最小化次模函数的组合强多项式时间算法

次模函数与完美图

Submodular functions and perfect graphs

本文提出了一种组合多项式时间算法，在不含棱柱或长度为4的洞且度数有界的完美图中，利用偶集构成的平衡分离子和次模函数最小化算法求解最大权独立集问题。

次模集合函数的单调分解

Monotonic Decompositions of Submodular Set Functions

本文将有限次模集合函数的单调分解结果推广到分析框架，引入无穷交替集合函数新类，证明其可分解为递增与递减次模函数之和或两个递增次模函数之差，并研究了无向图加权割函数的分解结构。

次模函数、强次模函数与多样性测度

Submodular and strongly submodular functions and diversities

本文以多样性测度为桥梁，探索次模函数、强次模函数与度量理论的联系，定义次模多样性测度并证明其几何嵌入结果，表明次模函数等可由广义外接球半径表示，丰富了相关领域的理论联系。

Submodular Minimization Under Congruency Constraints

Submodular function minimization (SFM) is a fundamental and efficiently solvable problem class in combinatorial optimization with a multitude of applications in various fields. Surprisingly, there is only very little known about constraint types under which SFM remains efficiently solvable. The arguably most relevant non-trivial constraint class for which polynomial SFM algorithms are known are parity constraints, i.e., optimizing only over sets of odd (or even) cardinality. Parity constraints capture classical combinatorial optimization problems like the odd-cut problem, and they are a key to…

Distributed Submodular Minimization And Motion Coordination Over Discrete State Space

We develop a framework for the distributed minimization of submodular functions. Submodular functions are a discrete analog of convex functions and are extensively used in large-scale combinatorial optimization problems. While there has been significant interest in the distributed formulations of convex optimization problems, distributed minimization of submodular functions has received relatively little research attention. Our framework relies on an equivalent convex reformulation of a submodular minimization problem, which is efficiently computable. We then use this relaxation to exploit met…

更快地最小化对称集函数

Minimizing Symmetric Set Functions Faster

我们描述了一个组合算法，给定有限集V上Rizzi意义下的单调一致对称集函数d，构造一个非平凡集S，使d（S，V-S）最小化。这包括对称次模函数的最小化的可能性。所提出的算法最多需要与Rizzi所描述的算法一样多的时间，但取决于函数d，它可以允许若干改进。

Weakly Submodular Functions

Submodular functions are well-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call {\em weakly submodular functions}. Our extension includes some (mildly) supermodular functions. We show that several natural functions belong to this class and relate our class to some other recent submodular function extensions. We consider the optimization problem of maximizing a weakly submodular function subject to uniform and general matroid c…

显式熵构造用于覆盖、设施选址和图割

Explicit Entropic Constructions for Coverage, Facility Location, and Graph Cuts

本文通过显式熵构造证明了实践中常用的子模函数如覆盖、设施选址和图割目标可精确表示为香农熵，建立了组合信息度量与经典信息论的联系。

Independence Properties of Generalized Submodular Information Measures

Recently a class of generalized information measures was defined on sets of items parametrized by submodular functions. In this paper, we propose and study various notions of independence between sets with respect to such information measures, and connections thereof. Since entropy can also be used to parametrize such measures, we derive interesting independence properties for the entropy of sets of random variables. We also study the notion of multi-set independence and its properties. Finally, we present optimization algorithms for obtaining a set that is independent of another given set, an…

Performance guarantees of forward and reverse greedy algorithms for minimizing nonsupermodular nonsubmodular functions on a matroid

This letter studies the problem of minimizing increasing set functions, or equivalently, maximizing decreasing set functions, over the base of a matroid. This setting has received great interest, since it generalizes several applied problems including actuator and sensor placement problems in control theory, multi-robot task allocation problems, video summarization, and many others. We study two greedy heuristics, namely, the forward and the reverse greedy algorithms. We provide two novel performance guarantees for the approximate solutions obtained by these heuristics depending on both the su…

包含谱函数的最大次模覆盖问题的近似及其在试验设计中的应用

Approximation of a Maximum-Submodular-Coverage problem involving spectral functions, with application to Experimental Design

本文研究一类由参数$p\in[0,1]$定义的组合最优化问题，它涉及到应用正半定矩阵的谱函数，在最优试验设计理论中有一定的应用.当$p$变为0时，这类问题往往是经典最大覆盖问题的一般化，而当$p$变为1时，这类问题往往是背包问题的平凡实例。在这篇文章中，我们建立了一个矩阵不等式，它表明，目标函数是次模的所有$p\in[0,1]$，从它遵循的贪婪的方法，这是经常用于这个问题，总是给一个设计内的最佳$1-1/e$。接下来，我们研究的设计发现，通过四舍五入的解决方案的连续放松的问题，一种方法，已被应用的几个作者。我们证明了一个不等式，它概括了经典的结果，从理论的最优设计，并允许我们给一个舍入过程的近似因子趋于1的$p$去1。

尝试不可能之事：枚举 n=6 时的极值次模函数

Attempting the impossible: enumerating extremal submodular functions for n=6

本文针对六元素基集上的极值次模函数枚举难题，提出两种新的不等式插入顺序以加速 Double Description 和 Adjacency Decomposition 方法，计算出 3600 亿个函数并首次给出总数估计。

Parallel Quasi-concave set optimization: A new frontier that scales without needing submodularity

Classes of set functions along with a choice of ground set are a bedrock to determine and develop corresponding variants of greedy algorithms to obtain efficient solutions for combinatorial optimization problems. The class of approximate constrained submodular optimization has seen huge advances at the intersection of good computational efficiency, versatility and approximation guarantees while exact solutions for unconstrained submodular optimization are NP-hard. What is an alternative to situations when submodularity does not hold? Can efficient and globally exact solutions be obtained? We i…