Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2009.04114

Adwords in a Panorama

Zhiyi Huang·Qiankun Zhang·Yuhao Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Three decades ago, Karp, Vazirani, and Vazirani (STOC 1990) defined the online matching problem and gave an optimal $1-\frac{1}{e} \approx 0.632$-competitive algorithm. Fifteen years later, Mehta, Saberi, Vazirani, and Vazirani (FOCS 2005) introduced the first generalization called AdWords driven by online advertising and obtained the optimal $1-\frac{1}{e}$ competitive ratio in the special case of small bids. It has been open ever since whether there is an algorithm for general bids better than the $0.5$-competitive greedy algorithm. This paper presents a $0.5016$-competitive algorithm for AdWords, answering this open question on the positive end. The algorithm builds on several ingredients, including a combination of the online primal dual framework and the configuration linear program of matching problems recently explored by Huang and Zhang (STOC 2020), a novel formulation of AdWords which we call the panorama view, and a generalization of the online correlated selection by Fahrbach, Huang, Tao, and Zadimorghaddam (FOCS 2020) which we call the panoramic online correlated selection.

Explore related subjects

cs.DScs.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhiyi Huang, Qiankun Zhang, Yuhao Zhang. 2020-11-27. Adwords in a Panorama. https://arxiv.org/abs/2009.04114

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

KEEP EXPLORING

Related papers

arXiv2026

Computational complexity of the recoverable robust shortest path problem in acyclic digraphs under interval budgeted uncertainty

In this paper, we consider the recoverable robust shortest path problem in acyclic digraphs, employing interval budgeted uncertainty to model uncertain second-stage costs. For the continuous budgeted uncertainty model, we prove that the problem is strongly NP-hard even in layered digraphs. Furthermore, we show that in general acyclic digraphs, the problem cannot be approximated within any constant factor unless $\mathrm{P} = \mathrm{NP}$, nor can it be approximated within a factor of $2^{\log^{1-ε} n}$ for any $ε> 0$ unless $\mathrm{NP} \subseteq \mathrm{DTIME}(n^{\mathrm{poly} \log n})$. For the discrete budgeted uncertainty model, we show that the problem is not approximable unless $\mathrm{P} = \mathrm{NP}$, even in layered digraphs. Finally, we establish that under continuous budgeted uncertainty, the integrality gap of a relaxation allowing a fractional first-stage solution is at least $Ω(\sqrt{n})$.

cs.DS↗
arXiv2026

Odd and Even Harder Problems on Cycle-Factors

For a graph (undirected or directed), a cycle-factor is a collection of vertex-disjoint cycles covering the entire vertex set. Cycle-factors subject to parity constraints arise naturally in the study of structural graph theory and algorithmic complexity. In this work, we study four variants of the problem of finding a cycle-factor subject to the following parity constraints: (1) all cycles are odd, (2) all cycles are even, (3) at least one cycle is odd, and (4) at least one cycle is even. We show that all the variants of the problem are NP-complete both in undirected and directed graphs, even if each vertex is incident to at most three edges. We also prove that the first two variants are NP-complete even for planar directed graphs.

cs.DS↗
arXiv2026

An $n^{8/5+o(1)}$-Time $Ω(λ^3)$-Approximation for Longest Common Subsequence

Let $λ$ denote the ratio of the length of a longest common subsequence of two length-$n$ strings to $n$. Rubinstein, Seddighin, Song and Sun [RSSS19] gave an $Ω(λ^3)$-approximation for LCS running in $\widetilde O(n^{39/20})$ time, where $39/20=1.95$. Song [Son19] mentioned that improving the $n^{1.95}$ running time is an interesting open question. We give an algorithm that computes an $Ω(λ^3)$-approximation of the longest common subsequence in $n^{8/5+o(1)}$ time. This improves the exponent $1.95$ to $1.6+o(1)$.

cs.DS↗