Exact Polynomial-time Algorithm Research Articles

On Constrained Mixed-Integer DR-Submodular Minimization

Diminishing returns (DR)–submodular functions encompass a broad class of functions that are generally nonconvex and nonconcave. We study the problem of minimizing any DR-submodular function with continuous and general integer variables under box constraints and, possibly, additional monotonicity constraints. We propose valid linear inequalities for the epigraph of any DR-submodular function under the constraints. We further provide the complete convex hull of such an epigraph, which, surprisingly, turns out to be polyhedral. We propose a polynomial-time exact separation algorithm for our proposed valid inequalities with which we first establish the polynomial-time solvability of this class of mixed-integer nonlinear optimization problems. Funding: This work was supported by the Office of Naval Research Global [Grant N00014-22-1-2602].

Zero sum subsequences and hidden subgroups

We propose a method for solving the hidden subgroup problem in nilpotent groups. The main idea is iteratively transforming the hidden subgroup to its images in the quotient groups by the members of a central series, eventually to its image in the commutative quotient of the original group, and then using an abelian hidden subgroup algorithm to determine this image. Knowing this image allows one to descend to a proper subgroup unless the hidden subgroup is the full group. The transformation relies on finding zero sum subsequences of sufficiently large sequences of vectors over finite prime fields. We present a new deterministic polynomial time algorithm for the latter problem in the case when the size of the field is constant. The consequence is a polynomial time exact quantum algorithm for the hidden subgroup problem in nilpotent groups having constant nilpotency class and whose order only have prime factors also bounded by a constant.

Matching vs. Individual Choice: How to Counter Regional Imbalance of Carsharing Demand

Among the most crucial organizational challenges of free-floating carsharing is the question how to cope with regional demand imbalance. Because users are allowed to leave a rented car anywhere in the service district, it regularly occurs that too many cars are left behind in low-demand regions whereas other regions face a demand surplus. In this paper, we consider a countermeasure that has been overlooked by previous research: an optimization-based matching of carsharing supply and demand that not only addresses the profit promised by the current matches but also targets future demand imbalance. To account for such imbalances, we define regional demand levels that specify the projected number of requested cars per region and aim to reduce the deviations of the regions’ actual car supply from these target levels. We present exact polynomial-time algorithms for this extended matching task that are suitable for real-time application on large carsharing platforms. In an extensive computational study, we compare optimization-based matching approaches with and without the consideration of demand imbalance and benchmark them with the status quo, the individual choice of carsharing users among available cars. Based on generated data with considerable demand variation among regions, our results indicate a clear advantage of our novel matching approach. In a further study based on a large carsharing data set, however, the proof of concept fails because the real-world regions are cut according to geographical characteristics instead of demand variation. To successfully relieve the strains of demand imbalance, our novel matching task thus requires a properly partitioned service district and reliable forecasts of the carsharing demands. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [Grants BO 3148/8-1 and BR 3873/10-1]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0067 .

Justifying groups in multiwinner approval voting

Justified representation (JR) is a standard notion of representation in multiwinner approval voting. Not only does a JR committee always exist, but previous work has also shown through experiments that the JR condition can typically be fulfilled by groups of fewer than k candidates, where k is the target size of the committee. In this paper, we study such groups—known as n/k-justifying groups—both theoretically and empirically. First, we show that under the impartial culture model, n/k-justifying groups of size less than k/2 are likely to exist, which implies that the number of JR committees is usually large. We then present efficient approximation algorithms that compute a small n/k-justifying group for any given instance, and a polynomial-time exact algorithm when the instance admits a tree representation. In addition, we demonstrate that small n/k-justifying groups can often be useful for obtaining a gender-balanced JR committee even though the problem is NP-hard.

WITCH-NG: efficient and accurate alignment of datasets with sequence length heterogeneity.

