arXiv subjects

Doan Dai Nguyen

Publications and source records attributed to Doan Dai Nguyen.

4 recordsLinked to original sources

arXiv2026

Stability in stochastic hypergraph matching II: weights, batch arrivals, and continuous time

Many real-life systems can be found as examples of stochastic matching on hypergraphs, such as production lines or assemble-to-order systems. Two common features are the number of items required may vary between matchings, and there may intermediary items which exist as a combination of other items and not of external arrivals. Both of these phenomena can be modelled by considering the weighted variant of stochastic matching. In this work, we formalise the notion of stochastic weighted matching on hypergraphs. We also allow batch arrivals, meaning multiple items of multiple classes may arrive at the same time, and in particular, the arrivals can be correlated between classes. Unlike the classical setting where items arrive at discrete time $t \in \mathbb{N}$, we allow arrival processes to take place in continuous time $t \in \mathbb{R}_{\geq 0}$. We then extend the results from Nguyen and Bušić (2026) to overcome the intricacies brought up by this new setting. This allows us to derive necessary and sufficient criteria as direct generalisations of those in the unweighted setting, which depend only on the per-class arrival rates. As such, the correlation between classes bear no differences. The constructive proofs also give a maximally stable, periodic-review, size-based, arrival-rate agnostic policy.

math.PR↗

arXiv2026

Stability in stochastic hypergraph matching III: general reneging

In many real-life matching problems, waiting agents might abandon before being matched, such as patients deceasing before receiving organs, passengers/drivers cancelling ride requests, or raw materials/intermediary products degrading in production lines. This poses the need for incorporating reneging in stochastic matching models. In this work, we consider matching models on hypergraphs with batch arrivals and general-weight matchings. Since our model allows fractional weights, we may not be able to talk about individual items, and thus reneging is not required to be independent between items of the same class. For the simplicity sake's, we assume items arrive at discrete time. We show that stability depends on the exact nature of reneging, in stark contrast with the non-reneging case where it depends on the arrivals only through the arrival rates. To our best knowledge, this is the first such sensitivity result in stochastic matching. We uncover a new stabilising mechanism which exists neither in the non-reneging case nor in the graph case, which explains why incorporating reneging in stochastic matching is not straightforward. Together with balancing mechanism as hinted in the online assignment framework for the non-reneging case, it gives a criterion necessary and sufficient for stability. Finally, whilst verifying stability is a hard problem, we give a family of MaxWeight-type policies parameterised by $\varepsilon > 0$, which are maximally stabilising for all $\varepsilon$ sufficiently small. Unfortunately there is no effective bound for $\varepsilon$, but we show how to adjust its value during implementation.

math.PR↗

arXiv2026

Stability in stochastic hypergraph matching I: necessary and sufficient criteria

Stochastic matching on hypergraphs is an important topic for its versatility in capturing real-life systems, from living donor transplant to ride-hailing. Nevertheless, finding necessary and sufficient criteria for stability is a long-standing problem. One of the key difficulties is the fact that greedy policies, whilst maximally stable for stochastic matching on graphs, no longer achieve maximal stability region on hypergraphs. So far, no alternative families of policies with similar properties have been known. In this work, we introduce online assignment policies, in which each item is assigned to a matching hyperedge type upon arrival. We show that this is a good generalisation to greedy policies, by proving that they are maximally stable. Their natural amenability to analysis allow us to derive several necessary and sufficient criteria for stability, which generalise the known criteria for graphs. Furthermore, the constructive proof gives a maximally stable arrival-rate agnostic policy.

cs.DM↗

arXiv2023

Polynomial-time parametric optimisation

In biology, predicting RNA secondary structures plays a vital role in determining its physical and chemical properties. Although we have powerful energy models to predict them as well as parametric analysis to understand the models themselves, the large number of parameters involved makes exploring the parameter space and effective fine-tuning complicated at best. The literature describes an approach via so-called RNA polytopes and several attempts to compute them entirely, but computing explicitly the polytopes is both practically and theoretically intractable. In this thesis, we demonstrate how to further modify the dynamic programming algorithms used in RNA secondary structure prediction, and more generally how to use only supporting functions to gather some information about the polytopes without explicit construction. We provide the mathematical frameworks with proofs or sketch thereof whenever necessary, and carry out some numerical experiments to show that our proposed methods are practical even when the number of parameters is large. As it turns out, one of our methods provides a solution to another problem in computational geometry previously unsolved to our knowledge, and we hope this thesis will accommodate future studies in RNA, as well as inspire further researches on the potential uses of polytopes' supporting functions in computational geometry.

q-bio.BM↗