Bicriteria Approximation Algorithm for Quarantining-vaccination-cure Problem

In this paper, we propose a model for the epidemic control problem, the goal of which is to minimize the total cost of quarantining, vaccination and cure under the constraint on the maximum number of infected people allowed. A (1+ε+ε3 , 1+ ε+1/ε )- bicriteria approximation algorithm is given.

Exact and Approximation Algorithms for the Multi-Depot Capacitated Arc Routing Problems

In this work,we investigate a generalization of the classical capacitated arc routing problem,called the Multi-depot Capacitated Arc Routing Problem(MCARP).We give exact and approximation algorithms for different variants of the MCARP.First,we obtain the first constant-ratio approximation algorithms for the MCARP and its nonfixed destination version.Second,for the multi-depot rural postman problem,i.e.,a special case of the MCARP where the vehicles have infinite capacity,we develop a(2-1/2k+1)-approximation algorithm(k denotes the number of depots).Third,we show the polynomial solvability of the equal-demand MCARP on a line and devise a 2-approximation algorithm for the multi-depot capacitated vehicle routing problem on a line.Lastly,we conduct extensive numerical experiments on the algorithms for the multi-depot rural postman problem to show their effectiveness.

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

In this paper,we consider the-prize-collecting minimum vertex cover problem with submodular penalties,which generalizes the well-known minimum vertex cover problem,minimum partial vertex cover problem and minimum vertex cover problem with submodular penalties.We are given a cost graph and an integer.This problem determines a vertex set such that covers at least edges.The objective is to minimize the total cost of the vertices in plus the penalty of the uncovered edge set,where the penalty is determined by a submodular function.We design a two-phase combinatorial algorithm based on the guessing technique and the primal-dual framework to address the problem.When the submodular penalty cost function is normalized and nondecreasing,the proposed algorithm has an approximation factor of.When the submodular penalty cost function is linear,the approximation factor of the proposed algorithm is reduced to,which is the best factor if the unique game conjecture holds.

Approximation Algorithms for Graph Partition into Bounded Independent Sets

The partition problem of a given graph into three independent sets of minimizing the maximum one is studied in this paper.This problem is NP-hard,even restricted to bipartite graphs.First,a simple 3/2-approximation algorithm for any 2-colorable graph is presented.An improved 7/5-approximation algorithm is then designed for a tree.The theoretical proof of the improved algorithm performance ratio is constructive,thus providing an explicit partition approach for each case according to the cardinality of two color classes.

An Approximation Algorithm for P-prize-collecting Set Cover Problem

In this paper,we consider the P-prize-collecting set cover(P-PCSC)problem,which is a generalization of the set cover problem.In this problem,we are given a set system(U,S),where U is a ground set and S⊆2U is a collection of subsets satisfying∪S∈SS=U.Every subset in S has a nonnegative cost,and every element in U has a nonnegative penalty cost and a nonnegative profit.Our goal is to find a subcollection C⊆S such that the total cost,consisting of the cost of subsets in C and the penalty cost of the elements not covered by C,is minimized and at the same time the combined profit of the elements covered by C is at least P,a specified profit bound.Our main work is to obtain a 2f+ε-approximation algorithm for the P-PCSC problem by using the primal-dual and Lagrangian relaxation methods,where f is the maximum frequency of an element in the given set system(U,S)andεis a fixed positive number.

