Quasi-Polynomial Algorithms for Submodular Tree Orienteering and Other Directed Network Design Problems

Ghuge, Rohan; Nagarajan, Viswanath

doi:10.1137/1.9781611975994.63

Cited by 18 publications

(6 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Grandoni, Laekhanukit, and Li [8] recently gave a tight quasi-polynomial time O(log 2 k/ log log k)-approximation for the Directed Steiner Tree problem, based on the Sherali-Adams hierarchy. Later, Ghuge and Nagarajan [6] showed that the same result can be obtained using a combinatorial algorithm, based on guessing what happens in the optimum directed Steiner tree. The guess-and-divide framework was also used in a recent result of Lokshtanov et al [11] to obtain a tight 2-approximation for the feedback vertex set on tournament graphs in polynomial time.…”

Section: The Power Of Linear Programming Hierarchy Vs Guessingmentioning

confidence: 82%

Towards PTAS for Precedence Constrained Scheduling via Combinatorial Algorithms

2020

Preprint

View full text Add to dashboard Cite

We study the classic problem of scheduling n precedence constrained unit-size jobs on m = O(1) machines so as to minimize the makespan. In a recent breakthrough, Levey and Rothvoss [10] developed a (1 + )-approximation for the problem with running time exp exp O m 2 2 log 2 log n , via the Sherali-Adams lift of the basic linear programming relaxation for the problem by exp O m 2 2 log 2 log n levels. Garg [5] recently improved the number of levels to log O(m 2 / 2 ) n, and thus the running time to exp log O(m 2 / 2 ) n , which is quasi-polynomial for constant m and .In this paper we present a (1 + )-approximation algorithm for the problem with running time, which is very close to a polynomial for constant m and . Unlike the algorithms of Levey-Rothvoss and Garg, which are based on the linear-programming hierarchy, our algorithm is purely combinatorial. We show that the conditioning operations on the lifted LP solution can be replaced by making guesses about the optimum schedule.Compared to the LP hierarchy framework, our guessing framework has two advantages, both playing important roles in deriving the improved running time. First, we can guess any information about the optimum schedule, as long as it can be described using a few bits, while in the conditioning framework, we can only condition on the variables in the basic LP. Second, the guessing framework can save a factor of log n in the exponent of running time. Roughly speaking, most of the time, the information we try to guess is binary and thus each nested guess only contributes to a multiplicative factor of 2 in the running time. In contrast, each conditioning operation in a sequence incurs a multiplicative factor of poly(n).

show abstract

Section: The Power Of Linear Programming Hierarchy Vs Guessingmentioning

confidence: 82%

Towards PTAS for Precedence Constrained Scheduling via Combinatorial Algorithms

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…In [27], an approximation algorithm is proposed that runs in quasi-polynomial time for the submodular tree orienteering problem. Differently from us, they work on directed graphs.…”

Section: Related Workmentioning

confidence: 99%

Speeding up Routing Schedules on Aisle-Graphs with Single Access

Sorbelli¹,

Carpin²,

Coró³

et al. 2021

Preprint

View full text Add to dashboard Cite

In this paper, we study the Orienteering Aislegraphs Single-access Problem (OASP), a variant of the orienteering problem for a robot moving in a so-called single-access aisle-graph, i.e., a graph consisting of a set of rows that can be accessed from one side only. Aisle-graphs model, among others, vineyards or warehouses. Each aisle-graph vertex is associated with a reward that a robot obtains when visits the vertex itself. As the robot's energy is limited, only a subset of vertices can be visited with a fully charged battery. The objective is to maximize the total reward collected by the robot with a battery charge. We first propose an optimal algorithm that solves OASP in O(m 2 n 2 ) time for aisle-graphs with a single access consisting of m rows, each with n vertices. With the goal of designing faster solutions, we propose four greedy sub-optimal algorithms that run in at most O(mn (m + n)) time. For two of them, we guarantee an approximation ratio of 1 2 (1 − 1 e ), where e is the base of the natural logarithm, on the total reward by exploiting the wellknown submodularity property. Experimentally, we show that these algorithms collect more than 80% of the optimal reward.

show abstract

“…The goal is to maximize f (S). Currently, the best result for this setting is due to Kesselheim and Tönnis [20], who achieve a 1/e-competitive ratio in exponential time in k, or 1 e (1 − 1 e ) in polynomial time in n and k. Submodular functions also has been used in the network design problems [15,14]. There are also some related online coloring problems in the literature [16,1].…”

Section: Related Workmentioning

confidence: 99%

“…In recent years, submodular optimization has found applications for different machine learning and data mining applications including data summarization, sparsity, active learning, recommendation, high-order graphical model inference, determinantal point processes [12,4,19], network inference, network design [15,13], and influence maximization in social networks [19].…”

Section: Introductionmentioning

confidence: 99%

Submodular Matroid Secretary Problem with Shortlists

Shadravan

2020

Preprint

View full text Add to dashboard Cite

In the matroid secretary problem, which is a generalization of the classic secretary problem, the elements of a matroid M arrive in random order. Once we observe an item we need to irrevocably decide whether or not to accept it. The set of selected elements should form an independent set of the matroid. The goal is to maximize the total sum of the values assigned to these elements. The existence of a constant competitive algorithm is a long standing open problem.In this paper, we introduce a version of this problem, which we refer to as submodular matroid secretary problem with shortlists (motivated by the shortlist model in [2]). In this setting, the algorithm is allowed to choose a subset of items as part of a shortlist, possibly more than k = rk(M) items. Then, after seeing the entire input, the algorithm can choose an independent subset from the shortlist. Furthermore we generalize the objective function to any monotone submodular function. The main question is that can an online algorithm achieve a constant competitive ratio using a shortlist of size O(k)?We design an algorithm that achieves a 1 2 (1 − 1/e 2 − − O(1/k)) competitive ratio for any constant > 0, using a shortlist of size O(k). This is especially surprising considering that the best known competitive ratio for the matroid secretary problem is O(log log k). We are also able to get a constant competitive algorithm using shortlist of size at most k and also a constant competitive algorithm in the preemption model.An important application of our algorithm is for the random order streaming of submodular functions. We show that our algorithm can be implemented in the streaming setting using O(k) memory. It achieves a 1 2 (1 − 1/e 2 − − O(1/k)) approximation. The previously best known approximation ratio for streaming submodular maximization under matroid constraint is 0.25 (adversarial order) due to Feldman et al. [12], Chekuri et al. [8] and Chakrabarti and Kale [7]. Moreover, we generalize our results to the case of p-matchoid constraints and give a 1 p+1 (1 − 1/e p+1 − − O(1/k)) approximation using O(k) memory, which asymptotically (as p and k increase) approaches the best known offline guarantee 1 p+1 [22].

show abstract

Quasi-Polynomial Algorithms for Submodular Tree Orienteering and Other Directed Network Design Problems

Cited by 18 publications

References 17 publications

Towards PTAS for Precedence Constrained Scheduling via Combinatorial Algorithms

Towards PTAS for Precedence Constrained Scheduling via Combinatorial Algorithms

Speeding up Routing Schedules on Aisle-Graphs with Single Access

Submodular Matroid Secretary Problem with Shortlists

Contact Info

Product

Resources

About