Develop Polynomial Time Algorithms Research Articles

The circular balancing problem

We propose a balancing problem with a minmax objective in a circular setting. This balancing problem involves the arrangement of an even number of items with different weights on a circle while minimizing the maximum total weight of items arranged on any half circle. Due to its generic structure, it may have applications in fair resource allocation schemes. We show the NP-hardness of the problem and develop polynomial-time algorithms when the number of distinct weights is a fixed constant. We propose for the general case a tight 7/6-approximation algorithm and show that it performs better than two existing algorithms designed for an equivalent problem in the literature. The worst-case performance ratio is derived through a linear combination of valid inequalities that are obtained from the problem definition, the properties of the proposed algorithm, and the optimal circular permutation structure. Furthermore, we formulate a more general problem of minimizing the maximum total weight of items on equally divided circular sectors and present its computational complexity and a tight approximation algorithm.

On Computing Makespan-Optimal Solutions for Generalized Sliding-Tile Puzzles

In the 15-puzzle game, 15 labeled square tiles are reconfigured on a 4 × 4 board through an escort, wherein each (time) step, a single tile neighboring it may slide into it, leaving the space previously occupied by the tile as the new escort. We study a generalized sliding-tile puzzle (GSTP) in which (1) there are 1+ escorts and (2) multiple tiles can move synchronously in a single time step. Compared with popular discrete multi-agent/robot motion models, GSTP provides a more accurate model for a broad array of high-utility applications, including warehouse automation and autonomous garage parking, but is less studied due to the more involved tile interactions. In this work, we analyze optimal GSTP solution structures, establishing that computing makespan optimal solutions for GSTP is NP-complete and developing polynomial time algorithms yielding makespans approximating the minimum with expected/high probability constant factors, assuming randomized start and goal configurations.

Joint Caching and Routing in Cache Networks With Arbitrary Topology

In-network caching and flexible routing are two of the most celebrated advantages of next generation network infrastructures. Yet few solutions are available for jointly optimizing caching and routing that provide performance guarantees for networks with arbitrary topology. We take a holistic approach towards this fundamental problem by analyzing its complexity in all the cases and developing polynomial-time algorithms with approximation guarantees in important special cases. We also reveal the fundamental challenge in achieving guaranteed approximation in the general case and propose an alternating optimization algorithm with good empirical performance and fast convergence. Our algorithms have demonstrated superior performance in both routing cost and congestion compared to the state-of-the-art solutions in evaluations based on real topology and request traces.

Conformance Checking and Pushdown Reactive Systems

Due to their asynchronous interactions, testing reactive systems is a laborious activity present in any software development project.
 In this setting, the finite memory formalism of Labeled Transition Systems has been used to generate test suites that can be applied to check ioco conformance of implementations to a given specification.
 In this work we turn to a more complex scenario where a stronger formalism is considered, the Visibly Pushdown Labeled Transition System (VPTS), which allows access to a potentially infinite pushdown memory.
 We study an extension of the ioco conformance relation to VPTS models and develop polynomial time algorithms to verify conformance for VPTS models in a white-box testing scenario.

The complexity of binary matrix completion under diameter constraints

We thoroughly study a novel but basic combinatorial matrix completion problem: Given a binary incomplete matrix, fill in the missing entries so that every pair of rows in the resulting matrix has a Hamming distance within a specified range. We obtain an almost complete picture of the complexity landscape regarding the distance constraints and the maximum number of missing entries in any row. We develop polynomial-time algorithms for maximum diameter three based on Deza's theorem (1973) [11] from extremal set theory. We also prove NP-hardness for diameter at least four. For the number of missing entries per row, we show polynomial-time solvability when there is only one and NP-hardness when there can be at least two. In many of our algorithms, we heavily rely on Deza's theorem to identify sunflower structures. This paves the way towards polynomial-time algorithms which are based on finding graph factors and solving 2-SAT instances.