Multiple sequence alignment is a basic part of many bioinformatics pipelines, including in phylogeny estimation, prediction of structure for both RNAs and proteins, and metagenomic sequence analysis. Yet many sequence datasets exhibit substantial sequence length heterogeneity, both because of large insertions and deletions in the evolutionary history of the sequences and the inclusion of unassembled reads or incompletely assembled sequences in the input. A few methods have been developed that can be highly accurate in aligning datasets with sequence length heterogeneity, with UPP one of the first methods to achieve good accuracy, and WITCH a recent improvement on UPP for accuracy. In this article, we show how we can speed up WITCH. Our improvement includes replacing a critical step in WITCH (currently performed using a heuristic search) by a polynomial time exact algorithm using Smith-Waterman. Our new method, WITCH-NG (i.e. 'next generation WITCH') achieves the same accuracy but is substantially faster. WITCH-NG is available at https://github.com/RuneBlaze/WITCH-NG. The datasets used in this study are from prior publications and are freely available in public repositories, as indicated in the Supplementary Materials. Supplementary data are available at Bioinformatics Advances online.

Exact algorithms for multi-module capacitated lot-sizing problem, and its generalizations with two-echelons and piecewise concave production costs

We study new generalizations of the classic capacitated lot-sizing problem with concave production (or transportation), holding, and subcontracting cost functions in which the total production (or transportation) capacity in each time period is the summation of capacities of a subset of n available modules (machines or vehicles) of different capacities. We refer to this problem as Multi-module Capacitated Lot-Sizing Problem without or with Subcontracting, and denote it by MCLS or MCLS-S, respectively. These are NP-hard problems if n is a part of the input and polynomially solvable for n = 1. In this article we address an open question: Does there exist a polynomial time exact algorithm for solving the MCLS or MCLS-S with fixed ? We present exact fixed-parameter tractable (polynomial) algorithms that solve MCLS and MCLS-S in time for a given It generalizes algorithm of Atamtürk and Hochbaum [Management Science 47(8):1081–1100, 2001] for MCLS-S with n = 1. We also present exact algorithms for two-generalizations of the MCLS and MCLS-S: (a) a lot-sizing problem with piecewise concave production cost functions (denoted by LS-PC-S) that takes time, where m is the number of breakpoints in these functions, and (b) two-echelon MCLS that takes time. The former reduces run time of algorithm of Koca et al. [INFORMS J. on Computing 26(4):767–779, 2014] for LS-PC-S by 93.6%, and the latter generalizes algorithm of van Hoesel et al. [Management Science 51(11):1706–1719, 2005] for two-echelon MCLS with n = 1. We perform computational experiments to evaluate the efficiency of our algorithms for MCLS and LS-PC-S and their parallel computing implementation, in comparison to Gurobi 9.1. The results of these experiments show that our algorithms are computationally efficient and stable. Our algorithm for MCLS-S addresses another open question related to the existence of a polynomial time algorithm for optimizing a linear function over n-mixing set (a generalization of the well-known 1-mixing set).

An approximation algorithm for a general class of multi-parametric optimization problems

In a widely-studied class of multi-parametric optimization problems, the objective value of each solution is an affine function of real-valued parameters. Then, the goal is to provide an optimal solution set, i.e., a set containing an optimal solution for each non-parametric problem obtained by fixing a parameter vector. For many multi-parametric optimization problems, however, an optimal solution set of minimum cardinality can contain super-polynomially many solutions. Consequently, no polynomial-time exact algorithms can exist for these problems even if textsf {P}=textsf {NP}. We propose an approximation method that is applicable to a general class of multi-parametric optimization problems and outputs a set of solutions with cardinality polynomial in the instance size and the inverse of the approximation guarantee. This method lifts approximation algorithms for non-parametric optimization problems to their parametric version and provides an approximation guarantee that is arbitrarily close to the approximation guarantee of the approximation algorithm for the non-parametric problem. If the non-parametric problem can be solved exactly in polynomial time or if an FPTAS is available, our algorithm is an FPTAS. Further, we show that, for any given approximation guarantee, the minimum cardinality of an approximation set is, in general, not ell -approximable for any natural number ell less or equal to the number of parameters, and we discuss applications of our results to classical multi-parametric combinatorial optimizations problems. In particular, we obtain an FPTAS for the multi-parametric minimum s-t-cut problem, an FPTAS for the multi-parametric knapsack problem, as well as an approximation algorithm for the multi-parametric maximization of independence systems problem.

