Optimal Polynomial Time Research Articles

Two-agent proportionate flowshop scheduling with deadlines: polynomial-time optimization algorithms

AbstractVolatility in the supply chain of critical products, notably the vaccine shortage during the pandemic, influences livelihoods and may lead to significant delays and long waiting times. Considering the capital- and time-intensive nature of capacity expansion plans, strategic operational production decisions are required best to address the supply-demand mismatches given the limited production resources. This research article investigates the production scenarios where the demand of one agent must be completed within a specified due date, hereinafter referred to as the deadline, while minimizing the maximum or total completion time of another agent's demand. For this purpose, the Two-Agent Proportionate Flowshop Scheduling Problem with deadlines is introduced. Two polynomial-time optimization algorithms are developed to solve these optimization problems. This study will serve as a basis for further developing this practical yet understudied scheduling problem.

OneMax is not the Easiest Function for Fitness Improvements.

We study the (1:s+1) success rule for controlling the population size of the (1,λ)- EA. It was shown by Hevia Fajardo and Sudholt that this parameter control mechanism can run into problems for large s if the fitness landscape is too easy. They conjectured that this problem is worst for the ONEMAX benchmark, since in some well-established sense ONEMAX is known to be the easiest fitness landscape. In this paper we disprove this conjecture. We show that there exist s and ɛ such that the self-adjusting (1,λ)-EA with the (1:s+1)-rule optimizes ONEMAX efficiently when started with ɛn zero-bits, but does not find the optimum in polynomial time on DYNAMIC BINVAL. Hence, we show that there are landscapes where the problem of the (1:s+1)-rule for controlling the population size of the (1,λ)-EA is more severe than for ONEMAX. The key insight is that, while ONEMAX is the easiest function for decreasing the distance to the optimum, it is not the easiest fitness landscape with respect to finding fitness-improving steps.

First-order reasoning and efficient semi-algebraic proofs

Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively; and using a reflection principle, the converse also holds.This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones.We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule.

Optimal Polynomial-Time Compression for Boolean Max CSP

In the Boolean maximum constraint satisfaction problem— Max CSPΓ —one is given a collection of weighted applications of constraints from a finite constraint language Γ, over a common set of variables, and the goal is to assign Boolean values to the variables so that the total weight of satisfied constraints is maximized. There exists a concise dichotomy theorem providing a criterion on Γ for the problem to be polynomial-time solvable and stating that otherwise, it becomes NP-hard. We study the NP-hard cases through the lens of kernelization and provide a complete characterization of Max CSPΓ with respect to the optimal compression size. Namely, we prove that Max CSPΓ parameterized by the number of variables n is either polynomial-time solvable, or there exists an integer d ≥ 2 depending on Γ, such that: (1) An instance of Max CSPΓ can be compressed into an equivalent instance with 𝒪( n d log n ) bits in polynomial time, (2) Max CSPΓ does not admit such a compression to 𝒪( n d-ε ) bits unless NP ⊆ co-NP / poly. Our reductions are based on interpreting constraints as multilinear polynomials combined with the framework of “constraint implementations”, formerly used in the context of APX-hardness. As another application of our reductions, we reveal tight connections between optimal running times for solving Max CSPΓ . More precisely, we show that obtaining a running time of the form 𝒪(2 (1-ε) n ) for particular classes of Max CSP s is as hard as breaching this barrier for Max d -SAT for some d .

Blockage‐free storage assignment and storage/retrieval scheduling in autonomous vehicle storage and retrieval systems

AbstractAutonomous vehicle storage and retrieval systems have greatly increased in popularity in the last decade. In such a system, at each tier multiple roaming vehicles transport totes between the storage locations and the lifts. However, this may lead to vehicle interference. We study in which order and by which vehicle the storage and retrieval requests should be executed to minimize the makespan, without vehicle interference. The optimal storage locations for incoming totes are also determined. A blocking mitigation protocol is proposed to address vehicle interference. We propose a two‐phase matheuristic, where in the first phase, the tier is divided into zones, with each zone assigned its own vehicle. The second phase focuses on reassigning requests between adjacent vehicles to obtain improved solutions. The models proposed in both phases are solved to optimality in polynomial time and pseudo‐polynomial time, respectively. Computational experiments show that the matheuristic produces high‐quality solutions within a few seconds, even for large‐sized instances, making it suitable for real‐time decisions. Compared to methods commonly used in practice, our matheuristic can reduce the makespan by up to 15%. Our results show that making integrated decisions that combine storage assignment and request scheduling, is more beneficial than sequential optimization in terms of throughput performance, space utilization and overall system cost. We also find that increasing the number of vehicles has a diminishing return effect on the makespan. Another finding is that the system with a large number of short storage aisles leads to higher throughput capacity than that with a small number of long storage aisles.

