7 research outputs found
Parameterized complexity of machine scheduling: 15 open problems
Machine scheduling problems are a long-time key domain of algorithms and
complexity research. A novel approach to machine scheduling problems are
fixed-parameter algorithms. To stimulate this thriving research direction, we
propose 15 open questions in this area whose resolution we expect to lead to
the discovery of new approaches and techniques both in scheduling and
parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc
Atomic Splittable Flow Over Time Games
In an atomic splittable flow over time game, finitely many players route flow dynamically through a network, in which edges are equipped with transit times, specifying the traversing time, and with capacities, restricting flow rates. Infinitesimally small flow particles controlled by the same player arrive at a constant rate at the player's origin and the player's goal is to maximize the flow volume that arrives at the player's destination within a given time horizon. Here, the flow dynamics are described by the deterministic queuing model, i.e., flow of different players merges perfectly, but excessive flow has to wait in a queue in front of the bottle-neck. In order to determine Nash equilibria in such games, the main challenge is to consider suitable definitions for the players' strategies, which depend on the level of information the players receive throughout the game. For the most restricted version, in which the players receive no information on the network state at all, we can show that there is no Nash equilibrium in general, not even for networks with only two edges. However, if the current edge congestions are provided over time, the players can adapt their route choices dynamically. We show that a profile of those strategies always lead to a unique feasible flow over time. Hence, those atomic splittable flow over time games are well-defined. For parallel-edge networks Nash equilibria exists and the total flow arriving in time equals the value of a maximum flow over time leading to a price of anarchy of 1.ISSN:1868-896
On the Fine-Grained Parameterized Complexity of Partial Scheduling to Minimize the Makespan
We study a natural variant of scheduling that we call partial scheduling: In this variant an instance of a scheduling problem along with an integer k is given and one seeks an optimal schedule where not all, but only k jobs, have to be processed.
Specifically, we aim to determine the fine-grained parameterized complexity of partial scheduling problems parameterized by k for all variants of scheduling problems that minimize the makespan and involve unit/arbitrary processing times, identical/unrelated parallel machines, release/due dates, and precedence constraints. That is, we investigate whether algorithms with runtimes of the type f(k)n^?(1) or n^?(f(k)) exist for a function f that is as small as possible.
Our contribution is two-fold: First, we categorize each variant to be either in ?, NP-complete and fixed-parameter tractable by k, or ?[1]-hard parameterized by k. Second, for many interesting cases we further investigate the run time on a finer scale and obtain run times that are (almost) optimal assuming the Exponential Time Hypothesis. As one of our main technical contributions, we give an ?(8^k k(|V|+|E|)) time algorithm to solve instances of partial scheduling problems minimizing the makespan with unit length jobs, precedence constraints and release dates, where G = (V,E) is the graph with precedence constraints
Approximations for Throughput Maximization
In this paper we study the classical problem of throughput maximization. In
this problem we have a collection of jobs, each having a release time
, deadline , and processing time . They have to be scheduled
non-preemptively on identical parallel machines. The goal is to find a
schedule which maximizes the number of jobs scheduled entirely in their
window. This problem has been studied extensively (even for the
case of ). Several special cases of the problem remain open. Bar-Noy et
al. [STOC1999] presented an algorithm with ratio for
machines, which approaches as increases. For ,
Chuzhoy-Ostrovsky-Rabani [FOCS2001] presented an algorithm with approximation
with ratio (for any ). Recently
Im-Li-Moseley [IPCO2017] presented an algorithm with ratio
for some absolute constant for any
fixed . They also presented an algorithm with ratio for general which approaches 1 as grows. The
approximability of the problem for remains a major open question. Even
for the case of and distinct processing times the problem is
open (Sgall [ESA2012]). In this paper we study the case of and show
that if there are distinct processing times, i.e. 's come from a set
of size , then there is a -approximation that runs in time
, where is the largest deadline.
Therefore, for constant and constant this yields a PTAS. Our algorithm
is based on proving structural properties for a near optimum solution that
allows one to use a dynamic programming with pruning
Recommended from our members
Metaheuristic approach for solving scheduling and financial derivative problems
This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University LondonThe objective of this thesis is to implement metaheuristic approaches to solve di erent
types of combinatorial problems. The thesis is focused on neighborhood heuristic optimisation
techniques such as Variable Neighborhood Search (VNS) and Ant Colony Optimisation
(ACO) algorithms. The thesis will focus on two diverse combinatorial problems.
A job shop scheduling problem, and a nancial derivative matching problem. The rst
is a NP-hard 2-stage assembly problem, where we will be focussing on the rst stage. It
consists of sequencing a set of jobs having multiple components to be processed. Each job
has to be worked on independently on a speci c machine. We consider these jobs to form
a vector of tasks. Our objective is to schedule jobs on the particular machines in order
to minimise the completion time before the second stage starts. In this thesis, we have
constructed a new hybrid metaheuristic approach to solve this unique job shop scheduling
problem.
The second problem has arisen in the nancial sector, where the nancial regulators collects
transaction data across regulated assets classes. Our focus is to identify any unhedged
derivative, Contract for Di erence (CFD), with its corresponding underlying asset that
has been reported to the corresponding component authorities. The underlying asset
and CFD transaction contain di erent variables, like volume and price. Therefore, we are
looking for a combination of underlying asset variables that may hedge the equivalent CFD
variables. Our aim is to identify unhedged or unmatched CFD's with their corresponding
underlying asset. This problem closely relates to the goal programming problem with
variable parameters. We have developed two new local search methods and embedded the
newly constructed local search methods with basic VNS, to attain a new modi ed variant
of the VNS algorithm. We then used these newly constructed VNS variants to solve this
nancial matching problem.
In tackling the Vector Job Scheduling problem, we developed a new hybrid optimisation
heuristic algorithm by combining VNS and ACO. We then compared the results of this hybrid algorithm with VNS and ACO on their own. We found that the hybrid algorithm
performance is better than the other two independent heuristic algorithms. In tackling
the nancial derivative problem, our objective is to match the CFD trades with their
corresponding underlying equity trades. Our goal is to identify the mismatched CFD
trades while optimising the search process. We have developed two new local search
techniques and we have implemented a VNS algorithm with the newly developed local
search techniques to attain better solutions