Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2207.04957

Submodular Dominance and Applications

Frederick Qiu·Sahil Singla
View original source ↗Read source document ↗Export metadata ↓

Abstract

In submodular optimization we often deal with the expected value of a submodular function $f$ on a distribution $\mathcal{D}$ over sets of elements. In this work we study such submodular expectations for negatively dependent distributions. We introduce a natural notion of negative dependence, which we call Weak Negative Regression (WNR), that generalizes both Negative Association and Negative Regression. We observe that WNR distributions satisfy Submodular Dominance, whereby the expected value of $f$ under $\mathcal{D}$ is at least the expected value of $f$ under a product distribution with the same element-marginals. Next, we give several applications of Submodular Dominance to submodular optimization. In particular, we improve the best known submodular prophet inequalities, we develop new rounding techniques for polytopes of set systems that admit negatively dependent distributions, and we prove existence of contention resolution schemes for WNR distributions.

Explore related subjects

cs.DScs.GT

Keep this discovery

BibTeXRIS

Frederick Qiu, Sahil Singla. 2022-07-11. Submodular Dominance and Applications. https://arxiv.org/abs/2207.04957

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