scholarly journals Revisiting Modified Greedy Algorithm for Monotone Submodular Maximization with a Knapsack Constraint

2021 ◽  
Vol 5 (1) ◽  
pp. 1-22
Author(s):  
Jing Tang ◽  
Xueyan Tang ◽  
Andrew Lim ◽  
Kai Han ◽  
Chongshou Li ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Greedy Algorithm ◽  
Branch And Bound ◽  
Upper Bound ◽  
Optimization Problem ◽  
Approximation Factor ◽  
Real World Application ◽  
Efficiency Of Algorithms ◽  
Knapsack Constraint ◽  
Submodular Maximization

Monotone submodular maximization with a knapsack constraint is NP-hard. Various approximation algorithms have been devised to address this optimization problem. In this paper, we revisit the widely known modified greedy algorithm. First, we show that this algorithm can achieve an approximation factor of 0.405, which significantly improves the known factors of 0.357 given by Wolsey and (1-1/e)/2\approx 0.316 given by Khuller et al. More importantly, our analysis closes a gap in Khuller et al.'s proof for the extensively mentioned approximation factor of (1-1/\sqrte )\approx 0.393 in the literature to clarify a long-standing misconception on this issue. Second, we enhance the modified greedy algorithm to derive a data-dependent upper bound on the optimum. We empirically demonstrate the tightness of our upper bound with a real-world application. The bound enables us to obtain a data-dependent ratio typically much higher than 0.405 between the solution value of the modified greedy algorithm and the optimum. It can also be used to significantly improve the efficiency of algorithms such as branch and bound.

Approximation Algorithms for Submodular Data Summarization with a Knapsack Constraint

2021 ◽  
Vol 5 (1) ◽  
pp. 1-31
Author(s):  
Kai Han ◽  
Shuang Cui ◽  
Tianshuai Zhu ◽  
Enpei Zhang ◽  
Benwei Wu ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Fundamental Problem ◽  
Randomized Algorithm ◽  
Deterministic Algorithm ◽  
Submodular Function ◽  
Approximation Ratio ◽  
Performance Bounds ◽  
Data Summarization ◽  
Submodular Optimization ◽  
Knapsack Constraint

Data summarization, i.e., selecting representative subsets of manageable size out of massive data, is often modeled as a submodular optimization problem. Although there exist extensive algorithms for submodular optimization, many of them incur large computational overheads and hence are not suitable for mining big data. In this work, we consider the fundamental problem of (non-monotone) submodular function maximization with a knapsack constraint, and propose simple yet effective and efficient algorithms for it. Specifically, we propose a deterministic algorithm with approximation ratio 6 and a randomized algorithm with approximation ratio 4, and show that both of them can be accelerated to achieve nearly linear running time at the cost of weakening the approximation ratio by an additive factor of ε. We then consider a more restrictive setting without full access to the whole dataset, and propose streaming algorithms with approximation ratios of 8+ε and 6+ε that make one pass and two passes over the data stream, respectively. As a by-product, we also propose a two-pass streaming algorithm with an approximation ratio of 2+ε when the considered submodular function is monotone. To the best of our knowledge, our algorithms achieve the best performance bounds compared to the state-of-the-art approximation algorithms with efficient implementation for the same problem. Finally, we evaluate our algorithms in two concrete submodular data summarization applications for revenue maximization in social networks and image summarization, and the empirical results show that our algorithms outperform the existing ones in terms of both effectiveness and efficiency.

Sorting permutations by fragmentation-weighted operations

2020 ◽  
Vol 18 (02) ◽  
pp. 2050006 ◽  
Author(s):  
Alexsandro Oliveira Alexandrino ◽  
Carla Negri Lintzmayer ◽  
Zanoni Dias
Keyword(s):  
Approximation Algorithms ◽  
Computational Biology ◽  
Cost Function ◽  
Traditional Approach ◽  
Upper Bounds ◽  
Evolutionary Distance ◽  
Lower And Upper Bounds ◽  
Approximation Factor ◽  
New Type ◽  
The Cost

