This paper studies fractional matching on general graphs in the fully online model of Huang et al. (JACM 2020), in which all vertices arrive online and remain available for only a limited time, and extends the classic Water-Filling algorithm to the fully online setting, establishing that Water-Filling is not optimal in the fully online setting.
Abstract
This paper studies fractional matching on general graphs in the fully online model of Huang et al. (JACM 2020), in which all vertices arrive online and remain available for only a limited time. The algorithm must make irrevocable fractional matching decisions while the relevant vertices are simultaneously available. We extend the classic Water-Filling algorithm, also known as Balance and originally introduced by Kalyanasundaram and Pruhs (TCS 2000), to the fully online setting. Using an online primal-dual framework, we prove that the generalized Water-Filling algorithm achieves a competitive ratio of $2-\sqrt{2}\approx 0.586$ in the fully online model, and that this analysis is tight. To surpass the $2-\sqrt{2}$ barrier, we incorporate the ideas of eager matching and history-based pricing into Water-Filling. We show that the resulting algorithm achieves an improved competitive ratio of $0.599$, thereby establishing that Water-Filling is not optimal in the fully online setting. On the hardness side, we further improve the known upper bound for fractional fully online matching, reducing the previous best bound of $0.6297$ due to Eckl et al. (ORL 2021) to $0.6132$.
This paper presents an $\frac{11}{6} \approx 1.83$-competitive algorithm for trees in the more general edge arrival model and gives a 1.5-competitive algorithm and provide a matching lower bound.
Júlia Baligács, B. Bosek, Y. Disser et al.· Embedded Systems and Applica...· 1 citation
We present a new online algorithm for the well-known Multi-Level Aggregation Problem (MLAP) with arbitrary delay functions, achieving a $2D$-competitive ratio, where $D$ is the depth of the underlying tree. This result improves the current best-known competitive ratio of $O(D^2)$ and asymptotically matches the $D$-competitive bound previously known only for the deadline variant, thereby closing the asymptotic gap between the two settings. Our key technical contribution is a novel dual fitting framework that provides a unified analysis for both settings; in particular, it also establishes a $D$-competitive ratio for MLAP with deadlines. Our analysis is built upon two new ideas: a hindsight dual construction, which resolves the infeasibility issues in traditional online primal-dual methods, and a time-dependent dual packing that maintains feasibility over dynamic request sets.
Sara Ahmadian, Shuchi Chawla, Ravi Kumar et al.· 0 citations
Competitive analysis is central to the study of online algorithms, but upper bounds are often highly problem-specific. We develop a more unifying methodology via the minimax viewpoint. Guided by Yao's principle, we reduce worst-case competitive analysis to Bayesian online design under an arbitrary correlated prior over arrival sequences. For such a prior, let $X^*$ be the hindsight-optimal fractional solution for the realized instance, and let $X^{(t)}=\mathbb E[X^*\mid \mathcal F_t]$ be its posterior process. Our guiding rule is posterior matching: at each time $t$, choose the feasible online action that tracks the current posterior $X^{(t)}$ as closely as the online constraints permit. We show that this single principle yields optimal or near-optimal guarantees for several classical online fractional problems, including set cover, load balancing, matching and more general resource-allocation problems, recovering or improving state-of-the-art bounds in these settings with norm/concave objectives. Via known rounding reductions, it also yields randomized integral guarantees for weighted paging, MTS on star metrics, and ski-rental. At a technical level, our analysis reduces competitive guarantees to key probabilistic inequalities for the vector martingales generated by the posterior of the offline optimum. The resulting framework gives a reusable route from Bayesian online design under arbitrary correlated priors to information-theoretic worst-case competitive guarantees.
Thomas Kesselheim, Marco Molinaro, Kalen Patton et al.· 1 citation
It is shown that predictions enable an algorithm that is simultaneously O (1)-consistent and O (log n )-robust for online sorting with predictions, and this result is extended to the setting of multiple predictions.
I. Bercea, G. Brodal, John Iacono et al.· Embedded Systems and Applica...· 0 citations
We study edge-weighted online bipartite matching under random arrival order, parameterized by the maximum offline degree $d$ and sampling fraction $\theta$. We analyze two sampling-based frameworks. For \emph{Deterministic Greedy Sampling}, which computes prices from a fixed-size initial sample and then applies a local threshold rule, we derive an explicit worst-case competitive ratio and prove it tight within this policy family for every fixed $d\ge2$ and $\theta\in[0,1]$. The optimal sampling choice interpolates between no sampling for $d=1,2$ and a dense-limit guarantee of approximately $0.2562$, improving on the classical $1/8$ analysis while retaining linear per-arrival time. We also derive worst-case bounds on the variance of the number of matched offline agents, including order-tight behavior as $\theta\to1_-$ and an $O(\theta)$ bound as $\theta\to0_+$ for fixed $m,d$. For \emph{Black-Box Sampling--Matching}, we introduce prefix-dependent reweighting followed by an arbitrary approximate offline matching solver and prove a transfer theorem whose guarantee is the offline approximation ratio times an explicit function of $d$ and $\theta$. With exact matching, the framework recovers the classical $1/e$ guarantee in the unbounded-degree limit.
The matroid secretary problem asks an online algorithm to select a high-weight independent set from elements arriving in uniformly random order, with immediate and irrevocable decisions. Singla (2026) recently gave a $4$-competitive algorithm for arbitrary matroids using only the number of elements and independence queries on already-arrived elements. Following his approach, we obtain an improved competitive ratio of $2+\sqrt3\approx3.7321$ in the same information model. Our algorithm accepts every element of a fixed canonical optimum with probability at least $2-\sqrt3$ and uses $O(n^2)$ independence queries. The algorithm modifies Singla's reversible reference process by retaining a randomly chosen part of the sample as a reserve whose membership in the reference greedy solution is not frozen. Balancing the remaining sample and post-sample elements preserves reversibility and allows an exact calculation of the probability that an exchange partner blocks a target element. The resulting guarantee has a direct analytic proof.
Hau Chan, Jia-Nan Lin, Chen-Hao Wang· 0 citations
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