Variable Neighborhood Search for precedence-constrained tasks optimization on heterogeneous systems

Supercomputing power is one of the fundamental pillars of the digital society, which depends on the accurate scheduling of parallel applications in High-Performance Computing (HPC) centers to minimize computing times. However, precedence-constraint task scheduling is a well-known NP-Hard optimization problem, and no optimal polynomial-time algorithm exists to solve it. Therefore, as accurate as possible to the optimal values, heuristic algorithms are of relevant interest. Our new scheduling proposal name is EFT-GVNS, which stands for Earliest Finish Time - General Variable Neighborhood Search. EFT-GVNS uses a Composite Local Search (CLS), making our proposal more efficient than traditional GVNS. EFT-GVNS accuracy against four high-performance algorithms from the state-of-the-art (EDA, EFT-ILS, GRASP-CPA, MPQGA) and one reference algorithm in the literature (HEFT) is studied. Experimental results over four real-world applications (Fpppp, LIGO, Robot, Sparse) and 14 synthetic instances from the literature show that EFT-GVNS outperforms in terms of the median achieved results, with a global improvement of 37.6%, 27.4%, 17.8%, 6.1%, 2.2% to HEFT, EDA, EFT-ILS, GRASP-CPA, and MPQGA, respectively. EFT-GVNS achieves all the 14 optimal values of the synthetic benchmark.

Vehicle routing for connected service areas - a versatile approach covering single, hierarchical, and bi-criteria objectives

Vehicle routing in urban areas or in-house tour planning is characterized by the fact that tours have to service orders in areas with limited road or aisle access. During last mile delivery, courier service providers often face the situation that subsets of customers in urban areas are located along a single street that can be accessed from only one or two directions. This also applies to warehouses, where, due to the given grid layout, pickers locate products on shelves only in specific areas positioned along an aisle between two neighboring cross aisles. By being confronted with a substantial time and/or cost pressure, those tour planning applications have to deal with the trade-off between service orientation and cost minimization. In order to cover many of those applications, this study proposes a general model that integrates the definition of customizable service areas with limited access and soft due dates of urgent orders, whereas the objective function can be defined in a versatile way covering to choose between a single objective, a hierarchical, and a bi-criteria objective system. The model is dubbed the Vehicle Routing Problem of Service Areas (VRPSA). A comprehensive complexity analysis of the VRPSA for different objective systems shows that a pure travel cost minimization variant can be solved to optimality in polynomial time, whereas the bi-criteria variant simultaneously pursuing travel cost and tardiness cost minimization is proven to be intractable. In order to generate tour plans, a customizable best-first branch-and-bound algorithm is developed and assessed through a computational study.

Load-optimization in Reconfigurable Data-center Networks: Algorithms and Complexity of Flow Routing

Emerging reconfigurable data centers introduce unprecedented flexibility in how the physical layer can be programmed to adapt to current traffic demands. These reconfigurable topologies are commonly hybrid, consisting of static and reconfigurable links, enabled by e.g., an Optical Circuit Switch (OCS) connected to top-of-rack switches in Clos networks. Even though prior work has showcased the practical benefits of hybrid networks, several crucial performance aspects are not well understood. For example, many systems enforce artificial segregation of the hybrid network parts, leaving money on the table. In this article, we study the algorithmic problem of how to jointly optimize topology and routing in reconfigurable data centers, in order to optimize a most fundamental metric, maximum link load. The complexity of reconfiguration mechanisms in this space is unexplored at large, especially for the following cross-layer network-design problem: given a hybrid network and a traffic matrix, jointly design the physical layer and the flow routing in order to minimize the maximum link load. We chart the corresponding algorithmic landscape in our work, investigating both un-/splittable flows and (non-)segregated routing policies. A topological complexity classification of the problem reveals NP-hardness in general for network topologies that are trees of depth at least two, in contrast to the tractability on trees of depth one. We moreover prove that the problem is not submodular for all these routing policies, even in multi-layer trees. However, networks that can be abstracted by a single packet switch (e.g., nonblocking Fat-Tree topologies) can be optimized efficiently, and we present optimal polynomial-time algorithms accordingly. We complement our theoretical results with trace-driven simulation studies, where our algorithms can significantly improve the network load in comparison to the state-of-the-art.