Approximation Algorithms for Multi-vehicle Stacker Crane Problems

We study a variety of multi-vehicle generalizations of the Stacker Crane Problem(SCP).The input consists of a mixed graph G=(V,E,A)with vertex set V,edge set E and arc set A,and a nonnegative integer cost function c on E∪A.We consider the following three problems:(1)k-depot SCP(k-DSCP).There is a depot set D⊆V containing k distinct depots.The goal is to determine a collection of k closed walks including all the arcs of A such that the total cost of the closed walks is minimized,where each closed walk corresponds to the route of one vehicle and has to start from a distinct depot and return to it.(2)k-SCP.There are no given depots,and each vehicle may start from any vertex and then go back to it.The objective is to find a collection of k closed walks including all the arcs of A such that the total cost of the closed walks is minimized.(3)k-depot Stacker Crane Path Problem(k-DSCPP).There is a depot set D⊆V containing k distinct depots.The aim is to find k(open)walks including all the arcs of A such that the total cost of the walks is minimized,where each(open)walk has to start from a distinct depot but may end at any vertex.We present the first constant-factor approximation algorithms for all the above three problems.To be specific,we give 3-approximation algorithms for the k-DSCP,the k-SCP and the k-DSCPP.If the costs of the arcs are symmetric,i.e.,for every arc there is a parallel edge of no greater cost,we develop better algorithms with approximation ratios max{9/5,2−1/2k+1},2,2,respectively.All the proposed algorithms have a time complexity of O(|V|3)except that the two 2-approximation algorithms run in O(|V|2log|V|)time.

Approximation Algorithms on k-Correlation Clustering

In this paper,we consider the k-correlation clustering problem.Given an edge-weighted graph G(V,E)where the edges are labeled either“+”(similar)or“−”(different)with nonnegative weights,we want to partition the nodes into at most k-clusters to maximize agreements—the total weights of“+”edges within clusters and“−”edges between clusters.This problem is NP-hard.We design an approximation algorithm with the approximation ratio{a,(2-k)a+k-1/k},where a is the weighted proportion of“+”edges in all edges.As a varies from 0 to 1,the approximation ratio varies from k-1/k to 1 and the minimum value is 1/2.

Approximation Algorithms for Discrete Polynomial Optimization

In this paper,we consider approximation algorithms for optimizing a generic multivariate polynomial function in discrete(typically binary)variables.Such models have natural applications in graph theory,neural networks,error-correcting codes,among many others.In particular,we focus on three types of optimization models:(1)maximizing a homogeneous polynomial function in binary variables;(2)maximizing a homogeneous polynomial function in binary variables,mixed with variables under spherical constraints;(3)maximizing an inhomogeneous polynomial function in binary variables.We propose polynomial-time randomized approximation algorithms for such polynomial optimizationmodels,and establish the approximation ratios(or relative approximation ratios whenever appropriate)for the proposed algorithms.Some examples of applications for these models and algorithms are discussed as well.

A Tight Approximation Algorithm for Multi-Vehicle CVRP with Unsplittable Demands on a Line

In this paper,the authors study the multi-vehicle capacitated vehicle routing problem on a line-shaped network with unsplittable demand.The objective is to find a transportation scheme to minimize the longest distance traveled by a single vehicle such that all the customers are served without violating the capacity constraint.The authors show that this problem has no polynomialtime algorithm with performance ratio less than 2 on condition that P≠NP,and then provide a 2-approximation algorithm.

Approximation Algorithms for Vertex Happiness

We investigate the maximum happy vertices(MHV)problem and its complement,the minimum unhappy vertices(MUHV)problem.In order to design better approximation algorithms,we introduce the supermodular and submodular multi-labeling(SUP-ML and SUB-ML)problems and show that MHV and MUHV are special cases of SUP-ML and SUB-ML,respectively,by rewriting the objective functions as set functions.The convex relaxation on the I ovasz extension,originally presented for the submodular multi-partitioning problem,can be extended for the SUB-ML problem,thereby proving that SUB-ML(SUP-ML,respectively)can be approximated within a factorof2-2/k(2/k,respectively),where k is the number of labels.These general results imply that MHV and MUHV can also be approximated within factors of 2/k and 2-2/k,respectively,using the same approximation algorithms.For the MUHV problem,we also show that it is approximation-equivalent to the hypergraph multiway cut problem;thus,MUHV is Unique Games-hard to achieve a(2-2/k-e)-approximation,for anyε>0.For the MHV problem,the 2/k-approximation improves the previous best approximation ratio max{1/k,1/(△+1/g(△)},where△is the maximum vertex degree of the input graph and g(△)=(√△+√△+1)2△>4△2.We also show that an existing LP relaxation for MHV is the same as the concave relaxation on the Lovasz extension for SUP-ML;we then prove an upper bound of 2/k on the integrality gap of this LP relaxation,which suggests that the 2/k-approximation is the best possible based on this LP relaxation.Lastly,we prove that it is Unique Games-hard to approximate the MHV problem within a factor of S2(log2 k/k).

