Related papers: Submodular Goal Value of Boolean Functions
Classification is a core topic in functional data analysis. A large number of functional classifiers have been proposed in the literature, most of which are based on functional principal component analysis or functional regression. In…
Maximizing a single submodular set function subject to a cardinality constraint is a well-studied and central topic in combinatorial optimization. However, finding a set that maximizes multiple functions at the same time is much less…
In this paper we simplify the definition of the weighted sum Boolean function which used to be inconvenient to compute and use. We show that the new function has essentially the same properties as the previous one. In particular, the bound…
The approximate degree of a Boolean function is the least degree of a real multilinear polynomial approximating it in the $\ell_\infty$-norm over the Boolean hypercube. We show that the approximate degree of the Bipartite Perfect Matching…
Submodularity is an important property of set functions and has been extensively studied in the literature. It models set functions that exhibit a diminishing returns property, where the marginal value of adding an element to a set…
In this paper we introduce the concept of modulus of regularity as a tool to analyze the speed of convergence, including the finite termination, for classes of Fej\'er monotone sequences which appear in fixed point theory, monotone operator…
We study the problem of {\sl certification}: given queries to a function $f : \{0,1\}^n \to \{0,1\}$ with certificate complexity $\le k$ and an input $x^\star$, output a size-$k$ certificate for $f$'s value on $x^\star$. This abstractly…
Submodular functions are at the core of many machine learning and data mining tasks. The underlying submodular functions for many of these tasks are decomposable, i.e., they are sum of several simple submodular functions. In many data…
We show a partial Boolean function $f$ together with an input $x\in f^{-1}\left(*\right)$ such that both $C_{\bar{0}}\left(f,x\right)$ and $C_{\bar{1}}\left(f,x\right)$ are at least $C\left(f\right)^{2-o\left(1\right)}$. Due to recent…
Horn functions form a subclass of Boolean functions and appear in many different areas of computer science and mathematics as a general tool to describe implications and dependencies. Finding minimum sized representations for such functions…
Block sensitivity ($bs(f)$), certificate complexity ($C(f)$) and fractional certificate complexity ($C^*(f)$) are three fundamental combinatorial measures of complexity of a boolean function $f$. It has long been known that $bs(f) \leq…
The maximum (or minimum) generalized eigenvalue of symmetric positive semidefinite matrices that depend on optimization variables often appears as objective or constraint functions in structural topology optimization when we consider…
Submodular functions -- functions exhibiting diminishing returns -- are central to machine learning. When the objective is monotone and non-negative, the greedy algorithm achieves a tight $63\%$ approximation. But many practical objectives…
The norm of the gradient $\nabla$f (x) measures the maximum descent of a real-valued smooth function f at x. For (nonsmooth) convex functions, this is expressed by the distance dist(0, $\partial$f (x)) of the subdifferential to the origin,…
Shapley values, originating in game theory and increasingly prominent in explainable AI, have been proposed to assess the contribution of facts in query answering over databases, along with other similar power indices such as Banzhaf…
The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically…
Stochastic Boolean Function Evaluation (SBFE) is the problem of determining the value of a given Boolean function $f$ on an unknown input $x$, when each bit of $x_i$ of $x$ can only be determined by paying a given associated cost $c_i$.…
Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In…
We consider the problem of allocating indivisible resources to players so as to maximize the minimum total value any player receives. This problem is sometimes dubbed the Santa Claus problem and its different variants have been subject to…
Submodular set-functions have many applications in combinatorial optimization, as they can be minimized and approximately maximized in polynomial time. A key element in many of the algorithms and analyses is the possibility of extending the…