Min-sum Objective Research Articles

An effective iterated two-stage heuristic algorithm for the multiple Traveling Salesmen Problem

The multiple Traveling Salesmen Problem (mTSP) is a general extension of the famous NP-hard Traveling Salesmen Problem (TSP), that there are m (m>1) salesmen to visit the cities. In this paper, we address the mTSP with both the minsum objective and minmax objective, which aims at minimizing the total length of the m tours and the length of the longest tour among all the m tours, respectively. We propose an iterated two-stage heuristic algorithm called ITSHA for the mTSP. Each iteration of ITSHA consists of an initialization stage and an improvement stage. The initialization stage aims to generate high-quality and diverse initial solutions. The improvement stage mainly applies the variable neighborhood search (VNS) approach based on our proposed effective local search neighborhoods to optimize the initial solution. Moreover, some local optima escaping approaches are employed to enhance the search ability of the algorithm. Extensive experimental results on a wide range of public benchmark instances show that ITSHA significantly outperforms the state-of-the-art heuristic algorithms in solving the mTSP on both the objectives.

Subsidy Allocations in the Presence of Income Shocks

Poverty and economic hardship are understood to be highly complex and dynamic phenomena. Due to the multi-faceted nature of welfare, assistance programs targeted at alleviating hardship can face challenges, as they often rely on simpler welfare measurements, such as income or wealth, that fail to capture to full complexity of each family's state. Here, we explore one important dimension – susceptibility to income shocks. We introduce a model of welfare that incorporates income, wealth, and income shocks and analyze this model to show that it can vary, at times substantially, from measures of welfare that only use income or wealth. We then study the algorithmic problem of optimally allocating subsidies in the presence of income shocks. We consider two well-studied objectives: the first aims to minimize the expected number of agents that fall below a given welfare threshold (a min-sum objective) and the second aims to minimize the likelihood that the most vulnerable agent falls below this threshold (a min-max objective). We present optimal and near-optimal algorithms for various general settings. We close with a discussion on future directions on allocating societal resources and ethical implications of related approaches.

On the approximability of Time Disjoint Walks

We introduce the combinatorial optimization problem Time Disjoint Walks (TDW), which has applications in collision-free routing of discrete objects (e.g., autonomous vehicles) over a network. This problem takes as input a digraph $G$ with positive integer arc lengths, and $k$ pairs of vertices that each represent a trip demand from a source to a destination. The goal is to find a walk and delay for each demand so that no two trips occupy the same vertex at the same time, and so that a min-max or min-sum objective over the trip durations is realized. We focus here on the min-sum variant of Time Disjoint Walks, although most of our results carry over to the min-max case. We restrict our study to various subclasses of DAGs, and observe that there is a sharp complexity boundary between Time Disjoint Walks on oriented stars and on oriented stars with the central vertex replaced by a path. In particular, we present a poly-time algorithm for min-sum and min-max TDW on the former, but show that min-sum TDW on the latter is NP-hard. Our main hardness result is that for DAGs with max degree $\Delta\leq3$, min-sum Time Disjoint Walks is APX-hard. We present a natural approximation algorithm for the same class, and provide a tight analysis. In particular, we prove that it achieves an approximation ratio of $\Theta(k/\log k)$ on bounded-degree DAGs, and $\Theta(k)$ on DAGs and bounded-degree digraphs.

Inverse Max + Sum spanning tree problem by modifying the sum-cost vector under weighted $$l_\infty $$ l ∞ Norm

The inverse max + sum spanning tree (IMSST) problem is studied, which is the first inverse problem on optimization problems with combined minmax---minsum objective functions. Given an edge-weighted undirected network $$G(V,E,c,w)$$ G ( V , E , c , w ) , the MSST problem is to find a spanning tree $$T$$ T which minimizes the combined weight $$\max _{e\in T}w(e)+\sum _{e\in T}c(e)$$ max e ? T w ( e ) + ? e ? T c ( e ) , which can be solved in $$O(m\log n)$$ O ( m log n ) time, where $$m:=|E|$$ m : = | E | and $$n:=|V|$$ n : = | V | . Whereas, in an IMSST problem, a spanning tree $$T_0$$ T 0 of $$G$$ G is given, which is not an optimal MSST. A new sum-cost vector $$\bar{c}$$ c ¯ is to be identified so that $$T_0$$ T 0 becomes an optimal MSST of the network $$G(V,E,\bar{c},w)$$ G ( V , E , c ¯ , w ) , where $$0\le c-l\le \bar{c} \le c+u$$ 0 ≤ c - l ≤ c ¯ ≤ c + u and $$l,u\ge 0$$ l , u ? 0 . The objective is to minimize the cost $$\max _{e\in E}q(e)|\bar{c}(e)-c(e)|$$ max e ? E q ( e ) | c ¯ ( e ) - c ( e ) | incurred by modifying the sum-cost vector $$c$$ c under weighted $$l_\infty $$ l ? norm, where $$q(e)\ge 1$$ q ( e ) ? 1 . We show that the unbounded IMSST problem is a linear fractional combinatorial optimization (LFCO) problem and develop a discrete type Newton method to solve it. Furthermore, we prove an $$O(m)$$ O ( m ) bound on the number of iterations, although most LFCO problems can be solved in $$O(m^2 \log m)$$ O ( m 2 log m ) iterations. Therefore, both the unbounded and bounded IMSST problems can be solved by solving $$O(m)$$ O ( m ) MSST problems. Computational results show that the algorithms can efficiently solve the problems.