An exact quantum hidden subgroup algorithm and applications to solvable groups

We present a polynomial time exact quantum algorithm for the hidden subgroup problem in $\Z_{m^k}^n$. The algorithm uses the quantum Fourier transform modulo $m$ and does not require factorization of $m$. For smooth $m$, i.e., when the prime factors of $m$ are of size $(\log m)^{O(1)}$, the quantum Fourier transform can be exactly computed using the method discovered independently by Cleve and Coppersmith, while for general $m$, the algorithm of Mosca and Zalka is available. Even for $m=3$ and $k=1$ our result appears to be new. We also present applications to compute the structure of abelian and solvable groups whose order has the same (but possibly unknown) prime factors as $m$. The applications for solvable groups also rely on an exact version of a technique proposed by Watrous for computing the uniform superposition of elements of subgroups.

An Exact Algorithm for the Linear Tape Scheduling Problem

Magnetic tapes are often considered as an outdated storage technology, yet they are still used to store huge amounts of data. Their main interests are a large capacity and a low price per gigabyte, which come at the cost of a much larger file access time than on disks. With tapes, finding the right ordering of multiple file accesses is thus key to performance. Moving the reading head back and forth along a kilometer long tape has a non-negligible cost and unnecessary movements thus have to be avoided. However, the optimization of tape request ordering has rarely been studied in the scheduling literature, much less than I/O scheduling on disks. For instance, minimizing the average service time for several read requests on a linear tape remains an open question. Therefore, in this paper, we aim at improving the quality of service experienced by users of tape storage systems, and not only the peak performance of such systems. To this end, we propose a reasonable polynomial-time exact algorithm while this problem and simpler variants have been conjectured NP-hard. We also refine the proposed model by considering U-turn penalty costs accounting for inherent mechanical accelerations. Then, we propose a low-cost variant of our optimal algorithm by restricting the solution space, yet still yielding an accurate suboptimal solution. Finally, we compare our algorithms to existing solutions from the literature on logs of the mass storage management system of a major datacenter. This allows us to assess the quality of previous solutions and the improvement achieved by our low-cost algorithm. Aiming for reproducibility, we make available the complete implementation of the algorithms used in our evaluation, alongside the dataset of tape requests that is, to the best of our knowledge, the first of its kind to be publicly released.

Scheduling with complete multipartite incompatibility graph on parallel machines: Complexity and algorithms

In this paper, the problem of scheduling on parallel machines with a presence of incompatibilities between jobs is considered. The incompatibility relation can be modeled as a complete multipartite graph in which each edge denotes a pair of jobs that cannot be scheduled on the same machine.The paper provides several results concerning schedules, optimal or approximate with respect to the two most popular criteria of optimality: Cmax (makespan) and ∑Cj (total completion time). The problems are considered for a variety of machine types: identical, uniform and unrelated. The results consist of delimitation of the easy (polynomial) and NP-hard problems within these constraints. Also, polynomial-time exact algorithms are provided for easier problems and algorithms with a guaranteed constant worst-case approximation ratio for harder ones.In particular, there is provided a polynomial-time approximation scheme (PTAS) for scheduling with respect to ∑Cj on uniform machines, when the number of parts (i.e. sets of vertices constituting the mentioned graph) is bounded. The problem is addressed by developing a linear programming relaxation technique with appropriate rounding. This technique together with an exhaustive search (albeit in a manner controlled by the precision parameter) allows to provide the desired algorithm. For Cmax a PTAS is provided for the case of unit time jobs, but when the number of parts is part of the input. Interestingly, the latter result shows a connection between the considered problem of scheduling and covering problems.