One of the main problems in Computational Biology is to find the evolutionary distance among species. In most approaches, such distance only involves rearrangements, which are mutations that alter large pieces of the species’ genome. When we represent genomes as permutations, the problem of transforming one genome into another is equivalent to the problem of Sorting Permutations by Rearrangement Operations. The traditional approach is to consider that any rearrangement has the same probability to happen, and so, the goal is to find a minimum sequence of operations which sorts the permutation. However, studies have shown that some rearrangements are more likely to happen than others, and so a weighted approach is more realistic. In a weighted approach, the goal is to find a sequence which sorts the permutations, such that the cost of that sequence is minimum. This work introduces a new type of cost function, which is related to the amount of fragmentation caused by a rearrangement. We present some results about the lower and upper bounds for the fragmentation-weighted problems and the relation between the unweighted and the fragmentation-weighted approach. Our main results are 2-approximation algorithms for five versions of this problem involving reversals and transpositions. We also give bounds for the diameters concerning these problems and provide an improved approximation factor for simple permutations considering transpositions.

scholarly journals Branch and Bound Method to Solve Multi Objectives Function

2016 ◽  
Vol 12 (3) ◽  
pp. 5964-5974
Author(s):  
Tahani Jabbar Kahribt ◽  
Mohammed Kadhim Al- Zuwaini
Keyword(s):  
Lower Bounds ◽  
Branch And Bound ◽  
Single Machine ◽  
Upper Bound ◽  
Completion Time ◽  
Heuristic Method ◽  
Total Weighted Completion Time ◽  
Branch And Bound Method ◽  
Special Cases ◽  
Weighted Completion Time

This paper  presents  a  branch  and  bound  algorithm  for  sequencing  a  set  of  n independent  jobs  on  a single  machine  to  minimize sum of the discounted total weighted completion time and maximum lateness,  this problems is NP-hard. Two lower bounds were proposed and heuristic method to get an upper bound. Some special cases were  proved and some dominance rules were suggested and proved, the problem solved with up to 50 jobs.

Two-stage submodular maximization problem beyond nonnegative and monotone

2021 ◽  
pp. 1-16
Author(s):  
Zhicheng Liu ◽  
Hong Chang ◽  
Ran Ma ◽  
Donglei Du ◽  
Xiaoyan Zhang
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Objective Function ◽  
Modular Function ◽  
Submodular Function ◽  
Maximization Problem ◽  
Cardinality Constraint ◽  
Two Stage ◽  
Time Efficiency ◽  
Submodular Maximization

Abstract We consider a two-stage submodular maximization problem subject to a cardinality constraint and k matroid constraints, where the objective function is the expected difference of a nonnegative monotone submodular function and a nonnegative monotone modular function. We give two bi-factor approximation algorithms for this problem. The first is a deterministic $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right),1} \right)$ -approximation algorithm, and the second is a randomized $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right) - \varepsilon ,1} \right)$ -approximation algorithm with improved time efficiency.

AN APPROXIMATE MORPHING BETWEEN POLYLINES

2005 ◽  
Vol 15 (02) ◽  
pp. 193-208 ◽  
Author(s):  
SERGEY BEREG
Keyword(s):  
Shortest Path ◽  
Optimization Problem ◽  
Linear Time ◽  
Medial Axis ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Geodesic Path ◽  
Minimum Width ◽  
Approximation Factor ◽  
Previous Algorithm

We consider the problem of continuously transforming or morphing one simple polyline into another polyline so that every point p of the initial polyline moves to a point q of the final polyline using the geodesic shortest path from p to q. The width of a morphing is defined as the longest geodesic path between corresponding points of the polylines. The optimization problem is to compute a morphing that minimizes the width. We present a linear-time algorithm for finding a morphing with width guaranteed to be at most two times the minimum width of a morphing. This improves the previous algorithm10 by a factor of log n. We develop a linear-time algorithm for computing a medial axis separator. We also show that the approximation factor is less than two for κ-straight polylines.

APPROXIMATION ALGORITHMS FOR A VARIANT OF DISCRETE PIERCING SET PROBLEM FOR UNIT DISKS

2013 ◽  
Vol 23 (06) ◽  
pp. 461-477 ◽  
Author(s):  
MINATI DE ◽  
GAUTAM K. DAS ◽  
PAZ CARMI ◽  
SUBHAS C. NANDY
Keyword(s):  
Approximation Algorithms ◽  
Simple Algorithm ◽  
Constant Factor ◽  
Performance Ratio ◽  
Approximation Result ◽  
Worst Case ◽  
Approximation Factor ◽  
Minimum Number ◽  
Unit Disks ◽  
Set Of Points

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio [Formula: see text] in nO(k) time.

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