Vehicle routing problems with regular objective functions on a path

AbstractWe investigate the problem of scheduling a fleet of vehicles to visit the customers located on a path to minimize some regular function of the visiting times of the customers. For the single‐vehicle problem, we prove that it is pseudopolynomially solvable for any minsum objective and polynomially solvable for any minmax objective. Also, we establish the NP‐hardness of minimizing the weighted number of tardy customers and the total weighted tardiness, and present polynomial algorithms for their special cases with a common due date. For the multivehicle problem involving n customers, we show that an optimal solution can be found by solving or O(n) single‐vehicle problems. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 61: 34–43, 2014

Clustering under approximation stability

A common approach to clustering data is to view data objects as points in a metric space, and then to optimize a natural distance-based objective such as the k -median, k -means, or min-sum score. For applications such as clustering proteins by function or clustering images by subject, the implicit hope in taking this approach is that the optimal solution for the chosen objective will closely match the desired “target” clustering (e.g., a correct clustering of proteins by function or of images by who is in them). However, most distance-based objectives, including those mentioned here, are NP-hard to optimize. So, this assumption by itself is not sufficient, assuming P ≠ NP, to achieve clusterings of low-error via polynomial time algorithms. In this article, we show that we can bypass this barrier if we slightly extend this assumption to ask that for some small constant c , not only the optimal solution, but also all c -approximations to the optimal solution, differ from the target on at most some ϵ fraction of points—we call this (c,ϵ)-approximation-stability . We show that under this condition, it is possible to efficiently obtain low-error clusterings even if the property holds only for values c for which the objective is known to be NP-hard to approximate. Specifically, for any constant c > 1, (c,ϵ) -approximation-stability of k -median or k -means objectives can be used to efficiently produce a clustering of error O (ϵ) with respect to the target clustering, as can stability of the min-sum objective if the target clusters are sufficiently large. Thus, we can perform nearly as well in terms of agreement with the target clustering as if we could approximate these objectives to this NP-hard value.

Scheduling an unbounded batching machine with family jobs and setup times

We consider the problem of scheduling n jobs on an unbounded batching machine that can process any number of jobs belonging to the same family simultaneously in the same batch. All jobs in the same batch complete at the same time. Jobs belonging to different families cannot be processed in the same batch, and setup times are required to switch between batches that process jobs from different families. For a fixed number of families m, we present a generic forward dynamic programming algorithm that solves the problem of minimizing an arbitrary regular cost function in pseudopolynomial time. We also present a polynomial-time backward dynamic programming algorithm with time complexity O (mn(n/m+1)m) for minimizing any additive regular minsum objective function and any incremental regular minmax objective function. The effectiveness of our dynamic programming algorithm is shown by computational experiments based on the scheduling of the heat-treating process in a steel manufacturing plant.

A time-indexed LP-based approach for min-sum job-shop problems

In this paper we propose two time-indexed IP formulations for job-shop scheduling problems with a min-sum objective. The first model has variables associated to job scheduling patterns. The exponential number of variables calls for a column generation scheme which is carried out by a dynamic programming procedure. The second model is of network flow type with side constraints. This model can be strengthened by adding cutting inequalities of clique type. It turns out that the two models are equivalent, since the dual of the second formulation is equivalent to the compact dual of the first model. However, they require significantly different solution approaches and may behave differently in terms of computing time and memory usage. Good upper bounds are found by a heuristic procedure that randomly generates schedules from fractional solutions. These features allow for an effective pruning of the branch-and-bound tree and narrowing the gap between lower and upper bounds. However, the size of both models is critically affected by the time-indexed formulation which may heavily slow down the computation.

Improved bounds for scheduling conflicting jobs with minsum criteria

We consider a general class of scheduling problems where a set of conflicting jobs needs to be scheduled (preemptively or nonpreemptively) on a set of machines so as to minimize the weighted sum of completion times. The conflicts among jobs are formed as an arbitrary conflict graph. Building on the framework of Queyranne and Sviridenko [2002b], we present a general technique for reducing the weighted sum of completion-times problem to the classical makespan minimization problem. Using this technique, we improve the best-known results for scheduling conflicting jobs with the min-sum objective, on several fundamental classes of graphs, including line graphs, ( k + 1)-claw-free graphs, and perfect graphs. In particular, we obtain the first constant-factor approximation ratio for nonpreemptive scheduling on interval graphs. We also improve the results of Kim [2003] for scheduling jobs on line graphs and for resource-constrained scheduling.

