arXiv subjects

Alison Hsiang-Hsuan Liu

Publications and source records attributed to Alison Hsiang-Hsuan Liu.

5 recordsLinked to original sources

arXiv2026

A Primal-Dual Approach to Randomized Online Bidding with Tail Constraints

Controlling tail risk in randomized algorithms has received increasing attention. Randomized online algorithms can achieve a lower expected competitive ratio than the competitive ratio of the best deterministic online algorithm. However, individual executions may still incur poor outcomes. Tail constraints address this risk by limiting the probability that the incurred ratio exceeds a prescribed threshold. We investigate how such constraints affect the structure and performance of randomized online bidding algorithms. By revisiting the classical primal-dual lower-bound framework for online bidding, we find that restricting the growth between consecutive bids can yield a stronger lower bound on the expected competitive ratio. We then show that tail constraints with zero exceedance probability impose an asymptotic upper bound on the ratios between consecutive bids. Incorporating this structural property into the primal-dual framework allows us to derive a stronger lower bound. Moreover, the dual analysis suggests a randomized geometric bidding algorithm whose performance matches this bound. The insight from our primal-dual analysis also guides the design of algorithms for general tail constraints allowing positive exceedance probabilities. Motivated by the geometric growth suggested by the dual solution, we study randomized periodic bidding algorithms. We show that, for any fixed geometric base, the uniform offset distribution is optimal in terms of both the expected competitive ratio and tail-constraint feasibility. We then characterize the exceedance probability of these algorithms and use this characterization to choose a geometric growth factor that satisfies the prescribed tail constraints, establishing parameter-dependent upper bounds on the optimal expected competitive ratio.

cs.DM↗

arXiv2026

Instance complexity of Boolean functions

In the area of query complexity of Boolean functions, the most widely studied cost measure of an algorithm is the worst-case number of queries made by it on an input. Motivated by the most natural cost measure studied in online algorithms, the competitive ratio, we consider a different cost measure for query algorithms for Boolean functions that captures the ratio of the cost of the algorithm and the cost of an optimal algorithm that knows the input in advance. The cost of an algorithm is its largest cost over all inputs. Grossman, Komargodski and Naor [ITCS'20] introduced this measure for Boolean functions, and dubbed it instance complexity. Grossman et al. showed, among other results, that monotone Boolean functions with instance complexity 1 are precisely those that depend on one or two variables. We complement the above-mentioned result of Grossman et al. by completely characterizing the instance complexity of symmetric Boolean functions. As a corollary we conclude that the only symmetric Boolean functions with instance complexity 1 are the Parity function and its complement. We also study the instance complexity of some graph properties like Connectivity and k-clique containment. In all the Boolean functions we study above, and those studied by Grossman et al., the instance complexity turns out to be the ratio of query complexity to minimum certificate complexity. It is a natural question to ask if this is the correct bound for all Boolean functions. We show a negative answer in a very strong sense, by analyzing the instance complexity of the Greater-Than and Odd-Max-Bit functions. We show that the above-mentioned ratio is linear in the input size for both of these functions, while we exhibit algorithms for which the instance complexity is a constant.

cs.CC↗

arXiv2026

The Power of Amortization on Minimizing Total Completion Time with Explorable Uncertainty

We study online scheduling to minimize total completion time with explorable uncertainty on single and multiple machines. Each job comes with an upper limit of its processing time, which could be potentially reduced by testing the job, which also takes time. The objective is to schedule all jobs with minimum total completion time. The challenge lies in deciding which jobs to test, the order of testing/processing jobs, and in multiple machine case which machine a job is allocated to. In multiple machine case, testing and processing of a job are allowed to be scheduled on different machines. Different settings have been studied before. In this work, we first consider the variable testing times setting. We enhance the analysis framework in Albers and Eckl (2020) and improve the analysis of the competitive ratio of their deterministic single machine algorithm from $4$ to $1+\sqrt{2} \approx 2.4143$. Using the new analysis framework, we propose a new deterministic algorithm that further improves the competitive ratio to $2.316513$. The new framework also enables us to develop a randomized algorithm improving the expected competitive ratio from $3.3794$ to $2.152271$. We further show that with $m$~machines, by extending the framework of Gong et al. (2024), there exists a deterministic $2.77629-(0.45977/m)$-competitive algorithm and a randomized $2.51098-(0.3587/m)$-competitive algorithm. The performance of the algorithms on multiple machines when $m = 1$ matches the current best algorithms on a single machine for variable testing times shown in this paper.

cs.DM↗

arXiv2026

Online Firefighting on Cactus Graphs

It is known that the online firefighting is 2-competitive on trees (Coupechoux et al. 2019), which suggests that the problem is relatively easy on trees. We extend the study to graphs containing cycles. We first show that the presence of cycles gives a strong advantage to the adversary: cycles create situations where the algorithm and the optimal solution operate on different game states, and the adversary can exploit the uncertainty in the firefighter sequence to trap the algorithm. Specifically, we prove that even on a tadpole graph (a cycle with a tail path), no deterministic online algorithm achieves a competitive ratio better than $Ω(\sqrt{n})$, where n is the number of vertices. We then propose an $O(\sqrt{n})$-competitive algorithm for 1-almost trees, which contain at most one cycle and generalize tadpole graphs. We further generalize this algorithm to cactus graphs, in which multiple cycles may appear, but no two share more than one vertex, and show that the online firefighting problem on cactus graphs remains $O(\sqrt{n})$-competitive. Finally, since cactus graphs have treewidth at most 2, we study a variant where firefighters are released in pairs, that is, each round an even number of firefighters is available. Surprisingly, in this setting the competitive complexity is significantly reduced, and we prove that the problem is at most 3-competitive. The main technical challenges lie in both algorithm design and analysis, since the algorithm and the optimal solution may break different cycles and thus operate on different residual graphs. To overcome this difficulty, we design a charging framework that carefully partitions the vertices saved by the optimal solution and charges them to the vertices saved by the algorithm. Namely, the charging scheme is carefully constructed to ensure that each vertex saved by the algorithm is charged at most a constant number of times.

cs.DS↗

arXiv2025

On the FirstFit Algorithm for Online Unit-Interval Coloring

In this paper, we study the performance of the FirstFit algorithm for the online unit-length intervals coloring problem where the intervals can be either open or closed, which serves a further investigation towards the actual performance of FirstFit. We develop a sophisticated counting method by generalizing the classic neighborhood bound, which limits the color FirstFit can assign an interval by counting the potential intersections. In the generalization, we show that for any interval, there is a critical interval intersecting it that can help reduce the overestimation of the number of intersections, and it further helps bound the color an interval can be assigned. The technical challenge then falls on identifying these critical intervals that guarantee the effectiveness of counting. Using this new mechanism for bounding the color that FirstFit can assign an interval, we provide a tight analysis of $2ω$ colors when all intervals have integral endpoints and an upper bound of $\lceil\frac{7}{3}ω\rceil-2$ colors for the general case, where $ω$ is the optimal number of colors needed for the input set of intervals.

cs.DS↗