Searching for network modules

When analyzing complex networks a key target is to uncover their modular structure, which means searching for a family of modules, namely node subsets spanning each a subnetwork more densely connected than the average. This work proposes a novel type of objective function for graph clustering, in the form of a multilinear polynomial whose coefficients are determined by network topology. It may be thought of as a potential function, to be maximized, taking its values on fuzzy clusterings or families of fuzzy subsets of nodes over which every node distributes a unit membership. When suitably parametrized, this potential is shown to attain its maximum when every node concentrates its all unit membership on some module. The output thus is a partition, while the original discrete optimization problem is turned into a continuous version allowing to conceive alternative search strategies. The instance of the problem being a pseudo-Boolean function assigning real-valued cluster scores to node subsets, modularity maximization is employed to exemplify a so-called quadratic form, in that the scores of singletons and pairs also fully determine the scores of larger clusters, while the resulting multilinear polynomial potential function has degree 2. After considering further quadratic instances, different from modularity and obtained by interpreting network topology in alternative manners, a greedy local-search strategy for the continuous framework is analytically compared with an existing greedy agglomerative procedure for the discrete case. Overlapping is finally discussed in terms of multiple runs, i.e. several local searches with different initializations.Comment: 10 page

Submodular Function Maximization for Group Elevator Scheduling

We propose a novel approach for group elevator scheduling by formulating it as the maximization of submodular function under a matroid constraint. In particular, we propose to model the total waiting time of passengers using a quadratic Boolean function. The unary and pairwise terms in the function denote the waiting time for single and pairwise allocation of passengers to elevators, respectively. We show that this objective function is submodular. The matroid constraints ensure that every passenger is allocated to exactly one elevator. We use a greedy algorithm to maximize the submodular objective function, and derive provable guarantees on the optimality of the solution. We tested our algorithm using Elevate 8, a commercial-grade elevator simulator that allows simulation with a wide range of elevator settings. We achieve significant improvement over the existing algorithms.Comment: 10 pages; 2017 International Conference on Automated Planning and Scheduling (ICAPS

Hypercontractive Inequality for Pseudo-Boolean Functions of Bounded Fourier Width

A function $f:\ \{-1,1\}^n\rightarrow \mathbb{R}$ is called pseudo-Boolean. It is well-known that each pseudo-Boolean function $f$ can be written as $f(x)=\sum_{I\in {\cal F}}\hat{f}(I)\chi_I(x),$ where ${\cal F}\subseteq \{I:\ I\subseteq [n]\}$,$[n]=\{1,2,...,n\}$, and$\chi_I(x)=\prod_{i\in I}x_i$and$\hat{f}(I)$are non-zero reals. The degree of$f$is$\max \{|I|:\ I\in {\cal F}\}$and the width of$f$is the minimum integer$\rho$such that every$i\in [n]$appears in at most$\rho$sets in$\cal F$. For$i\in [n]$, let$\mathbf{x}_i$be a random variable taking values 1 or -1 uniformly and independently from all other variables$\mathbf{x}_j$,$j\neq i.$Let$\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_n)$. The$p$-norm of$f$is$||f||_p=(\mathbb E[|f(\mathbf{x})|^p])^{1/p}$for any$p\ge 1$. It is well-known that$||f||_q\ge ||f||_p$whenever$q> p\ge 1$. However, the higher norm can be bounded by the lower norm times a coefficient not directly depending on$f$: if$f$is of degree$d$and$q> p>1$then$ ||f||_q\le (\frac{q-1}{p-1})^{d/2}||f||_p.$This inequality is called the Hypercontractive Inequality. We show that one can replace$d$by$\rho$in the Hypercontractive Inequality for each$q> p\ge 2$as follows:$ ||f||_q\le ((2r)!\rho^{r-1})^{1/(2r)}||f||_p,$where$r=\lceil q/2\rceil$. For the case$q=4$and$p=2$, which is important in many applications, we prove a stronger inequality:$ ||f||_4\le (2\rho+1)^{1/4}||f||_2.

Quadratization of Symmetric Pseudo-Boolean Functions

A pseudo-Boolean function is a real-valued function $f(x)=f(x_1,x_2,\ldots,x_n)$ of $n$ binary variables; that is, a mapping from $\{0,1\}^n$ to $\mathbb{R}$. For a pseudo-Boolean function $f(x)$ on $\{0,1\}^n$, we say that $g(x,y)$ is a quadratization of $f$ if $g(x,y)$ is a quadratic polynomial depending on $x$ and on $m$ auxiliary binary variables $y_1,y_2,\ldots,y_m$ such that $f(x)= \min \{g(x,y) : y \in \{0,1\}^m \}$ for all $x \in \{0,1\}^n$. By means of quadratizations, minimization of $f$ is reduced to minimization (over its extended set of variables) of the quadratic function $g(x,y)$. This is of some practical interest because minimization of quadratic functions has been thoroughly studied for the last few decades, and much progress has been made in solving such problems exactly or heuristically. A related paper \cite{ABCG} initiated a systematic study of the minimum number of auxiliary $y$-variables required in a quadratization of an arbitrary function $f$ (a natural question, since the complexity of minimizing the quadratic function $g(x,y)$ depends, among other factors, on the number of binary variables). In this paper, we determine more precisely the number of auxiliary variables required by quadratizations of symmetric pseudo-Boolean functions $f(x)$, those functions whose value depends only on the Hamming weight of the input $x$ (the number of variables equal to $1$).Comment: 17 page

Particle algorithms for optimization on binary spaces

We discuss a unified approach to stochastic optimization of pseudo-Boolean objective functions based on particle methods, including the cross-entropy method and simulated annealing as special cases. We point out the need for auxiliary sampling distributions, that is parametric families on binary spaces, which are able to reproduce complex dependency structures, and illustrate their usefulness in our numerical experiments. We provide numerical evidence that particle-driven optimization algorithms based on parametric families yield superior results on strongly multi-modal optimization problems while local search heuristics outperform them on easier problems