Preferences Single-Peaked on a Tree: Multiwinner Elections and Structural Results


 
 
 A preference profile is single-peaked on a tree if the candidate set can be equipped with a tree structure so that the preferences of each voter are decreasing from their top candidate along all paths in the tree. This notion was introduced by Demange (1982), and subsequently Trick (1989b) described an efficient algorithm for deciding if a given profile is single-peaked on a tree. We study the complexity of multiwinner elections under several variants of the Chamberlin–Courant rule for preferences single-peaked on trees. We show that in this setting the egalitarian version of this rule admits a polynomial-time winner determination algorithm. For the utilitarian version, we prove that winner determination remains NP-hard for the Borda scoring function; indeed, this hardness results extends to a large family of scoring functions. However, a winning committee can be found in polynomial time if either the number of leaves or the number of internal vertices of the underlying tree is bounded by a constant. To benefit from these positive results, we need a procedure that can determine whether a given profile is single-peaked on a tree that has additional desirable properties (such as, e.g., a small number of leaves). To address this challenge, we develop a structural approach that enables us to compactly represent all trees with respect to which a given profile is single-peaked. We show how to use this representation to efficiently find the best tree for a given profile for use with our winner determination algorithms: Given a profile, we can efficiently find a tree with the minimum number of leaves, or a tree with the minimum number of internal vertices among trees on which the profile is single-peaked. We then explore the power and limitations of this framework: we develop polynomial-time algorithms to find trees with the smallest maximum degree, diameter, or pathwidth, but show that it is NP-hard to check whether a given profile is single-peaked on a tree that is isomorphic to a given tree, or on a regular tree.
 
 

Pickup and delivery problems with autonomous vehicles on rings

In this paper we introduce a new class of Pickup and Delivery problems on circles (or rings). These problems arise in the field of public transportation systems where autonomous (i.e. driverless) vehicles travel on circular networks. We consider a set of stations arranged in a circle and a set of transportation requests. Each request asks for the transportation of a certain quantity from a pickup station to a delivery station. A fleet of capacitated vehicles is available at the depot. In the first part of the paper we propose a classification scheme for these problems. In the second part, we address the variants in which the vehicles are allowed to move in a single direction of the circle (either clockwise or counterclockwise) and the objective is to minimize the number of tours on the ring while serving all the requests. We provide a complexity analysis for this class of problems. We develop polynomial time algorithms for the variants that are polynomially solvable and proofs of NP-hardness for the variants that are NP-hard. In addition, for the latter, we provide mathematical formulations and perform computational tests that show the effectiveness of these formulations. Finally, we compare optimal solutions with those obtained using a straightforward greedy algorithm.

Rescheduling production and outbound deliveries when transportation service is disrupted

Unexpected service disruptions in transportation systems caused by accidents, system breakdowns, poor weather conditions, etc., are quite common. When disruptions occur, rescheduling of vehicles is often needed in order to mitigate the damage caused by the disruptions. In integrated production and outbound distribution systems, a disruption in outbound distribution operation affects not only the delivery plan but also the production schedule. In this paper, we consider a simple integrated scheduling model of production and outbound deliveries with a minimum headway constraint between vehicle departures, and study the situation where an optimal solution of the integrated scheduling model has been obtained but the delivery service is suddenly unavailable for a certain time period due to some unexpected incidents. We would like to determine a new production and delivery schedule in which no delivery takes place during the unavailable period. The objective is to simultaneously maintain a low cost schedule and control the magnitude of changes in the delivery times of the finished goods. We consider three different ways to control the time disruption, and develop polynomial-time algorithms for the corresponding problems.

Multiechelon Lot Sizing: New Complexities and Inequalities

We study a multiechelon lot-sizing problem for a serial supply chain that consists of a production level and several transportation levels, where the demands can exist in the production echelon as well as in any transportation echelons. With the presence of stationary production capacity and general cost functions, our model integrates production, inventory, and transportation decisions and generalizes existing literature on many multiechelon lot-sizing models. First, we answer an open question in the literature by showing that multiechelon lot sizing with intermediate demands (MLS) is NP-hard. Second, we develop polynomial time algorithms for both uncapacitated and capacitated MLS with a fixed number of echelons. The run times of our algorithms improve on those of many known algorithms for different MLS models. Third, we present families of valid inequalities for MLS that generalize known inequalities. For the uncapacitated case, we develop a polynomial-time separation algorithm and efficient separation heuristics. Finally, we demonstrate the effectiveness of a branch-and-cut algorithm using proposed inequalities to solve large multi-item MLS problems.

