k-Submodular maximization with two kinds of constraints

2020 ◽  
pp. 2150036
Author(s):  
Ganquan Shi ◽  
Shuyang Gu ◽  
Weili Wu
Keyword(s):  
Approximation Algorithm ◽  
Submodular Function ◽  
Approximation Ratio ◽  
Size Difference ◽  
Total Size ◽  
Difference Constraints ◽  
Submodular Maximization ◽  
Individual Size

[Formula: see text]-submodular maximization is a generalization of submodular maximization, which requires us to select [Formula: see text] disjoint subsets instead of one subset. Attracted by practical values and applications, we consider [Formula: see text]-submodular maximization with two kinds of constraints. For total size and individual size difference constraints, we present a [Formula: see text]-approximation algorithm for maximizing a nonnegative k-submodular function, running in time [Formula: see text] at worst. Specially, if [Formula: see text] is multiple of [Formula: see text], the approximation ratio can reduce to [Formula: see text], running in time [Formula: see text] at worst. Besides, this algorithm can be applied to [Formula: see text]-bisubmodular achieving [Formula: see text]-approximation running in time [Formula: see text]. Furthermore, if [Formula: see text] is multiple of 2, the approximation ratio can reduce to [Formula: see text], running in time [Formula: see text] at worst. For individual size constraint, there is a [Formula: see text]-approximation algorithm for maximizing a nonnegative [Formula: see text]-submodular function and an nonnegative [Formula: see text]-bisubmodular function, running in time [Formula: see text] and [Formula: see text] respectively, at worst.

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.

Monotone submodular maximization over the bounded integer lattice with cardinality constraints

2019 ◽  
Vol 11 (06) ◽  
pp. 1950075
Author(s):  
Lei Lai ◽  
Qiufen Ni ◽  
Changhong Lu ◽  
Chuanhe Huang ◽  
Weili Wu
Keyword(s):  
Approximation Algorithm ◽  
Greedy Algorithm ◽  
Deterministic Algorithm ◽  
Submodular Function ◽  
Integer Lattice ◽  
Cardinality Constraint ◽  
Cardinality Constraints ◽  
Definition Of ◽  
Deterministic Formula ◽  
Submodular Maximization

We consider the problem of maximizing monotone submodular function over the bounded integer lattice with a cardinality constraint. Function [Formula: see text] is submodular over integer lattice if [Formula: see text], [Formula: see text], where ∨ and ∧ represent elementwise maximum and minimum, respectively. Let [Formula: see text], and [Formula: see text], we study the problem of maximizing submodular function [Formula: see text] with constraints [Formula: see text] and [Formula: see text]. A random greedy [Formula: see text]-approximation algorithm and a deterministic [Formula: see text]-approximation algorithm are proposed in this paper. Both algorithms work in value oracle model. In the random greedy algorithm, we assume the monotone submodular function satisfies diminishing return property, which is not an equivalent definition of submodularity on integer lattice. Additionally, our random greedy algorithm makes [Formula: see text] value oracle queries and deterministic algorithm makes [Formula: see text] value oracle queries.

scholarly journals The Power of Subsampling in Submodular Maximization

2021 ◽  
Author(s):  
Christopher Harshaw ◽  
Ehsan Kazemi ◽  
Moran Feldman ◽  
Amin Karbasi
Keyword(s):  
Local Search ◽  
State Of The Art ◽  
Video Summarization ◽  
Submodular Function ◽  
Approximation Ratio ◽  
Present Sample ◽  
Movie Recommendation ◽  
Algorithmic Technique ◽  
Line Setting ◽  
Submodular Maximization

We propose subsampling as a unified algorithmic technique for submodular maximization in centralized and online settings. The idea is simple: independently sample elements from the ground set and use simple combinatorial techniques (such as greedy or local search) on these sampled elements. We show that this approach leads to optimal/state-of-the-art results despite being much simpler than existing methods. In the usual off-line setting, we present SampleGreedy, which obtains a [Formula: see text]-approximation for maximizing a submodular function subject to a p-extendible system using [Formula: see text] evaluation and feasibility queries, where k is the size of the largest feasible set. The approximation ratio improves to p + 1 and p for monotone submodular and linear objectives, respectively. In the streaming setting, we present Sample-Streaming, which obtains a [Formula: see text]-approximation for maximizing a submodular function subject to a p-matchoid using O(k) memory and [Formula: see text] evaluation and feasibility queries per element, and m is the number of matroids defining the p-matchoid. The approximation ratio improves to 4p for monotone submodular objectives. We empirically demonstrate the effectiveness of our algorithms on video summarization, location summarization, and movie recommendation tasks.

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.

AN APPROXIMATION ALGORITHM FOR LOCATING MAXIMAL DISKS WITHIN CONVEX POLYGONS

2011 ◽  
Vol 21 (06) ◽  
pp. 661-684
Author(s):  
HIROFUMI AOTA ◽  
TAKURO FUKUNAGA ◽  
HIROSHI NAGAMOCHI
Keyword(s):  
Approximation Algorithm ◽  
Convex Polygon ◽  
Computation Time ◽  
Approximation Ratio ◽  
Convex Polygons ◽  
The Given

This paper considers a problem of locating the given number of disks into a container so that the area covered by the disks is maximized. In the problem, the radii of the disks can be changed arbitrarily unless they overlap outside of the container, and the disks are allowed to overlap with each other. We present an approximation algorithm for this problem assuming that the container is a convex polygon. Our algorithm achieves approximation ratio (0.78 - ϵ) for any small ϵ > 0. Since the computation time of our algorithm depends on the number of corners of the convex polygon exponentially, we also give a heuristic to reduce the number of corners.