Trimmed estimator for circular–circular regression: breakdown properties and an exact algorithm for computation

The paper attempts to address the robustness issues in circular–circular regression. The Möbius transformation-based link function for circular–circular regression is considered. Defining the concept of breakdown point in this context, the robustness issues of the estimators in this model are discussed. Maximum trimmed cosine estimator in this context is considered and the breakdown point of the estimator is calculated. An exact polynomial time algorithm is then proposed for the computation of the estimator which makes the methodology useful and readily applicable for empirical datasets. Simulation studies show that the estimator is robust with respect to the outliers. An analysis of real data is performed to illustrate the proposed methodology.

An Exact Quantum Polynomial-Time Algorithm for Solving k-Junta Problem with One Uncomplemented Product

An Exact Quantum Polynomial-Time Algorithm for Solving k-Junta Problem with One Uncomplemented Product

Dynamic Inventory Relocation in Disaster Relief

This study investigates dynamic inventory relocation to respond proactively to the changing relief demand forecasts over time. In particular, we examine how to relocate mobile inventory optimally to serve nonstationary stochastic demand at several potential disaster sites. We propose a dynamic relocation model using dynamic programming (DP) and develop both analytical and numerical results regarding optimal relocation policies, the minimum cost‐to‐go function, and the value of inventory mobility over traditional warehouse pre‐positioning. Given the computational complexity of the backwards DP algorithm, we develop a base state heuristic (BSH) for general problems by exploiting the real‐world disaster pattern of occurrence. For problems with temporally independent demand, we propose a polynomial time exact algorithm based on a spatial–temporal graph. For problems with spatially independent demand, we design a speedup technique to implement BSH in polynomial time. The proposed model and algorithms are further extended to consider the impact of transportation uncertainties. Numerical experiments show that the proposed algorithms return high‐quality decisions only in a small fraction of the time required by an exact algorithm and a myopic algorithm. The proposed model and algorithms are applicable to any type of mobile inventory, facility, or server in similar settings.

Exact and approximate methods for the score-constrained packing problem

This paper investigates a packing problem related to the one-dimensional bin packing problem in which the order and orientation of items influences the feasibility of a solution. We give an exact polynomial-time algorithm for the Constrained Ordering Problem, explaining how it can be used to find a feasible packing of items in a single bin. We then introduce an evolutionary algorithm for the multi-bin version of the problem, which incorporates the exact algorithm along with a local search procedure and three recombination operators. The mechanisms and results produced by each of the recombination operators are compared, and we discuss the circumstances in which each approach proves most advantageous.

Phase transition in the bipartite z-matching

We study numerically the maximum z-matching problems on ensembles of bipartite random graphs. The z-matching problems describes the matching between two types of nodes, users and servers, where each server may serve up to z users at the same time. Using a mapping to standard maximum-cardinality matching, and because for the latter there exists a polynomial-time exact algorithm, we can study large system sizes of up to 10^6 nodes. We measure the capacity and the energy of the resulting optimum matchings. First, we confirm previous analytical results for bipartite regular graphs. Next, we study the finite-size behaviour of the matching capacity and find the same scaling behaviour as before for standard matching, which indicates the universality of the problem. Finally, we investigate for bipartite Erdős–Rényi random graphs the saturability as a function of the average degree, i.e. whether the network allows as many customers as possible to be served, i.e. exploiting the servers in an optimal way. We find phase transitions between unsaturable and saturable phases. These coincide with a strong change of the running time of the exact matching algorithm, as well with the point where a minimum-degree heuristic algorithm starts to fail.Graphical

A Polynomial-Time Exact Algorithm for k -Depot Capacitated Vehicle Routing Problem on a Tree

This paper proposes a polynomial-time exact algorithm for the k -depot capacitated vehicle routing problem on a tree for fixed k ( k -depot CVRPT for short), which involves dispatching a fixed number of k capacitated vehicles in depots on a tree-shaped graph to serve customers with the objective of minimizing total distance traveled. The polynomial-time exact algorithm improves the 2-approximation algorithm when k is a constant.

