Dimitris Fotakis, Loukas Kavouras, Grigorios Koumoutsos, Stratis Skoulakis, Manolis Vardas
备注：
A preliminary version of this article appeared in the Proceedings of the 47th International Colloquium on Automata, Languages and Programming (ICALP 2020)
We consider the online Min-Sum Set Cover (MSSC), a natural and intriguing generalization of the classical list update problem. In Online MSSC, the algorithm maintains a permutation on $n$ elements based on subsets $S_1, S_2, \ldots$ arriving online. The algorithm serves each set $S_t$ upon arrival, using its current permutation $π_{t}$, incurring an access cost equal to the position of the first element of $S_t$ in $π_{t}$. Then, the algorithm may update its permutation to $π_{t+1}$, incurring a moving cost equal to the Kendall tau distance of $π_{t}$ to $π_{t+1}$. The objective is to minimize the total access and moving cost for serving the entire sequence. We consider the $r$-uniform version, where each $S_t$ has cardinality $r$. List update is the special case where $r = 1$.
We obtain tight bounds on the competitive ratio of deterministic online algorithms for MSSC against a static adversary, that serves the entire sequence by a single permutation. First, we show a lower bound of $(r+1)(1-\frac{r}{n+1})$ on the competitive ratio. Then, we consider several natural generalizations of successful list update algorithms and show that they fail to achieve any interesting competitive guarantee. On the positive side, we obtain a $O(r)$-competitive deterministic algorithm using ideas from online learning and the multiplicative weight updates (MWU) algorithm.
Furthermore, we consider efficient algorithms. We propose a memoryless online algorithm, called Move-All-Equally, which is inspired by the Double Coverage algorithm for the $k$-server problem. We show that its competitive ratio is $Ω(r^2)$ and $2^{O(\sqrt{\log n \cdot \log r})}$, and conjecture that it is $f(r)$-competitive. We also compare Move-All-Equally against the dynamic optimal solution and obtain (almost) tight bounds by showing that it is $Ω(r \sqrt{n})$ and $O(r^{3/2} \sqrt{n})$-competitive.
We consider fractional online covering problems with $\ell_q$-norm objectives. The problem of interest is of the form $\min\{ f(x) \,:\, Ax\ge 1, x\ge 0\}$ where $f(x)=\sum_{e} c_e \|x(S_e)\|_{q_e} $ is the weighted sum of $\ell_q$-norms and $A$ is a non-negative matrix. The rows of $A$ (i.e. covering constraints) arrive online over time. We provide an online $O(\log d+\log ρ)$-competitive algorithm where $ρ= \frac{\max a_{ij}}{\min a_{ij}}$ and $d$ is the maximum of the row sparsity of $A$ and $\max |S_e|$. This is based on the online primal-dual framework where we use the dual of the above c…
We design the first online algorithm with poly-logarithmic competitive ratio for the edge-weighted degree-bounded Steiner forest(EW-DB-SF) problem and its generalized variant. We obtain our result by demonstrating a new generic approach for solving mixed packing/covering integer programs in the online paradigm. In EW-DB-SF we are given an edge-weighted graph with a degree bound for every vertex. Given a root vertex in advance we receive a sequence of terminal vertices in an online manner. Upon the arrival of a terminal we need to augment our solution subgraph to connect the new terminal to the…
The bin covering problem asks for covering a maximum number of bins with an online sequence of $n$ items of different sizes in the range $(0,1]$; a bin is said to be covered if it receives items of total size at least 1. We study this problem in the advice setting and provide tight bounds for the size of advice required to achieve optimal solutions. Moreover, we show that any algorithm with advice of size $o(\log \log n)$ has a competitive ratio of at most 0.5. In other words, advice of size $o(\log \log n)$ is useless for improving the competitive ratio of 0.5, attainable by an online algorit…
Online models that allow recourse are highly effective in situations where classical models are too pessimistic. One such problem is the online machine covering problem on identical machines. In this setting, jobs arrive one by one and must be assigned to machines with the objective of maximizing the minimum machine load. When a job arrives, we are allowed to reassign some jobs as long as their total size is (at most) proportional to the processing time of the arriving job. The proportionality constant is called the migration factor of the algorithm. Using a rounding procedure with useful stru…
