with membership…
Efficient Minimization of Higher Order Submodular Functions using Monotonic Boolean Functions
University of Utah（犹他大学）；Alan Turing Institute（艾伦图灵研究所）；University of Edinburgh（爱丁堡大学）；ETH Zurich（苏黎世联邦理工学院）；University of Oxford（牛津大学）
次模函数最小化是在机器学习、经济学、博弈论、计算机视觉和许多其他领域中的各种应用中的关键问题。通用求解器的复杂度为$O(n^3 \log^2 n . E +n^4 {\log}^{O(1)} n)$，其中$E$是计算函数所需的时间，$n$是变量数\cite{Lee2015}。另一方面，许多计算机视觉和机器学习问题是在子模函数的特殊子类上定义的，这些子模函数可以被写为在包含很少变量的团上定义的许多子模代价函数的总和。在这样的函数中，这些子类的伪布尔（或多项式）表示\cite{BorosH02}具有度（或阶数，或团大小）$k$，其中$k \ll n$。在这项工作中，我们开发了有效的算法最小化这个有用的子类的次模函数。为了做到这一点，我们定义了新的映射，转换子模函数的顺序$k$到二次。其基本思想是使用辅助变量来模拟高阶项，并且使用精心构造的线性规划来找到变换。特别是，我们建模的辅助变量作为单调布尔函数，使我们能够获得一个紧凑的变换使用尽可能少的辅助变量。