Approximation Algorithm for the Balanced 2-Correlation Clustering Problem

The Correlation Clustering Problem(CorCP) is a significant clustering problem based on the similarity of data.It has significant applications in different fields,such as machine learning,biology,and data mining,and many different problems in other areas.In this paper,the Balanced 2-CorCP(B2-CorCP) is introduced and examined,and a new interesting variant of the CorCP is described.The goal of this clustering problem is to partition the vertex set into two clusters with equal size,such that the number of disagreements is minimized.We first present a polynomial time algorithm for the B2-CorCP on M-positive edge dominant graphs(M≥ 3).Then,we provide a series of numerical experiments,and the results show the effectiveness of our algorithm.

Improved Approximation Algorithm for Scheduling on a Serial Batch Machine with Split-Allowed Delivery

This paper considers the integrated production and delivery scheduling on a serial batch machine,in which split is allowed in the delivery of the jobs.The objective is to minimize the makespan,i.e.,the maximum delivery completion time of the jobs.Lu et al.(Theor Comput Sci 572:50–57,2015)showed that this problem is strongly NP-hard,and presented a 32-approximation algorithm.In this paper,we present an improved 43-approximation algorithm for this problem.We also present a polynomial-time algorithm for the special case when all jobs have the identical weight.

An Improved Approximation Algorithm for the k-Level Facility Location Problem with Soft Capacities

We consider the k-level facility location problem with soft capacities(k-LFLPSC). In the kLFLPSC, each facility i has a soft capacity u_i along with an initial opening cost f_i ≥ 0, i.e., the capacity of facility i is an integer multiple of u_i incurring a cost equals to the corresponding multiple of f_i. We firstly propose a new bifactor(ln(1/β)/(1-β), 1 + 2/(1-β))-approximation algorithm for the k-level facility location problem(k-LFLP), where β∈(0, 1) is a fixed constant. Then, we give a reduction from the k-LFLPSC to the k-LFLP. The reduction together with the above bifactor approximation algorithm for the k-LFLP imply a5.5053-approximation algorithm for the k-LFLPSC which improves the previous 6-approximation.

An Approximation Algorithm for the Risk-Adjusted Two-Stage Stochastic Facility Location Problem with Penalties

In this paper,we consider the risk-adjusted two-stage stochastic facility location problem with penalties(RSFLPP).Using the monotonicity and positive homogeneity of the risk measure function,we present an LP-rounding-based 6-approximation algorithm.

Approximation Algorithms for Stochastic Combinatorial Optimization Problems

Stochastic optimization has established itself as a major method to handle uncertainty in various optimization problems by modeling the uncertainty by a probability distribution over possible realizations.Traditionally,the main focus in stochastic optimization has been various stochastic mathematical programming(such as linear programming,convex programming).In recent years,there has been a surge of interest in stochastic combinatorial optimization problems from the theoretical computer science community.In this article,we survey some of the recent results on various stochastic versions of classical combinatorial optimization problems.Since most problems in this domain are NP-hard(or#P-hard,or even PSPACE-hard),we focus on the results which provide polynomial time approximation algorithms with provable approximation guarantees.Our discussions are centered around a few representative problems,such as stochastic knapsack,stochastic matching,multi-armed bandit etc.We use these examples to introduce several popular stochastic models,such as the fixed-set model,2-stage stochastic optimization model,stochastic adaptive probing model etc,as well as some useful techniques for designing approximation algorithms for stochastic combinatorial optimization problems,including the linear programming relaxation approach,boosted sampling,content resolution schemes,Poisson approximation etc.We also provide some open research questions along the way.Our purpose is to provide readers a quick glimpse to the models,problems,and techniques in this area,and hopefully inspire new contributions.

