arXiv subjects

Shahin Kamali

Publications and source records attributed to Shahin Kamali.

32 records · Page 2Linked to original sources

arXiv2021

Competitive Sequencing with Noisy Advice

Several well-studied online resource allocation problems can be formulated in terms of infinite, increasing sequences of positive values, in which each element is associated with a corresponding allocation value. Examples include problems such as online bidding, searching for a hidden target on an unbounded line, and designing interruptible algorithms based on repeated executions. The performance of the online algorithm, in each of these problems, is measured by the competitive ratio, which describes the multiplicative performance loss due to the absence of full information on the instance. We study such competitive sequencing problems in a setting in which the online algorithm has some (potentially) erroneous information, expressed as a $k$-bit advice string, for some given $k$. We first consider the untrusted advice setting of [Angelopoulos et al, ITCS 2020], in which the objective is to quantify performance considering two extremes: either the advice is either error-free, or it is generated by a (malicious) adversary. Here, we show a Pareto-optimal solution, using a new approach for applying the functional-based lower-bound technique due to [Gal, Israel J. Math. 1972]. Next, we study a nascent noisy advice setting, in which a number of the advice bits may be erroneous; the exact error is unknown to the online algorithm, which only has access to a pessimistic estimate (i.e., an upper bound on this error). We give improved upper bounds, but also the first lower bound on the competitive ratio of an online problem in this setting. To this end, we combine ideas from robust query-based search in arrays, and fault-tolerant contract scheduling. Last, we demonstrate how to apply the above techniques in robust optimization without predictions, and discuss how they can be applicable in the context of more general online problems.

cs.DS↗

arXiv2021

An improved bound on the burning number of graphs

The burning number conjecture states that the burning number of a connected graph is at most $\lceil \sqrt{n} \rceil.$ While the conjecture is unresolved, Land and Lu proved that the burning number of a connected graph is at most $ \sqrt{(3/2)n}+O(1).$ Using an algorithmic approach, we provide an improved upper bound for the burning number of a connected graph: $$\bigg \lceil \frac{\sqrt{12n+64}+8}{3} \bigg \rceil = \sqrt{(4/3)n} +O(1) .$$

math.CO↗

arXiv2021

On the Fault-Tolerant Online Bin Packing Problem

We study the fault-tolerant variant of the online bin packing problem. Similar to the classic bin packing problem, an online sequence of items of various sizes should be packed into a minimum number of bins of uniform capacity. For applications such as server consolidation, where bins represent servers and items represent jobs of different loads, it is required to maintain fault-tolerant solutions. In a fault-tolerant packing, any job is replicated into f+1 servers, for some integer f > 1, so that the failure of up to f servers does not interrupt service. We build over a practical model introduced by Li and Tang [SPAA 2017] in which each job of load $x$ has a primary replica of load $x$ and $f$ standby replicas, each of load $x/η$, where $η>1$ is a parameter of the problem. Upon failure of up to $f$ servers, any primary replica in a failed bin should be replaced by one of its standby replicas so that the extra load of the new primary replica does not cause an overflow in its bin. We study a general setting in which bins might fail while the input is still being revealed. Our main contribution is an algorithm, named Harmonic-Stretch, which maintains fault-tolerant packings under this general setting. We prove that Harmonic-Stretch has an asymptotic competitive ratio of at most 1.75. This is an improvement over the best existing asymptotic competitive ratio 2 of an algorithm by Li and Tang [TPDS 2020], which works under a model in which bins fail only after all items are packed.

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arXiv2020

Contract Scheduling With Predictions

Contract scheduling is a general technique that allows to design a system with interruptible capabilities, given an algorithm that is not necessarily interruptible. Previous work on this topic has largely assumed that the interruption is a worst-case deadline that is unknown to the scheduler. In this work, we study the setting in which there is a potentially erroneous prediction concerning the interruption. Specifically, we consider the setting in which the prediction describes the time that the interruption occurs, as well as the setting in which the prediction is obtained as a response to a single or multiple binary queries. For both settings, we investigate tradeoffs between the robustness (i.e., the worst-case performance assuming adversarial prediction) and the consistency (i.e, the performance assuming that the prediction is error-free), both from the side of positive and negative results.

cs.AI↗

arXiv2020

Randomized Two-Valued Bounded Delay Online Buffer Management

In the bounded delay buffer management problem unit size packets arrive online to be sent over a network link. The objective is to maximize the total weight of packets sent before their deadline. In this paper we are interested in the two-valued variant of the problem, where every packet has either low (1) or high priority weight ($α$ > 1). We show that its randomized competitive ratio against an oblivious adversary is 1 + ($α$ -- 1)/($α$ 2 + $α$).

