Submodular Penalties Research Articles

An approximation algorithm for the -prize-collecting multicut problem in trees with submodular penalties

AbstractLet $T=(V,E)$ be a tree in which each edge is assigned a cost; let $\mathcal{P}$ be a set of source–sink pairs of vertices in V in which each source–sink pair produces a profit. Given a lower bound K for the profit, the K-prize-collecting multicut problem in trees with submodular penalties is to determine a partial multicut $M\subseteq E$ such that the total profit of the disconnected pairs after removing M from T is at least K, and the total cost of edges in M plus the penalty of the set of still-connected pairs is minimized, where the penalty is determined by a nondecreasing submodular function. Based on the primal-dual scheme, we present a combinatorial polynomial-time algorithm by carefully increasing the penalty. In the theoretical analysis, we prove that the approximation factor of the proposed algorithm is $(\frac{8}{3}+\frac{4}{3}\kappa+\varepsilon)$ , where $\kappa$ is the total curvature of the submodular function and $\varepsilon$ is any fixed positive number. Experiments reveal that the objective value of the solutions generated by the proposed algorithm is less than 130% compared with that of the optimal value in most cases.

A Combinatorial Approximation Algorithm for the Vector Scheduling with Submodular Penalties on Parallel Machines

In this paper, we focus on solving the vector scheduling problem with submodular penalties on parallel machines. We are given n jobs and m parallel machines, where each job is associated with a d -dimensional vector. Each job can either be rejected, incurring a rejection penalty, or accepted and processed on one of the m parallel machines. The objective is to minimize the sum of the maximum load overall dimensions of the total vector for all accepted jobs, along with the total penalty for rejected jobs. The penalty is determined by a submodular function. Our main work is to design a 2 − 1 / m min r , d -approximation algorithm to solve this problem. Here, r denotes the maximum ratio of the maximum load to the minimum load on the d -dimensional vectors among all jobs.

A combinatorial approximation algorithm for k-level facility location problem with submodular penalties

A combinatorial approximation algorithm for k-level facility location problem with submodular penalties

Algorithms for single machine scheduling problem with release dates and submodular penalties

Algorithms for single machine scheduling problem with release dates and submodular penalties

A primal-dual approximation algorithm for the k-prize-collecting minimum vertex cover problem with submodular penalties

A primal-dual approximation algorithm for the k-prize-collecting minimum vertex cover problem with submodular penalties

An approximation algorithm for the group prize-collecting Steiner tree problem with submodular penalties

An approximation algorithm for the group prize-collecting Steiner tree problem with submodular penalties

An Approximation Algorithm for the Parallel-Machine Customer Order Scheduling with Delivery Time and Submodular Rejection Penalties

An Approximation Algorithm for the Parallel-Machine Customer Order Scheduling with Delivery Time and Submodular Rejection Penalties

${\boldsymbol~k}$-prize-collecting minimum power cover problem with submodular penalties on a plane

Given a set of $n$ points, a set of $m$ sensors on a plane, and a positive integer $k$ ($\leq~n$).Each sensor $s$ can adjust its power $p(s)$ and covering range, which is a disk of radius $r(s)$ satisfying $p(s)=r(s)^{\alpha}$, where $\alpha\geq~1$ is called the attenuation factor of power. The $k$-prize-collecting minimum power cover problem with submodular penalties on a plane determines a power assignmentsuch that at least $k$ users are covered. The goal is to minimize the overall power of the power assignment plus the penalty of the uncovered user set, where the penalty is determined by a submodular function. This problem generalizes the well-known minimum power cover problem, minimum power partial cover problem, and $k$-prize-collecting minimum power cover problem.Based on the geometric properties of the semi-disjoint disk set and primal-dual method, we present a two-phase $(5\cdot~2^{\alpha}+1)$-approximation algorithm. When the penalty function is linear, the approximation ratio of our algorithm is at most $5\cdot~2^{\alpha}$.

