Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1708.04903

Online Primal-Dual Algorithms with Configuration Linear Programs

Nguyen Kim Thang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Non-linear, especially convex, objective functions have been extensively studied in recent years in which approaches relies crucially on the convexity property of cost functions. In this paper, we present primal-dual approaches based on configuration linear programs to design competitive online algorithms for problems with arbitrarily-grown objective. This approach is particularly appropriate for non-linear (non-convex) objectives in online setting. We first present a simple greedy algorithm for a general cost-minimization problem. The competitive ratio of the algorithm is characterized by the mean of a notion, called smoothness, which is inspired by a similar concept in the context of algorithmic game theory. The algorithm gives optimal (up to a constant factor) competitive ratios while applying to different contexts such as network routing, vector scheduling, energy-efficient scheduling and non-convex facility location. Next, we consider the online $0-1$ covering problems with non-convex objective. Building upon the resilient ideas from the primal-dual framework with configuration LPs, we derive a competitive algorithm for these problems. Our result generalizes the online primal-dual algorithm developed recently by Azar et al. for convex objectives with monotone gradients to non-convex objectives. The competitive ratio is now characterized by a new concept, called local smoothness --- a notion inspired by the smoothness. Our algorithm yields tight competitive ratio for the objectives such as the sum of $\ell_{k}$-norms and gives competitive solutions for online problems of submodular minimization and some natural non-convex minimization under covering constraints.

Explore related subjects

cs.DScs.DM

Keep this discovery

BibTeXRIS

Nguyen Kim Thang. 2017-08-16. Online Primal-Dual Algorithms with Configuration Linear Programs. https://arxiv.org/abs/1708.04903

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↗