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arXiv2020

Beyond Worst-case Analysis of Multicore Caching Strategies

Every processor with multiple cores sharing a cache needs to implement a cache-replacement algorithm. Previous work demonstrated that the competitive ratio of a large class of online algorithms, including Least-Recently-Used (LRU), grows with the length of the input. Furthermore, even offline algorithms like Furthest-In-Future, the optimal algorithm in single-core caching, cannot compete in the multicore setting. These negative results motivate a more in-depth comparison of multicore caching algorithms via alternative analysis measures. Specifically, the power of the adversary to adapt to online algorithms suggests the need for a direct comparison of online algorithms to each other. In this paper, we introduce cyclic analysis, a generalization of bijective analysis introduced by Angelopoulos and Schweitzer [JACM'13]. Cyclic analysis captures the advantages of bijective analysis while offering flexibility that makes it more useful for comparing algorithms for a variety online problems. In particular, we take the first steps beyond worst-case analysis for analysis of multicore caching algorithms. We use cyclic analysis to establish relationships between multicore caching algorithms, including the advantage of LRU over all other multicore caching algorithms in the presence of locality of reference.

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arXiv2020

Online Bin Covering with Advice

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 algorithm without advice. This result highlights a difference between the bin covering and the bin packing problems in the advice model: for the bin packing problem, there are several algorithms with advice of constant size that outperform online algorithms without advice. Furthermore, we show that advice of size $O(\log \log n)$ is sufficient to achieve a competitive ratio that is arbitrarily close to $0.53\bar{3}$ and hence strictly better than the best ratio $0.5$ attainable by purely online algorithms. The technicalities involved in introducing and analyzing this algorithm are quite different from the existing results for the bin packing problem and confirm the different nature of these two problems. Finally, we show that a linear number of bits of advice is necessary to achieve any competitive ratio better than 15/16 for the online bin covering problem.

cs.DS↗

arXiv2019

Burning Two Worlds: Algorithms for Burning Dense and Tree-like Graphs

Graph burning is a simple model for the spread of social influence in networks. The objective is to measure how quickly a fire (e.g., a piece of fake news) can be spread in a network. The burning process takes place in discrete rounds. In each round, a new fire breaks out at a selected vertex and burns it. Meanwhile, the old fires extend to their neighbours and burn them. A burning schedule selects where the new fire breaks out in each round, and the burning problem asks for a schedule that burns all vertices in a minimum number of rounds, termed the burning number of the graph. The burning problem is known to be NP-hard even when the graph is a tree or a disjoint set of paths. For connected graphs, it has been conjectured that burning takes at most $\lceil \sqrt{n} \rceil$ rounds. We approach the algorithmic study of graph burning from two directions. First, we consider graphs with minimum degree $δ$. We present an algorithm that burns any graph of size $n$ in at most $\sqrt{\frac{24n}{δ+1}}$ rounds. In particular, for dense graphs with $δ\in Θ(n)$, all vertices are burned in a constant number of rounds. More interestingly, even when $δ$ is a constant that is independent of the graph size, our algorithm answers the graph-burning conjecture in the affirmative by burning the graph in at most $\lceil \sqrt{n} \rceil$ rounds. Next, we consider burning graphs with bounded path-length or tree-length. These include many graph families including connected interval graphs and connected chordal graphs. We show that any graph with path-length $pl$ and diameter $d$ can be burned in $\lceil \sqrt{d-1} \rceil + pl$ rounds. Our algorithm ensures an approximation ratio of $1+o(1)$ for graphs of bounded path-length. We introduce another algorithm that achieves an approximation ratio of $2+o(1)$ for burning graphs of bounded tree-length.

math.CO↗

arXiv2019

Approximation Algorithms for Graph Burning

Numerous approaches study the vulnerability of networks against social contagion. Graph burning studies how fast a contagion, modeled as a set of fires, spreads in a graph. The burning process takes place in synchronous, discrete rounds. In each round, a fire breaks out at a vertex, and the fire spreads to all vertices that are adjacent to a burning vertex. The selection of vertices where fires start defines a schedule that indicates the number of rounds required to burn all vertices. Given a graph, the objective of an algorithm is to find a schedule that minimizes the number of rounds to burn graph. Finding the optimal schedule is known to be NP-hard, and the problem remains NP-hard when the graph is a tree or a set of disjoint paths. The only known algorithm is an approximation algorithm for disjoint paths, which has an approximation ratio of 1.5.

math.CO↗

arXiv2016

On the List Update Problem with Advice

We study the online list update problem under the advice model of computation. Under this model, an online algorithm receives partial information about the unknown parts of the input in the form of some bits of advice generated by a benevolent offline oracle. We show that advice of linear size is required and sufficient for a deterministic algorithm to achieve an optimal solution or even a competitive ratio better than $15/14$. On the other hand, we show that surprisingly two bits of advice are sufficient to break the lower bound of $2$ on the competitive ratio of deterministic online algorithms and achieve a deterministic algorithm with a competitive ratio of $5/3$. In this upper-bound argument, the bits of advice determine the algorithm with smaller cost among three classical online algorithms, TIMESTAMP and two members of the MTF2 family of algorithms. We also show that MTF2 algorithms are $2.5$-competitive.