A Combinatorial 2-Approximation Algorithm for the Parallel-Machine Scheduling with Release Times and Submodular Penalties

In this paper, we consider parallel-machine scheduling with release times and submodular penalties (P|rj,reject|Cmax+π(R)), in which each job can be accepted and processed on one of m identical parallel machines or rejected, but a penalty must paid if a job is rejected. Each job has a release time and a processing time, and the job can not be processed before its release time. The objective of P|rj,reject|Cmax+π(R) is to minimize the makespan of the accepted jobs plus the penalty of the rejected jobs, where the penalty is determined by a submodular function. This problem generalizes a multiprocessor scheduling problem with rejection, the parallel-machine scheduling with submodular penalties, and the single machine scheduling problem with release dates and submodular rejection penalties. In this paper, inspired by the primal-dual method, we present a combinatorial 2-approximation algorithm to P|rj,reject|Cmax+π(R). This ratio coincides with the best known ratio for the parallel-machine scheduling with submodular penalties and the single machine scheduling problem with release dates and submodular rejection penalties.

An Approximation Algorithm for the Generalized Prize-Collecting Steiner Forest Problem with Submodular Penalties

An Approximation Algorithm for the Generalized Prize-Collecting Steiner Forest Problem with Submodular Penalties

Approximation algorithms for the submodular edge cover problem with submodular penalties

In this paper, we consider the submodular edge cover problem with submodular penalties. In this problem, we are given an undirected graph G=(V,E) with vertex set V and edge set E. Assume the covering cost function c:2E→R+ and the penalty function p:2V→R+ are both submodular with p non-decreasing, c(∅)=0 and p(∅)=0. The goal of the submodular edge cover problem with submodular penalties is to select an edge subset to cover some vertices and penalize the vertex subset containing uncovered vertices such that the total cost of covering and penalty is minimized. For this problem, we first give a 2Δ-approximation algorithm by using a primal-dual technique, where Δ is the maximal degree of the graph G. Then we transform this problem into a submodular set cover problem, and by applying a known result for the submodular set cover problem we conclude that there is an approximation algorithm with an approximation ratio Δ+1.

Approximation algorithms for the multiprocessor scheduling with submodular penalties

In this paper, we consider multiprocessor scheduling with submodular penalties to extend multiprocessor scheduling with rejection to submodular function. An instance of the problem is given by n jobs and m machines with each job having a certain processing time on a machine. We aim to find a subset R of rejected jobs, and assign each of other jobs to one of the m machines. The objective is to minimize the sum of the makespan of the m machines and the rejection penalty R, where the rejection penalty is determined by a submodular function. For this problem, we design a non-combinatorial Lovasz rounding algorithm that achieves a worst-case guarantee of $$\frac{3+\sqrt{5}}{2}$$ . Then, we consider a special case of this problem in which all the machines are identical, i.e. each job has the same processing time on any machine, and we design a combinatorial $$(2-\frac{1}{m})$$ -approximation algorithm based on the greedy method and list scheduling (LS) algorithm.

Approximation Algorithms for the Submodular Load Balancing with Submodular Penalties

In this paper, we study the submodular load balancing problem with submodular penalties. The objective of this problem is to balance the load among sets, while some elements can be rejected by paying some penalties. Officially, given an element set V, we want to find a subset R of rejected elements, and assign other elements to one of m sets A1,A2,⋯,Am. The objective is to minimize the sum of the maximum load among A1,A2,⋯,Am and the rejection penalty of R, where the load and rejection penalty are determined by different submodular functions. We study the submodular load balancing problem with submodular penalties under two settings: heterogenous setting (load functions are not identical) and homogenous setting (load functions are identical). Moreover, we design a Lovász rounding algorithm achieving a worst-case guarantee of m+1 under the heterogenous setting and a min{m,⌈nm⌉+1}=O(n)-approximation combinatorial algorithm under the homogenous setting.

Approximation algorithms for the dynamic k-level facility location problems

In this paper, we first consider a dynamic k-level facility location problem, which is a generalization of the k-level facility location problem when considering time factor. We present a combinatorial primal-dual approximation algorithm for this problem which finds a constant factor approximate solution. Then, we investigative the dynamic k-level facility location problem with submodular penalties and outliers, which extend the existing problem on two fronts, namely from static to dynamic and from without penalties (outliers) to penalties (outliers) allowed. Based on primal-dual technique and the triangle inequality property, we also give two constant factor approximation algorithms for the dynamic problem with submodular penalties and outliers, respectively.