Uniform and most uniform partitions of trees

This paper addresses centered and non centered equipartition tree problems into p connected components (p-partitions). In the former case, each partition must contain exactly one special vertex called center, whereas in the latter, partitions are not required to fulfill this condition. Among the different equipartition problems considered in the literature, we focus on: (1) Most Uniform Partition (MUP) and (2) Uniform Partition (UP). Both criteria are defined either w.r.t. weights assigned to each vertex or to costs assigned to each vertex–center pair. Costs are assumed to be flat, i.e., they are independent of the topology of the tree. With respect to costs, MUP minimizes the difference between the maximum and minimum cost of the components of a partition and UP resorts to optimal min–max or max–min partitions. We provide polynomial time algorithms for centered and non centered versions of the MUP problem on trees when weights are assigned to each vertex. In the non centered case, our results set as polynomial the complexity of this problem which was an open question for several years. On the contrary, we prove that the centered version of MUP with flat costs is NP-complete even on trees. For the UP problem, we develop polynomial time algorithms for the max–min and min–max centered p-partition problems on trees both in the case of weight and cost-based objective functions.

Single-machine scheduling problems with a batch-dependent aging effect and variable maintenance activities

We consider single-machine scheduling problems with a batch-dependent ageing effect and variable maintenance activities between batches. The machine can process several jobs as a batch. It requires maintenance activities where the maintenance time depends on the flow time of the pre-batch, i.e. the batch processed before a batch. A job’s actual processing time is an increasing exponential function of its operation time within a batch. The objectives are to minimise the makespan and the total completion time. We develop polynomial time algorithms for the makespan minimisation problem and the total completion time minimisation problem under the condition that the ageing factor is greater than one. We also provide a mathematical programming approach and two heuristic algorithms to analyse the total completion time minimisation problem when the ageing factor is less than one for even one batch. The computational analysis indicates that the proposed heuristic algorithms are more efficient for the smaller ageing factor, whereas the Modified Shortest Processing Time algorithm is more efficient than the proposed heuristic algorithms for the larger ageing factor.

Bicriteria Scheduling on a Single Batching Machine with Transportation and Deterioration to Minimize Total Completion Time and Production Costs

In this paper, we consider a coordinated scheduling problem on a single batching machine with transportation and deterioration. The jobs are transported by a vehicle to the single batching machine for further processing. The processing time of a job is a step increasing function of its exposure time which is equivalent to the time interval from the beginning of the transportation to the starting of the job on the machine. The objective is to find Pareto-optimal schedules with two performance criteria (total completion time and the number of batches) to balance the inventory level and the production costs. We prove that the general problem is strongly NP-hard. We further develop polynomial-time algorithms for two special cases with a fixed job sequence and without exposure time limit, respectively. For the general problem, we develop a heuristic algorithm and a branch and bound algorithm. Computational experiments show that the heuristic algorithms perform well on randomly generated problem instances, and the branch and bound algorithm can obtain Pareto-optimal solutions for the small-scaled instances.

The Destination-Loader-Door Assignment Problem for Automated Package Sorting Centers

This paper presents a new model and solution procedure for a problem that arises in configuring package sorting centers that perform multiple automated sorts per day. For a given set of loading bays, the first objective is to assign destinations to consecutive doors so that the number of changes of destination-to-door assignments from one sort to the next is minimized. The second and third objectives are to minimize the number of loaders who work the doors and to evenly distribute the volume of packages assigned to each loader. A variety of constraints vastly complicates these assignments and leads to a mixed-integer programming (MIP) model, which we significantly strengthened with structurally derived cuts. A novel feature of the formulation is the use of pattern variables to represent the door assignments. Taking a multiobjective programming approach, solutions are obtained by solving a series of MIPs, each addressing one of the three objectives. The effectiveness of our approach is demonstrated using data provided by a well-known package carrier for 24 work centers in four facilities. An additional contribution is a complexity analysis of the destination-to-door and the loader-to-door assignment subproblems. Both are shown to be strongly NP-hard. We also examine special cases of the loader subproblem and develop polynomial-time algorithms for them.

Augmenting approach for some Maximum Set problems

The method of augmenting graphs is a general approach to solve the Maximum Independent Set problem. Our objective is to employ this approach to develop polynomial-time algorithms for some so-called Maximum Set problems, i.e. problems which can be stated as follows. Given a (simple) graph G, find a maximum vertex subset S of G such that the subgraph induced by S satisfies a given property Π. Such problems were shown to be NP-hard in general if the properties considered are non-trivial and hereditary Lewis and Yannakakis (1980) and Yannakakis (1978). In this paper, using the augmenting graph technique, we describe a graph class, in which some problems can be solved in polynomial time. We also prove the NP-hardness of some Maximum Set problems where the considered properties are not hereditary.

