Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1807.05529

Online Submodular Maximization: Beating 1/2 Made Simple

Niv Buchbinder·Moran Feldman·Yuval Filmus·Mohit Garg
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Submodular Welfare Maximization problem (SWM) captures an important subclass of combinatorial auctions and has been studied extensively from both computational and economic perspectives. In particular, it has been studied in a natural online setting in which items arrive one-by-one and should be allocated irrevocably upon arrival. In this setting, it is well known that the greedy algorithm achieves a competitive ratio of 1/2, and recently Kapralov et al. (2013) showed that this ratio is optimal for the problem. Surprisingly, despite this impossibility result, Korula et al. (2015) were able to show that the same algorithm is 0.5052-competitive when the items arrive in a uniformly random order, but unfortunately, their proof is very long and involved. In this work, we present an (arguably) much simpler analysis that provides a slightly better guarantee of 0.5096-competitiveness for the greedy algorithm in the random-arrival model. Moreover, this analysis applies also to a generalization of online SWM in which the sets defining a (simple) partition matroid arrive online in a uniformly random order, and we would like to maximize a monotone submodular function subject to this matroid. Furthermore, for this more general problem, we prove an upper bound of 0.576 on the competitive ratio of the greedy algorithm, ruling out the possibility that the competitiveness of this natural algorithm matches the optimal offline approximation ratio of 1-1/e.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Niv Buchbinder, Moran Feldman, Yuval Filmus, Mohit Garg. 2018-11-19. Online Submodular Maximization: Beating 1/2 Made Simple. https://arxiv.org/abs/1807.05529

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

KEEP EXPLORING

Related papers

arXiv2026

An efficient implementation for solving the all pairs minimax path problem in an undirected dense graph

We provide an efficient $ O(n^2) $ implementation for solving the all pairs minimax path problem or widest path problem in an undirected dense graph. It is a code implementation of the Algorithm 4 (MMJ distance by Calculation and Copy) in a previous paper. The distance matrix is also called the all points path distance (APPD). We conducted experiments to test the implementation and algorithm, compared it with several other algorithms for solving the APPD matrix. Result shows Algorithm 4 works good for solving the widest path or minimax path APPD matrix. It can drastically improve the efficiency for computing the APPD matrix. There are several theoretical outcomes which claim the APPD matrix can be solved accurately in $ O(n^2) $ . However, they are impractical because there is no code implementation of these algorithms. It seems Algorithm 4 is the first algorithm that has an actual code implementation for solving the APPD matrix of minimax path or widest path problem in $ O(n^2) $, in an undirected dense graph.

cs.DS↗
arXiv2026

Approximating Partition in Deterministic Near-Linear Time

We propose a deterministic $\widetilde{O}(n + \frac{1}ε)$-time FPTAS (Fully Polynomial-Time Approximation Scheme) for the classical Partition problem. This is the best possible (up to a polylogarithmic factor) assuming SETH (Strong Exponential Time Hypothesis) [Abboud, Bringmann, Hermelin, and Shabtay'22]. Prior to our work, the best known FPTAS for Partition runs in $\widetilde{O}(n + (\frac{1}ε)^{5/4})$ time [Deng, Jin and Mao'23, Wu and Chen'22]. Our result is obtained by solving a more general problem of weakly approximating Subset Sum.

cs.DS↗
arXiv2026

Dynamic Treewidth in Logarithmic Time

We present a dynamic data structure that maintains a tree decomposition of width at most $9k+8$ of a dynamic graph with treewidth at most $k$, which is updated by edge insertions and deletions. The amortized update time of our data structure is $2^{O(k)} \log n$, where $n$ is the number of vertices. The data structure also supports maintaining any ``dynamic programming scheme'' on the tree decomposition, providing, for example, a dynamic version of Courcelle's theorem with $O_{k}(\log n)$ amortized update time; the $O_{k}(\cdot)$ notation hides factors that depend on $k$. This improves upon a result of Korhonen, Majewski, Nadara, Pilipczuk, and Sokołowski [FOCS 2023], who gave a similar data structure but with amortized update time $2^{k^{O(1)}} n^{o(1)}$. Furthermore, our data structure is arguably simpler. Our main novel idea is to maintain a tree decomposition that is ``downwards well-linked'', which allows us to implement local rotations and analysis similar to those for splay trees.

cs.DS↗