Evolution through segmental duplications and losses: a Super-Reconciliation approach

The classical gene and species tree reconciliation, used to infer the history of gene gain and loss explaining the evolution of gene families, assumes an independent evolution for each family. While this assumption is reasonable for genes that are far apart in the genome, it is not appropriate for genes grouped into syntenic blocks, which are more plausibly the result of a concerted evolution. Here, we introduce the Super-Reconciliation problem which consists in inferring a history of segmental duplication and loss events (involving a set of neighboring genes) leading to a set of present-day syntenies from a single ancestral one. In other words, we extend the traditional Duplication-Loss reconciliation problem of a single gene tree, to a set of trees, accounting for segmental duplications and losses. Existency of a Super-Reconciliation depends on individual gene tree consistency. In addition, ignoring rearrangements implies that existency also depends on gene order consistency. We first show that the problem of reconstructing a most parsimonious Super-Reconciliation, if any, is NP-hard and give an exact exponential-time algorithm to solve it. Alternatively, we show that accounting for rearrangements in the evolutionary model, but still only minimizing segmental duplication and loss events, leads to an exact polynomial-time algorithm. We finally assess time efficiency of the former exponential time algorithm for the Duplication-Loss model on simulated datasets, and give a proof of concept on the opioid receptor genes.

Defending with Shared Resources on a Network

In this paper we consider a defending problem on a network. In the model, the defender holds a total defending resource of R, which can be distributed to the nodes of the network. The defending resource allocated to a node can be shared by its neighbors. There is a weight associated with every edge that represents the efficiency defending resources are shared between neighboring nodes. We consider the setting when each attack can affect not only the target node, but its neighbors as well. Assuming that nodes in the network have different treasures to defend and different defending requirements, the defender aims at allocating the defending resource to the nodes to minimize the loss due to attack. We give polynomial time exact algorithms for two important special cases of the network defending problem. For the case when an attack can only affect the target node, we present an LP-based exact algorithm. For the case when defending resources cannot be shared, we present a max-flow-based exact algorithm. We show that the general problem is NP-hard, and we give a 2-approximation algorithm based on LP-rounding. Moreover, by giving a matching lower bound of 2 on the integrality gap on the LP relaxation, we show that our rounding is tight.

Determining matchdays in sports league schedules to minimize rest differences

Many sports leagues first announce the games to be played in each round and then determine their matchdays as the season progresses. This study focuses on the fairness criterion of minimizing the total rest difference among opposing teams to find the matchdays for an announced schedule. We show that the problem is decomposable into optimizing the rounds separately. We also provide a polynomial-time exact algorithm for canonical schedules.

Broadcasting a file in a communication network

We study the problem of distributing a file, initially located at a server, among a set of n nodes. The file is divided into mge 1 equally sized packets. After downloading a packet, nodes can upload it to other nodes, possibly to multiple nodes in parallel. Each node, however, may receive each packet from a single source node only. The upload and download rates between nodes are constrained by node- and server-specific upload and download capacities. The objective is to minimize the makespan. This problem has been proposed and analyzed first by Mundinger et al. (J Sched 11:105–120, 2008. https://doi.org/10.1007/s10951-007-0017-9) under the assumption that uploads obey the fair sharing principle, that is, concurrent upload rates from a common source are equal at any point in time. Under this assumption, the authors devised an optimal polynomial time algorithm for the case where the upload capacity of the server and the nodes’ upload and download capacities are all equal. In this work, we drop the fair sharing assumption and derive an exact polynomial time algorithm for the case when upload and download capacities per node and among nodes are equal. We further show that the problem becomes strongly NP-hard for equal upload and download capacities per node that may differ among nodes, even for a single packet. For this case, we devise a polynomial time smash {(1+2sqrt{2})}-approximation algorithm. Finally, we devise two polynomial time algorithms with approximation guarantees of 5 and 2 + lceil log _2 lceil n/mrceil rceil /m, respectively, for the general case of m packets.