Combinatorial approximation algorithms for the submodular multicut problem in trees with submodular penalties

In this paper, we introduce the submodular multicut problem in trees with submodular penalties, which generalizes the prize-collecting multicut problem in trees and the submodular vertex cover with submodular penalties. We present a combinatorial approximation algorithm, based on the primal-dual algorithm for the submodular set cover problem. In addition, we present a combinatorial 3-approximation algorithm for a special case where the edge cost is a modular function, based on the primal-dual scheme for the multicut problem in trees.

Approximation Algorithm for Resource Allocation Problems with Time Dependent Penalties

The Resource Allocation Problem with Time Dependent Penalties (RAPTP) is a variant of uncapacitated resource allocation problems generally referred as uncapacitated facility allocation problems or uncapacitated facility location problem (UFLP). Work done in this paper is motivated by the work of Du, Lu and Xu [7] in which authors considered facility location problems with submodular penalties and presented a 3-approximation primal dual algorithm. This paper considers that each unallocated demand point adds to penalty that increases as time passes and is thus represented by function [Formula: see text] where [Formula: see text] and [Formula: see text] are elapse time and priority of demand point [Formula: see text]. As this problem has been considered for emergency service allocation, all demand points should be allocated to some facility or resource within some stipulated time limit beyond which it may lose its purpose. Thus penalty incurred by a demand point is considered till that threshold value only. Thus it is assumed that penalty contribution by a demand point remains constant after a specified threshold value. By exploiting the properties of time dependent penalties, a 4-approximation primal-dual algorithm is proposed which is based on LP framework, and is the first constant-factor approximation algorithm for RAPTP.

A note on the submodular vertex cover problem with submodular penalties

In this paper, we prove that there exists a combinatorial 3-approximation algorithm for the submodular vertex cover problem with submodular penalties introduced by Xu, Wang, Du, and Wu.

Approximation algorithms for submodular vertex cover problems with linear/submodular penalties using primal-dual technique

The notion of penalty has been introduced into many combinatorial optimization models. In this paper, we consider the submodular vertex cover problems with linear and submodular penalties, which are two variants of the submodular vertex cover problem where not all the edges are required to be covered by a vertex cover, and the uncovered edges are penalized. The problem is to determine a vertex subset to cover some edges and penalize the uncovered edges such that the total cost including covering and penalty is minimized. To overcome the difficulty of implementing the primal-dual framework directly, we relax the two dual programs to slightly weaker versions. We then present two primal-dual approximation algorithms with approximation ratios of 2 and 4, respectively.

Improved Approximation Algorithms for the Facility Location Problems with Linear/Submodular Penalties

We consider the facility location problem with submodular penalties (FLPSP) and the facility location problem with linear penalties (FLPLP), two extensions of the classical facility location problem (FLP). First, we introduce a general algorithmic framework for a class of covering problems with submodular penalties, extending the recent result of Geunes et al. (Math Program 130:85---106, 2011) with linear penalties. This framework leverages existing approximation results for the original covering problems to obtain corresponding results for their counterparts with submodular penalties. Specifically, any LP-based $$\alpha $$?-approximation for the original covering problem can be leveraged to obtain an $$\left( 1-e^{-1/\alpha }\right) ^{-1}$$1-e-1/?-1-approximation algorithm for the counterpart with submodular penalties. Consequently, any LP-based approximation algorithm for the classical FLP (as a covering problem) can yield, via this framework, an approximation algorithm for the counterpart with submodular penalties. Second, by exploiting some special properties of submodular/linear penalty function, we present an LP rounding algorithm which has the currently best $$2$$2-approximation ratio over the previously best $$2.375$$2.375 by Li et al. (Theoret Comput Sci 476:109---117, 2013) for the FLPSP and another LP-rounding algorithm which has the currently best $$1.5148$$1.5148-approximation ratio over the previously best $$1.853$$1.853 by Xu and Xu (J Comb Optim 17:424---436, 2008) for the FLPLP, respectively.

An approximation algorithm for the dynamic facility location problem with submodular penalties

In this paper, we study the dynamic facility location problem with submodular penalties (DFLPSP). We present a combinatorial primal-dual 3-approximation algorithm for the DFLPSP.