Semi-online models where decisions may be revoked in a limited way have been studied extensively in the last years. This is motivated by the fact that the pure online model is often too restrictive to model real-world applications, where some changes might be allowed. A well-studied measure of the amount of decisions that can be revoked is the migration factor $β$: When an object $o$ of size $s(o)$ arrives, the decisions for objects of total size at most $β\cdot s(o)$ may be revoked. Usually $β$ should be a constant. This means that a small object only leads to small changes. This measure has …
In this paper, we study the set cover problem in the fully dynamic model. In this model, the set of active elements, i.e., those that must be covered at any given time, can change due to element arrivals and departures. The goal is to maintain an algorithmic solution that is competitive with respect to the current optimal solution. This model is popular in both the dynamic algorithms and online algorithms communities. The difference is in the restriction placed on the algorithm: in dynamic algorithms, the running time of the algorithm making updates (called update time) is bounded, while in on…
Set cover and hitting set are fundamental problems in combinatorial optimization which are well-studied in the offline, online, and dynamic settings. We study the geometric versions of these problems and present new online and dynamic algorithms for them. In the online version of set cover (resp. hitting set), $m$ sets (resp.~$n$ points) are give $n$ points (resp.~$m$ sets) arrive online, one-by-one. In the dynamic versions, points (resp. sets) can arrive as well as depart. Our goal is to maintain a set cover (resp. hitting set), minimizing the size of the computed solution. For online set cov…
The Adwords and Online Bipartite Matching problems have enjoyed a renewed attention over the past decade due to their connection to Internet advertising. Our community has contributed, among other things, new models (notably stochastic) and extensions to the classical formulations to address the issues that arise from practical needs. In this paper, we propose a new generalization based on matroids and show that many of the previous results extend to this more general setting. Because of the rich structures and expressive power of matroids, our new setting is potentially of interest both in th…
Slovak Academy of Sciences（斯洛伐克科学院）；Toronto Metropolitan University（多伦多都会大学）；Carleton University（卡尔顿大学）；Wesleyan University（卫斯理安大学）；Concordia University（康考迪亚大学）；Rutgers University（罗格斯大学）
We consider the online $k$-taxi problem, a generalization of the $k$-server problem, in which $k$ servers are located in a metric space. A sequence of requests is revealed one by one, where each request is a pair of two points, representing the start and destination of a travel request by a passenger. The goal is to serve all requests while minimizing the distance traveled without carrying a passenger. We show that the classic Double Coverage algorithm has competitive ratio $2^k-1$ on HSTs, matching a recent lower bound for deterministic algorithms. For bounded depth HSTs, the competitive rati…
The design of online algorithms has tended to focus on algorithms with worst-case guarantees, e.g., bounds on the competitive ratio. However, it is well-known that such algorithms are often overly pessimistic, performing sub-optimally on non-worst-case inputs. In this paper, we develop an approach for data-driven design of online algorithms that maintain near-optimal worst-case guarantees while also performing learning in order to perform well for typical inputs. Our approach is to identify policy classes that admit global worst-case guarantees, and then perform learning using historical data …
We introduce the batched set cover problem, which is a generalization of the online set cover problem. In this problem, the elements of the ground set that need to be covered arrive in batches. Our main technical contribution is a tight $Ω(H_{m - 2^z + 1})$ lower bound on the competitive ratio of any fractional batched algorithm given an adversary that is required to produce batches of VC-dimension at least $z$, for some $z \in \mathbb{N}^0$. This restriction on the adversary is motivated by the fact that, in some real world applications, decisions are made after collecting batches of data of …
Recently, the first online complexity class, AOC, was introduced. The class consists of many online problems where each request must be either accepted or rejected, and the aim is to either minimize or maximize the number of accepted requests, while maintaining a feasible solution. All AOC-complete problems (including Independent Set, Vertex Cover, Dominating Set, and Set Cover) have essentially the same advice complexity. In this paper, we study weighted versions of problems in AOC, i.e., each request comes with a weight and the aim is to either minimize or maximize the total weight of the ac…