Approximation Algorithms on k-Cycle Transversal and k-Clique Transversal

Given a weighted graph G=(V,E)with weight w:E→Z+,a k-cycle transversal is an edge subset A of E such that G−A has no k-cycle.The minimum weight of kcycle transversal is the weighted transversal number on k-cycle,denoted byτk(Gw).In this paper,we design a(k−1/2)-approximation algorithm for the weighted transversal number on k-cycle when k is odd.Given a weighted graph G=(V,E)with weight w:E→Z+,a k-clique transversal is an edge subset A of E such that G−A has no k-clique.The minimum weight of k-clique transversal is the weighted transversal number on k-clique,denoted byτapproximation algorithm for the weighted transversal number on k(Gw).In this paper,we design a(k2−k−1)/2-k-clique.Last,we discuss the relationship between k-clique covering and k-clique packing in complete graph Kn.

An Approximation Algorithm for the Stochastic Fault-Tolerant Facility Location Problem

In this paper,we study a stochastic version of the fault-tolerant facility location problem.By exploiting the stochastic structure,we propose a 5-approximation algorithm which uses the LP-rounding technique based on the revised optimal solution to the linear programming relaxation of the stochastic fault-tolerant facility location problem.

Distributed dynamic stochastic approximation algorithm over time-varying networks

In this paper,a distributed stochastic approximation algorithm is proposed to track the dynamic root of a sum of time-varying regression functions over a network.Each agent updates its estimate by using the local observation,the dynamic information of the global root,and information received from its neighbors.Compared with similar works in optimization area,we allow the observation to be noise-corrupted,and the noise condition is much weaker.Furthermore,instead of the upper bound of the estimate error,we present the asymptotic convergence result of the algorithm.The consensus and convergence of the estimates are established.Finally,the algorithm is applied to a distributed target tracking problem and the numerical example is presented to demonstrate the performance of the algorithm.

The Quantum Approximate Algorithm for Solving Traveling Salesman Problem

The Quantum Approximate Optimization Algorithm(QAOA)is an algorithmic framework for finding approximate solutions to combinatorial optimization problems.It consists of interleaved unitary transformations induced by two operators labelled the mixing and problem Hamiltonians.To fit this framework,one needs to transform the original problem into a suitable form and embed it into these two Hamiltonians.In this paper,for the well-known NP-hard Traveling Salesman Problem(TSP),we encode its constraints into the mixing Hamiltonian rather than the conventional approach of adding penalty terms to the problem Hamiltonian.Moreover,we map edges(routes)connecting each pair of cities to qubits,which decreases the search space significantly in comparison to other approaches.As a result,our method can achieve a higher probability for the shortest round-trip route with only half the number of qubits consumed compared to IBM Q’s approach.We argue the formalization approach presented in this paper would lead to a generalized framework for finding,in the context of QAOA,high-quality approximate solutions to NP optimization problems.

Two-Agent Makespan Minimization Problem on Parallel Machines

A two-agent scheduling problem on parallel machines is considered in this paper. Our objective is to minimize the makespan for agent A, subject to an upper bound on the makespan for agent B. In this paper, we provide a new approximation algorithm called CLPT. On the one hand, we compare the performance between the CLPT algorithm and the optimal solution and find that the solution obtained by the CLPT algorithm is very close to the optimal solution. On the other hand, we design different experimental frameworks to compare the CLPT algorithm and the A-LS algorithm for a comprehensive performance evaluation. A large number of numerical simulation results show that the CLPT algorithm outperformed the A-LS algorithm.