cs.DS↗

arXiv2014

Efficient Online Strategies for Renting Servers in the Cloud

In Cloud systems, we often deal with jobs that arrive and depart in an online manner. Upon its arrival, a job should be assigned to a server. Each job has a size which defines the amount of resources that it needs. Servers have uniform capacity and, at all times, the total size of jobs assigned to a server should not exceed the capacity. This setting is closely related to the classic bin packing problem. The difference is that, in bin packing, the objective is to minimize the total number of used servers. In the Cloud, however, the charge for each server is proportional to the length of the time interval it is rented for, and the goal is to minimize the cost involved in renting all used servers. Recently, certain bin packing strategies were considered for renting servers in the Cloud [Li et al. SPAA'14]. There, it is proved that all Any-Fit bin packing strategy has a competitive ratio of at least $μ$, where $μ$ is the max/min interval length ratio of jobs. It is also shown that First Fit has a competitive ratio of $2μ+ 13$ while Best Fit is not competitive at all. We observe that the lower bound of $μ$ extends to all online algorithms. We also prove that, surprisingly, Next Fit algorithm has competitive ratio of at most $2 μ+1$. We also show that a variant of Next Fit achieves a competitive ratio of $K \times max\{1,μ/(K-1)\}+1$, where $K$ is a parameter of the algorithm. In particular, if the value of $μ$ is known, the algorithm has a competitive ratio of $μ+2$; this improves upon the existing upper bound of $μ+8$. Finally, we introduce a simple algorithm called Move To Front (MTF) which has a competitive ratio of at most $6μ+ 7$ and also promising average-case performance. We experimentally study the average-case performance of different algorithms and observe that the typical behaviour of MTF is distinctively better than other algorithms.

cs.DS↗

arXiv2014

An All-Around Near-Optimal Solution for the Classic Bin Packing Problem

In this paper we present the first algorithm with optimal average-case and close-to-best known worst-case performance for the classic on-line problem of bin packing. It has long been observed that known bin packing algorithms with optimal average-case performance were not optimal in the worst-case sense. In particular First Fit and Best Fit had optimal average-case ratio of 1 but a worst-case competitive ratio of 1.7. The wasted space of First Fit and Best Fit for a uniform random sequence of length $n$ is expected to be $Θ(n^{2/3})$ and $Θ(\sqrt{n} \log ^{3/4} n)$, respectively. The competitive ratio can be improved to 1.691 using the Harmonic algorithm; further variations of this algorithm can push down the competitive ratio to 1.588. However, Harmonic and its variations have poor performance on average; in particular, Harmonic has average-case ratio of around 1.27. In this paper, first we introduce a simple algorithm which we term Harmonic Match. This algorithm performs as well as Best Fit on average, i.e., it has an average-case ratio of 1 and expected wasted space of $Θ(\sqrt{n} \log ^{3/4} n)$. Moreover, the competitive ratio of the algorithm is as good as Harmonic, i.e., it converges to $ 1.691$ which is an improvement over 1.7 of Best Fit and First Fit. We also introduce a different algorithm, termed as Refined Harmonic Match, which achieves an improved competitive ratio of $1.636$ while maintaining the good average-case performance of Harmonic Match and Best Fit. Finally, our extensive experimental evaluation of the studied bin packing algorithms shows that our proposed algorithms have comparable average-case performance with Best Fit and First Fit, and this holds also for sequences that follow distributions other than the uniform distribution.

cs.DS↗

arXiv2013

Online Bin Packing with Advice

We consider the online bin packing problem under the advice complexity model where the 'online constraint' is relaxed and an algorithm receives partial information about the future requests. We provide tight upper and lower bounds for the amount of advice an algorithm needs to achieve an optimal packing. We also introduce an algorithm that, when provided with log n + o(log n) bits of advice, achieves a competitive ratio of 3/2 for the general problem. This algorithm is simple and is expected to find real-world applications. We introduce another algorithm that receives 2n + o(n) bits of advice and achieves a competitive ratio of 4/3 + ε. Finally, we provide a lower bound argument that implies that advice of linear size is required for an algorithm to achieve a competitive ratio better than 9/8.

cs.DS↗

arXiv2013

On Advice Complexity of the k-server Problem under Sparse Metrics

We consider the k-server problem under the advice model of computation when the underlying metric space is sparse. On one side, we show that an advice of size Ω(n) is required to obtain a 1-competitive algorithm for sequences of size n, even for the 2-server problem on a path metric of size N >= 5. Through another lower bound argument, we show that at least (n/2)(log α - 1.22) bits of advice is required to obtain an optimal solution for metric spaces of treewidth α, where 4 <= α < 2k. On the other side, we introduce Θ(1)-competitive algorithms for a wide range of sparse graphs, which require advice of (almost) linear size. Namely, we show that for graphs of size N and treewidth α, there is an online algorithm which receives $O(n (log α + log log N))$ bits of advice and optimally serves a sequence of length n. With a different argument, we show that if a graph admits a system of μ collective tree (q,r)-spanners, then there is a (q+r)-competitive algorithm which receives O(n (log μ + log log N)) bits of advice. Among other results, this gives a 3-competitive algorithm for planar graphs, provided with O(n log log N) bits of advice.

cs.DS↗