This paper studies online algorithms augmented with multiple machine-learned predictions. While online algorithms augmented with a single prediction have been extensively studied in recent years, the literature for the multiple predictions setting is sparse. In this paper, we give a generic algorithmic framework for online covering problems with multiple predictions that obtains an online solution that is competitive against the performance of the best predictor. Our algorithm incorporates the use of predictions in the classic potential-based analysis of online algorithms. We apply our algorit…
We study the online maximum coverage problem on a line, in which, given an online sequence of sub-intervals (which may intersect among each other) of a target large interval and an integer $k$, we aim to select at most $k$ of the sub-intervals such that the total covered length of the target interval is maximized. The decision to accept or reject each sub-interval is made immediately and irrevocably (no preemption) right at the release timestamp of the sub-interval. We comprehensively study different settings of this problem regarding both the length of a released sub-interval and the total nu…
In the bin covering problem, the goal is to fill as many bins as possible up to a certain minimal level with a given set of items of different sizes. Online variants, in which the items arrive one after another and have to be packed immediately on their arrival without knowledge about the future items, have been studied extensively in the literature. We study the simplest possible online algorithm Dual Next-Fit, which packs all arriving items into the same bin until it is filled and then proceeds with the next bin in the same manner. The competitive ratio of this and any other reasonable onlin…
University of North Florida（北佛罗里达大学）；Kent State University（肯特州立大学）
In this paper, we study the problem of coverage planning by a mobile robot with a limited energy budget. The objective of the robot is to cover every point in the environment while minimizing the traveled path length. The environment is initially unknown to the robot. Therefore, it needs to avoid the obstacles in the environment on-the-fly during the exploration. As the robot has a specific energy budget, it might not be able to cover the complete environment in one traversal. Instead, it will need to visit a static charging station periodically in order to recharge its energy. To solve the st…
Purdue University（普渡大学）；University of Melbourne（墨尔本大学）；Massachusetts Institute of Technology（麻省理工学院）；UC Berkeley（加州大学伯克利分校）；Rice University（莱斯大学）；Carnegie Mellon University（卡内基梅隆大学）
Semidefinite programming (SDP) is a unifying framework that generalizes both linear programming and quadratically-constrained quadratic programming, while also yielding efficient solvers, both in theory and in practice. However, there exist known impossibility results for approximating the optimal solution when constraints for covering SDPs arrive in an online fashion. In this paper, we study online covering linear and semidefinite programs in which the algorithm is augmented with advice from a possibly erroneous predictor. We show that if the predictor is accurate, we can efficiently bypass t…
University of Wrocław（弗罗茨瓦夫大学）；University of Warsaw（华沙大学）
We study a fundamental model of online preference aggregation, where an algorithm maintains an ordered list of $n$ elements. An input is a stream of preferred sets $R_1, R_2, \dots, R_t, \dots$. Upon seeing $R_t$ and without knowledge of any future sets, an algorithm has to rerank elements (change the list ordering), so that at least one element of $R_t$ is found near the list front. The incurred cost is a sum of the list update costs (the number of swaps of neighboring list elements) and access costs (position of the first element of $R_t$ on the list). This scenario occurs naturally in appli…
We consider Online Minimum Bipartite Matching under the uniform metric. We show that Randomized Greedy achieves a competitive ratio equal to $(1+1/n) (H_{n+1}-1)$, which matches the lower bound. Comparing with the fact that RG achieves an optimal ratio of $Θ(\ln n)$ for the same problem but under the adversarial order, we find that the weaker arrival assumption of random order doesn't offer any extra algorithmic advantage for RG, or make the model strictly more tractable.
