Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1804.07458

Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals

Zhiyi Huang·Zhihao Gavin Tang·Xiaowei Wu·Yuhao Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhiyi Huang, Zhihao Gavin Tang, Xiaowei Wu, Yuhao Zhang. 2019-09-11. Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals. https://arxiv.org/abs/1804.07458

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

KEEP EXPLORING

Related papers

arXiv2026

Online Flexible Busy Time Scheduling on Heterogeneous Machines

We study the online busy time scheduling model on heterogeneous machines. In our setting, jobs with uniform processing time arrive online with a deadline that becomes known to the algorithm at the job's arrival time. An algorithm has access to machines, each with different associated capacities and costs. The goal is to schedule jobs on machines by their deadline, so that the total cost incurred by the scheduling algorithm is minimized. While busy time scheduling has been well-studied, relatively little is known when machines are heterogeneous (i.e., have different costs and capacities), despite this natural theoretical generalization being the most practical model for clients using cloud computing services. We make significant progress in understanding this model by designing a deterministic online algorithm with competitive ratio 8(2p-1)/p < 16 when all jobs have uniform processing time p. A randomized version of this algorithm is 4(2p-1)/(p \ln 2)-competitive against an oblivious adversary. For unit-processing-time jobs, we give lower bounds of 4 and e (where e is Euler's number) on the competitive ratio of deterministic and randomized online algorithms, respectively. For unit-processing-time jobs with agreeable deadlines, we provide a deterministic 2-competitive online algorithm and a matching lower bound.

cs.DS↗
arXiv2026

The Binary Tree Mechanism is Optimal for Differentially Private Continual Counting

Private continual counting is a fundamental problem in differential privacy: given a binary stream of length $n$, where each $1$ corresponds to the contribution of one individual, the goal is to release all running counts while protecting the privacy of each individual. For fixed privacy parameters, the standard binary tree mechanism achieves expected $\ell_\infty$ error $O(\log^{3/2} n)$ under approximate differential privacy and $O(\log^2 n)$ under pure differential privacy. Whether these dependences on the stream length are necessary has remained a central open problem. For fixed $\varepsilon\in(0,1)$, we prove a lower bound of $Ω(\log^{3/2} n)$ under approximate DP with sufficiently small fixed $δ>0$, and a lower bound of $Ω(\log^2 n)$ under pure DP. These bounds establish the optimality of the binary tree mechanism in both settings. The bounds hold for arbitrary mechanisms, even when the entire stream is available in advance. Both proofs use the same decomposition and accumulation of residual noise along a tree. As a consequence of the approximate-DP bound, we also obtain a largest-possible separation between hereditary discrepancy and private $\ell_\infty$ error for linear queries, showing that the known general upper bound in terms of hereditary discrepancy has the optimal dependence on the number of queries.

cs.DS↗
arXiv2026

Directed Hamiltonian-Cycle Parity in $O^*((3/2)^n)$ Deterministic Time and Polynomial Space

We give a deterministic algorithm that computes the parity of the number of Hamiltonian cycles in an $n$-vertex directed graph in $O(n^4(3/2)^n)$ time and $O(n^2)$ bits of working space, improving the $O^*(φ^n)$ bound of Björklund and Husfeldt. Their local-degree formula reduces the problem to a weighted sum over solutions of structured quadratic equations. We cover the corresponding ternary state space by binary subcubes, each inducing an affine system. The Kuang--Wang cover can be regenerated within the target bound; canonical ownership resolves its overlaps, while self-loop conditional expectations bound every affine solution visit. Rollback elimination shares the work across cover prefixes. The same cover gives a Las Vegas algorithm listing all $L$ solutions of $m$ affine product constraints in $N$ Boolean variables in expected time $\operatorname{poly}(N,m)((3/2)^m+L)$ and polynomial space. Finally, we show that complete enumeration can require $Ω((3/2)^n)$ visits even on strongly connected digraphs after an optimal self-loop choice. This is a limitation of the enumeration method, not a general lower bound for Hamiltonian-cycle parity.

cs.DS↗