Approximation algorithms for shop scheduling problems with minsum objective: A correction

We present a correction to the paper, "Approximation algorithms for shop scheduling problems with minsum objective" (Journal of Scheduling 2002; 5:287---305) by Queyranne and Sviridenko. This correction provides a correct derivation of its 2e? approximation result.

A note on scheduling to meet two min-sum objectives

We consider a single machine scheduling problem with two min-sum objective functions: the sum of completion times and the sum of weighted completion times. We propose a simple polynomial time ( 1 + ( 1 / γ ) , 1 + γ ) -approximation algorithm, and show that for γ > 1 , there is no ( x , y ) -approximation with 1 < x < 1 + ( 1 / γ ) and 1 < y < 1 + ( γ - 1 ) / ( 2 + γ ) .

A (2+ ε)-approximation algorithm for the generalized preemptive open shop problem with minsum objective

In this paper we consider a generalized version of the classical preemptive open shop problem with sum of weighted job completion times objective. The main result is a (2+ ε)-approximation algorithm for this problem. In the last section we also discuss the possibility of improving our algorithm.

Approximation algorithms for shop scheduling problems with minsum objective

We consider a general class of multiprocessor shop scheduling problems, preemptive or non-preemptive, with precedence constraints between operations, with job or operation release dates, and with a class of objective functions including weighted sums of job, operations and stage completion times. We present a general approximation method combining a linear programming relaxation in the operation completion times, with any algorithm for the makespan version of these problems without release dates. If the latter produces a schedule with makespan no larger than ρ times the ‘trivial lower bound’ consisting of the largest of all stage average loads (or ‘congestion’) and job lengths (or ‘dilation’), then our method produces a feasible schedule with minsum objective no larger than 2eρ times the optimum where 2e≈5.44. This leads, in particular, to a polynomial time algorithm with polylogarithmic performance guarantee for the minsum multiprocessor dag-shop problem J(P)∣rij, dagj∣ΣSwSCS where ΣSwSCS is a general minsum objective including weighted sum of operation and job completion times, stages makespans and others, whereas the best known earlier performance guarantees were O(m) (where m is the number of stages) for the special cases J∥ΣCj, F(P)∣rj∣ ΣwjCj and O∥ΣCj. We also obtain a O(1) performance guarantee for the acyclic job shop problem J∣pij=1, acyclic-dagj∣ΣSwSCS with unit processing times and weighted sum of operation (or job) completion time objective. Our results extend to a broad class of minsum objective functions including some convex objectives related to load balancing. We then present an improved 5.83-approximation algorithm for the open shop problem O∣rj∣ΣwjCj with total weighted job completion time objective. We conclude with a very simple method which yields O(m)-approximation algorithms for various job shop problems (preemptive, non-preemptive, and no-wait) with m single-processor stages and total weighted job completion time objective. Copyright © 2002 John Wiley & Sons, Ltd.

Solving Discrete Optimization Problems When Element Costs are Random

In a general class of discrete optimization problems with min-sum objective function, some of the elements may have random costs associated with them. In such a situation, the notion of optimality needs to be suitably modified. We define an optimal solution to be a feasible solution with the minimum risk. It is shown that the knowledge of the means of these random costs is enough to reduce such a problem into one with no random costs.

A PTAS for Minimizing the Total Weighted Completion Time on Identical Parallel Machines

We consider the problem of scheduling a set of n jobs on m identical parallel machines so as to minimize the weighted sum of job completion times. This problem is NP-hard in the strong sense. The best approximation result known so far was a [Formula: see text]-approximation algorithm that has been derived by Kawaguchi and Kyan back in 1986. The contribution of this paper is a polynomial time approximation scheme for this setting, which settles a problem that was open for a long time. Moreover, our result constitutes the first known approximation scheme for a strongly NP-hard scheduling problem with minsum objective.

Algorithms and codes for dense assignment problems: the state of the art

The paper considers the classic linear assignment problem with a min-sum objective function, and the most efficient and easily available codes for its solution. We first give a survey describing the different approaches in the literature, presenting their implementations, and pointing out similarities and differences. Then we select eight codes and we introduce a wide set of dense instances containing both randomly generated and benchmark problems. Finally we discuss the results of extensive computational experiments obtained by solving the above instances with the eight codes, both on a workstation with Unix operating system and on a personal computer running under Windows 95.

A note on pre-emptive scheduling with earliness and tardiness costs

Abstract The option of pre-emption seems to have escaped the attention of existing literature on scheduling problems with earliness and tardiness costs. This note presents an attempt to accommodate pre-emption in such settings. The focus is on models with a common due date for all jobs under different cost structures ( job dependent versus job independent), and different objectives ( minsum and minmax). It appears that only one of these cases is not polynomially solvable: the case of minsum objective and job-dependent ( linear) cost. An exact dynamic programming algorithm is introduced for this problem, as well as an efficient heuristic, which is shown to perform extremely well in our numerical tests.