Two-dimensional drift analysis: Optimizing two functions simultaneously can be hard

In this paper we show how to use drift analysis in the case of two random variables X1,X2, when the drift is approximatively given by A⋅(X1,X2)T for a matrix A. The non-trivial case is that X1 and X2 impede each other's progress, and we give a full characterization of this case. As an application, we develop and analyze a minimal example TwoLin of a dynamic environment that can be hard. The environment consists of two linear functions f1 and f2 with positive weights, and in each generation selection is based on one of them at random. They only differ in the set of positions that have weight 1 and n. We show that the (1+1)-EA with mutation rate χ/n is efficient for small χ on TwoLin, but does not find the shared optimum in polynomial time for large χ.

Index Coding Algorithms: Cooperative Caching and Delivery for F-RANs

In a Fog Radio Access Network (F-RAN), fog access points (F-APs) are equipped with caches that can store popular files during off-peak hours. Besides, they are densely deployed to have overlapping radio coverage so that requested files can be delivered cooperatively using beamforming. The bottleneck of the network is typically in the bandwidth-limited wireless fronthaul, which connects a cloud server to the F-APs. This work studies index coding design for cooperative caching and delivery in F-RAN to minimize fronthaul traffic and transmit energy. Index coding algorithms are designed considering the cached content at the F-APs and the possibility of beamforming in the access network under coded and uncoded caching schemes. An optimal polynomial-time index coding algorithm for uncoded and repetition caching and an efficient heuristic for Maximum Distance Separable (MDS) coded caching are designed, and their superior performance is verified by simulations. The study is further extended to consider the tradeoff between the traffic load of the fronthaul link and the transmit energy consumed in the access network. At the expense of more fronthaul traffic, beamforming opportunities can be increased, significantly reducing energy consumption. Algorithms to achieve the tradeoff are crafted, and simulation results show that uncoded caching well balances the tradeoff.

Accurate Identification of Transcription Regulatory Sequences and Genes in Coronaviruses.

Transcription regulatory sequences (TRSs), which occur upstream of structural and accessory genes as well as the end of a coronavirus genome, play a critical role in discontinuous transcription in coronaviruses. We introduce two problems collectively aimed at identifying these regulatory sequences as well as their associated genes. First, we formulate the TRS Identification problem of identifying TRS sites in a coronavirus genome sequence with prescribed gene locations. We introduce CORSID-A, an algorithm that solves this problem to optimality in polynomial time. We demonstrate that CORSID-A outperforms existing motif-based methods in identifying TRS sites in coronaviruses. Second, we demonstrate for the first time how TRS sites can be leveraged to identify gene locations in the coronavirus genome. To that end, we formulate the TRS and Gene Identification problem of simultaneously identifying TRS sites and gene locations in unannotated coronavirus genomes. We introduce CORSID to solve this problem, which includes a web-based visualization tool to explore the space of near-optimal solutions. We show that CORSID outperforms state-of-the-art gene finding methods in coronavirus genomes. Furthermore, we demonstrate that CORSID enables de novo identification of TRS sites and genes in previously unannotated coronavirus genomes. CORSID is the first method to perform accurate and simultaneous identification of TRS sites and genes in coronavirus genomes without the use of any prior information.

Result diversification by multi-objective evolutionary algorithms with theoretical guarantees

Given a ground set of items, the result diversification problem aims to select a subset with high “quality” and “diversity” while satisfying some constraints. It arises in various real-world artificial intelligence applications, such as web-based search, document summarization and feature selection, and also has applications in other areas, e.g., computational geometry, databases, finance and operations research. Previous algorithms are mainly based on greedy or local search. In this paper, we propose to reformulate the result diversification problem as a bi-objective maximization problem, and solve it by a multi-objective evolutionary algorithm (EA), i.e., the GSEMO. We theoretically prove that the GSEMO can achieve the (asymptotically) optimal theoretical guarantees under both static and dynamic environments. For cardinality constraints, the GSEMO can achieve the optimal polynomial-time approximation ratio, 1/2. For more general matroid constraints, the GSEMO can achieve an asymptotically optimal polynomial-time approximation ratio, 1/2−ϵ/(4n), where ϵ>0 and n is the size of the ground set of items. Furthermore, when the objective function (i.e., a linear combination of quality and diversity) changes dynamically, the GSEMO can maintain this approximation ratio in polynomial running time, addressing the open question proposed by Borodin et al. [7]. This also theoretically shows the superiority of EAs over local search for solving dynamic optimization problems for the first time, and discloses the robustness of the mutation operator of EAs against dynamic changes. Experiments on the applications of web-based search, multi-label feature selection and document summarization show the superior performance of the GSEMO over the state-of-the-art algorithms (i.e., the greedy algorithm and local search) under both static and dynamic environments.