scholarly journals ON THE PIPAGE ROUNDING ALGORITHM FOR SUBMODULAR FUNCTION MAXIMIZATION — A VIEW FROM DISCRETE CONVEX ANALYSIS

2009 ◽  
Vol 01 (01) ◽  
pp. 1-23 ◽  
Author(s):  
AKIYOSHI SHIOURA
Keyword(s):  
Approximation Algorithm ◽  
Convex Analysis ◽  
Submodular Function ◽  
Submodular Functions ◽  
Concave Functions ◽  
Discrete Convex Analysis ◽  
Discrete Convex ◽  
Weighted Rank ◽  
Rounding Algorithm ◽  
Rank Functions

We consider the problem of maximizing a nondecreasing submodular set function under a matroid constraint. Recently, Calinescu et al. (2007) proposed an elegant framework for the approximation of this problem, which is based on the pipage rounding technique by Ageev and Sviridenko (2004), and showed that this framework indeed yields a (1 - 1/e)-approximation algorithm for the class of submodular functions which are represented as the sum of weighted rank functions of matroids. This paper sheds a new light on this result from the viewpoint of discrete convex analysis by extending it to the class of submodular functions which are the sum of M ♮-concave functions. M ♮-concave functions are a class of discrete concave functions introduced by Murota and Shioura (1999), and contain the class of the sum of weighted rank functions as a proper subclass. Our result provides a better understanding for why the pipage rounding algorithm works for the sum of weighted rank functions. Based on the new observation, we further extend the approximation algorithm to the maximization of a nondecreasing submodular function over an integral polymatroid. This extension has an application in multi-unit combinatorial auctions.

An Approximation Algorithm for a Heterogeneous Traveling Salesman Problem

2011 ◽  
Author(s):  
Jungyun Bae ◽  
Sivakumar Rathinam
Keyword(s):  
Approximation Algorithm ◽  
Traveling Salesman Problem ◽  
Approximation Ratio ◽  
Traveling Salesman ◽  
Dual Algorithm ◽  
Routing Problem ◽  
Primal Dual ◽  
Surveillance Applications

Surveillance applications require a collection of heterogeneous vehicles to visit a set of targets. In this article, we consider a fundamental routing problem that arises in these applications involving two vehicles. Specifically, we consider a routing problem where there are two heterogeneous vehicles that start from distinct initial locations, and a set of targets. The objective is to find a tour for each vehicle such that each of the targets is visited at least once by a vehicle and the sum of the distances traveled by the vehicles is a minimum. We present a novel primal-dual algorithm for the same that provides an approximation ratio of 2.

scholarly journals Simple and Efficient Budget Feasible Mechanisms for Monotone Submodular Valuations

2021 ◽  
Vol 9 (1) ◽  
pp. 1-20
Author(s):  
Pooya Jalaly ◽  
Éva Tardos
Keyword(s):  
Approximation Algorithm ◽  
Mechanism Design ◽  
Design Problem ◽  
Optimization Problem ◽  
Submodular Function ◽  
Individual Item ◽  
Large Market ◽  
Total Budget ◽  
Approximation Guarantee ◽  
Budget Feasible

We study the problem of a budget limited buyer who wants to buy a set of items, each from a different seller, to maximize her value. The budget feasible mechanism design problem requires the design a mechanism that incentivizes the sellers to truthfully report their cost and maximizes the buyer’s value while guaranteeing that the total payment does not exceed her budget. Such budget feasible mechanisms can model a buyer in a crowdsourcing market interested in recruiting a set of workers (sellers) to accomplish a task for her. This budget feasible mechanism design problem was introduced by Singer in 2010. We consider the general case where the buyer’s valuation is a monotone submodular function. There are a number of truthful mechanisms known for this problem. We offer two general frameworks for simple mechanisms, and by combining these frameworks, we significantly improve on the best known results, while also simplifying the analysis. For example, we improve the approximation guarantee for the general monotone submodular case from 7.91 to 5 and for the case of large markets (where each individual item has negligible value) from 3 to 2.58. More generally, given an r approximation algorithm for the optimization problem (ignoring incentives), our mechanism is a r + 1 approximation mechanism for large markets, an improvement from 2 r 2 . We also provide a mechanism without the large market assumption, where we achieve a 4 r + 1 approximation guarantee. We also show how our results can be used for the problem of a principal hiring in a Crowdsourcing Market to select a set of tasks subject to a total budget.

A parametric approximation algorithm for spatial group keyword queries

2021 ◽  
Vol 25 (2) ◽  
pp. 305-319
Author(s):  
Jincao Li ◽  
Ming Xu
Keyword(s):  
Approximation Algorithm ◽  
Large Scale ◽  
Goal Function ◽  
Approximate Solutions ◽  
Approximation Ratio ◽  
Query Point ◽  
Database Applications ◽  
Benchmark Datasets ◽  
Goal Value ◽  
Parametric Approximation

With the application of big data, various queries arise for information retrieval. Spatial group keyword queries aim to find a set of spatial objects that cover the query keywords and minimize a goal function such as the total distance between the objects and the query point. This problem is widely found in database applications and is known to be NP-hard. Efficient algorithms for solving this problem can only provide approximate solutions, and most of these algorithms achieve a fixed approximation ratio (the upper bound of the ratio of an approximate goal value to the optimal goal value). Thus, to obtain a self-adjusting algorithm, we propose an approximation algorithm for achieving a parametric approximation ratio. The algorithm makes a trade-off between the approximation ratio and time consumption enabling the users to assign arbitrary query accuracy. Additionally, it runs in an on-the-fly manner, making it scalable to large-scale applications. The efficiency and scalability of the algorithm were further validated using benchmark datasets.

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