Low-Rank Approximation and Completion of Positive Tensors

Unlike the matrix case, computing low-rank approximations of tensors is NP-hard and numerically ill-posed in general. Even the best rank-1 approximation of a tensor is NP-hard. In this paper, we use convex optimization to develop polynomial-time algorithms for low-rank approximation and completion of positive tensors. Our approach is to use algebraic topology to define a new (numerically well-posed) decomposition for positive tensors, which we show is equivalent to the standard tensor decomposition in important cases. Though computing this decomposition is a nonconvex optimization problem, we prove it can be exactly reformulated as a convex optimization problem. This allows us to construct polynomial-time randomized algorithms for computing this decomposition and for solving low-rank tensor approximation problems. Among the consequences is that best rank-1 approximations of positive tensors can be computed in polynomial time. Our framework is next extended to the tensor completion problem, where noisy ent...

Parallel-Machine Scheduling with Delivery Times and Deteriorating Maintenance

In this paper, we consider parallel-machine scheduling with past-sequence-dependent (p-s-d) delivery times and deteriorating maintenance. The delivery time of a job is proportional to its waiting time in the system. Each machine has a deteriorating maintenance activity, i.e., delaying the maintenance increases the time required to perform it. We consider three versions of the problem to minimize the total absolute deviation of job completion times, the total load on all the machines, and the total completion time. We develop polynomial-time algorithms to solve them.

No-wait scheduling of position dependent jobs and adjustable processing intervals

In this paper, we consider a single machine no-wait scheduling model whereby job processing times are general functions of their position in the job sequence. We assume that the single machine must operate at a certain cycle, which can be determined by the scheduler. Furthermore, exactly one job has to be completed by the end of each cycle. Using different variations of the Linear Assignment Problem formulation, we develop polynomial time algorithms for minimizing the following objectives: makespan, total completion time, maximum earliness and total earliness.

Decision support for unrelated parallel machine scheduling with discrete controllable processing times

In a manufacturing or service system, the actual processing time of a job can be controlled by the amount of an indivisible resource allocated, such as workers or auxiliary facilities. In this paper, we consider unrelated parallel-machine scheduling problems with discrete controllable processing times. The processing time of a job is discretely controllable by the allocation of indivisible resources. The planner must make decisions on whether or how to allocate resources to jobs during the scheduling horizon to optimize the performance measures. The objective is to minimize the total cost including the cost measured by a standard criterion and the total processing cost. We first consider three scheduling criterions: the total completion time, the total machine load, and the total earliness and tardiness penalties. If the number of machines and the number of possible processing times are fixed, we develop polynomial time algorithms for the considered problems. We then consider the minimization problem of the makespan cost plus the total processing cost and present an integer programming method and a heuristic method to solve the studied problem.

Bicriteria scheduling on a single batching machine with job transportation and deterioration considerations

AbstractWe study a single batching machine scheduling problem with transportation and deterioration considerations arising from steel production. A set of jobs are transported, one at a time, by a vehicle from a holding area to the single batching machine. The machine can process several jobs simultaneously as a batch. The processing time of a job will increase if the duration from the time leaving the holding area to the start of its processing exceeds a given threshold. The time needed to process a batch is the longest of the job processing times in the batch. The problem is to determine the job sequence for transportation and the job batching for processing so as to minimize the makespan and the number of batches. We study four variations (P1, P2, P3, P4) of the problem with different treatments of the two criteria. We prove that all the four variations are strongly NP‐hard and further develop polynomial time algorithms for their special cases. For each of the first three variations, we propose a heuristic algorithm and analyze its worst‐case performance. For P4, which is to find the Pareto frontier, we provide a heuristic algorithm and an exact algorithm based on branch and bound. Computational experiments show that all the heuristic algorithms perform well on randomly generated problem instances, and the exact algorithm for P4 can obtain Pareto optimal schedules for small‐scale instances. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 269–285, 2014

Weighted Independent Directed Acyclic Graphs for Resilient and Efficient Multipath Routing

In order to achieve resilient multipath routing with efficient path choosing dynamically, we introduce the concept of Weighted Independent Directed Acyclic Graphs (WIDAGs). In a given network we develop polynomial-time algorithms to compute link-independent and node-independent DAGs. We use Heuristic methods to dynamically choose the maximum profitable (least cost) route upon a single link failure. The algorithm developed in this paper takes the advantage of the resiliency provided by IDAG and in addition, improves efficiency by choosing least cost path dynamically.