Scheduling with a processing time oracle

In this paper we study a single machine scheduling problem with the objective of minimizing the sum of completion times. Each of the given jobs is either short or long. However the processing times are initially hidden to the algorithm, but can be tested. This is done by executing a processing time oracle, which reveals the processing time of a given job. Each test occupies a time unit in the schedule, therefore the algorithm must decide for which jobs it will call the processing time oracle. The objective value of the resulting schedule is compared with the objective value of an optimal schedule, which is computed using full information. The resulting competitive ratio measures the price of hidden processing times, and the goal is to design an algorithm with minimal competitive ratio. Two models are studied in this paper. In the non-adaptive model, the algorithm needs to decide beforehand which jobs to test, and which jobs to execute untested. However in the adaptive model, the algorithm can make these decisions adaptively depending on the outcomes of the job tests. In both models we provide optimal polynomial time algorithms following a two-phase strategy, which consist of a first phase where jobs are tested, and a second phase where jobs are executed obliviously. Experiments give strong evidence that optimal algorithms have this structure. Proving this property is left as an open problem.

An almost optimal approximation algorithm for monotone submodular multiple knapsack

We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint. The input is a set I of items, each has a non-negative weight, and a set of bins of arbitrary capacities. Also, we are given a submodular, monotone and non-negative function f over subsets of the items. The objective is to find a packing of a subset of items A⊆I in the bins such that f(A) is maximized. Our main result is an almost optimal polynomial time (1−e−1−ε)-approximation algorithm for the problem, for any ε>0. The algorithm relies on a structuring technique which converts a general multiple knapsack constraint to a constraint in which the bins are partitioned into groups of exponentially increasing cardinalities, each consisting of bins of uniform capacity. We derive the result by combining structuring with a refined analysis of techniques for submodular optimization subject to knapsack constraints.

Minimum Length Scheduling for Multi-Cell Full Duplex Wireless Powered Communication Networks.

Wireless powered communication networks (WPCNs) will be a major enabler of massive machine type communications (MTCs), which is a major service domain for 5G and beyond systems. These MTC networks will be deployed by using low-power transceivers and a very limited set of transmission configurations. We investigate a novel minimum length scheduling problem for multi-cell full-duplex wireless powered communication networks to determine the optimal power control and scheduling for constant rate transmission model. The formulated optimization problem is combinatorial in nature and, thus, difficult to solve for the global optimum. As a solution strategy, first, we decompose the problem into the power control problem (PCP) and scheduling problem. For the PCP, we propose the optimal polynomial time algorithm based on the evaluation of Perron–Frobenius conditions. For the scheduling problem, we propose a heuristic algorithm that aims to maximize the number of concurrently transmitting users by maximizing the allowable interference on each user without violating the signal-to-noise-ratio (SNR) requirements. Through extensive simulations, we demonstrate a 50% reduction in the schedule length by using the proposed algorithm in comparison to unscheduled concurrent transmissions.

Towards Minimizing the $R$ Metric for Measuring Network Robustness

With the prevalence of cyber-attacks on infrastructure networks such as the Internet backbone, measuring the robustness of network topologies has become a critical issue. To this end, a metric called ‘R’ has been proposed and widely used. `R assumes that an attack on a network consists of n rounds of node removals with one node removed each time, where n is the number of nodes in the network. Since the value of R depends on the sequence of node removals, finding the sequence minimizing R becomes an interesting problem, denoted as Min-R hereafter. Although many heuristic approaches have been proposed for Min-R, there has been no optimal polynomial-time algorithm given so far, nor is Min-R proved NP-hard. To fill this gap, this paper presents the following fundamental results. First, to minimize R, a node always needs to be removed from the largest connected component of the current network in each round. Second, the Min-R problem has an Optimal Substructure, which allows it to be solved by Dynamic Programming in exponential time instead of the naive factorial time on general graphs. Finally, an efficient and optimal logarithmic-time algorithm is given for Min-R on path graphs, with its complexity and optimality proved.

Learning how to dynamically route autonomous vehicles on shared roads

Road congestion induces significant costs across the world, and road network disturbances, such as traffic accidents, can cause highly congested traffic patterns. If a planner had control over the routing of all vehicles in the network, they could easily reverse this effect. In a more realistic scenario, we consider a planner that controls autonomous cars, which are a fraction of all present cars. We study a dynamic routing game, in which the route choices of autonomous cars can be controlled and the human drivers react selfishly and dynamically. As the problem is prohibitively large, we use deep reinforcement learning to learn a policy for controlling the autonomous vehicles. This policy indirectly influences human drivers to route themselves in such a way that minimizes congestion on the network. To gauge the effectiveness of our learned policies, we establish theoretical results characterizing equilibria and empirically compare the learned policy results with best possible equilibria. We prove properties of equilibria on parallel roads and provide a polynomial-time optimization for computing the most efficient equilibrium. Moreover, we show that in the absence of these policies, high demand and network perturbations would result in large congestion, whereas using the policy greatly decreases the travel times by minimizing the congestion. To the best of our knowledge, this is the first work that employs deep reinforcement learning to reduce congestion by indirectly influencing humans’ routing decisions in mixed-autonomy traffic.

Fitness–Distance Balance based adaptive guided differential evolution algorithm for security-constrained optimal power flow problem incorporating renewable energy sources

One of the most difficult types of problems computationally is the security-constrained optimal power flow (SCOPF), a non-convex, nonlinear, large-scale, nondeterministic polynomial time optimization problem. With the use of renewable energy sources in the SCOPF process, the uncertainties of operating conditions and stress on power systems have increased even more. Thus, finding a feasible solution for the problem has become a still greater challenge. Even modern powerful optimization algorithms have been unable to find realistic solutions for the problem. In order to solve this kind of difficult problem, an optimization algorithm needs to have an unusual exploration ability as well as exploitation–exploration balance. In this study, we have presented an optimization model of the SCOPF problem involving wind and solar energy systems. This model has one problem space and innumerable local solution traps, plus a high level of complexity and discrete and continuous variables. To enable the optimization model to find the solution effectively, the adaptive guided differential evolution (AGDE) algorithm was improved by using the Fitness–Distance Balance (FDB) method with its balanced searching and high-powered diversity abilities. By using the FDB method, solution candidates guiding the search process in the AGDE algorithm could be selected more effectively as in nature. In this way, AGDE’s exploration and balanced search capabilities were improved. To solve the SCOPF problem involving wind and solar energy systems, the developed algorithm was tested on an IEEE 30-bus test system under different operational conditionals. The simulation results obtained from the proposed algorithm were effective in finding the optimal solution compared to the results of the metaheuristics algorithms and reported in the literature.

Optimal algorithms for scheduling under time-of-use tariffs

We consider a natural generalization of classical scheduling problems to a setting in which using a time unit for processing a job causes some time-dependent cost, the time-of-use tariff, which must be paid in addition to the standard scheduling cost. We focus on preemptive single-machine scheduling and two classical scheduling cost functions, the sum of (weighted) completion times and the maximum completion time, that is, the makespan. While these problems are easy to solve in the classical scheduling setting, they are considerably more complex when time-of-use tariffs must be considered. We contribute optimal polynomial-time algorithms and best possible approximation algorithms. For the problem of minimizing the total (weighted) completion time on a single machine, we present a polynomial-time algorithm that computes for any given sequence of jobs an optimal schedule, i.e., the optimal set of time slots to be used for preemptively scheduling jobs according to the given sequence. This result is based on dynamic programming using a subtle analysis of the structure of optimal solutions and a potential function argument. With this algorithm, we solve the unweighted problem optimally in polynomial time. For the more general problem, in which jobs may have individual weights, we develop a polynomial-time approximation scheme (PTAS) based on a dual scheduling approach introduced for scheduling on a machine of varying speed. As the weighted problem is strongly NP-hard, our PTAS is the best possible approximation we can hope for. For preemptive scheduling to minimize the makespan, we show that there is a comparably simple optimal algorithm with polynomial running time. This is true even in a certain generalized model with unrelated machines.

Load-Optimization in Reconfigurable Networks

Emerging reconfigurable data centers introduce the unprecedented flexibility in how the physical layer can be programmed to adapt to current traffic demands. These reconfigurable topologies are commonly hybrid, consisting of static and reconfigurable links, enabled by e.g. an Optical Circuit Switch (OCS) connected to top-of-rack switches in Clos networks. Even though prior work has showcased the practical benefits of hybrid networks, several crucial performance aspects are not well understood. In this paper, we study the algorithmic problem of how to jointly optimize topology and routing in reconfigurable data centers with a known traffic matrix, in order to optimize a most fundamental metric, maximum link load. We chart the corresponding algorithmic landscape by investigating both un-/splittable flows and (non-)segregated routing policies. We moreover prove that the problem is not submodular for all these routing policies, even in multi-layer trees, where a topological complexity classification of the problem reveals that already trees of depth two are intractable. However, networks that can be abstracted by a single packet switch (e.g., nonblocking Fat-Tree topologies) can be optimized efficiently, and we present optimal polynomialtime algorithms accordingly. We complement our theoretical results with trace-driven simulation studies, where our algorithms can significantly improve the network load in comparison to the state of the art.