222 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
Scheduling Kernels via Configuration LP
Makespan minimization (on parallel identical or unrelated machines) is arguably the most natural and studied scheduling problem. A common approach in practical algorithm design is to reduce the size of a given instance by a fast preprocessing step while being able to recover key information even after this reduction. This notion is formally studied as kernelization (or simply, kernel) - a polynomial time procedure which yields an equivalent instance whose size is bounded in terms of some given parameter. It follows from known results that makespan minimization parameterized by the longest job processing time p_max has a kernelization yielding a reduced instance whose size is exponential in p_max. Can this be reduced to polynomial in p_max?
We answer this affirmatively not only for makespan minimization, but also for the (more complicated) objective of minimizing the weighted sum of completion times, also in the setting of unrelated machines when the number of machine kinds is a parameter.
Our algorithm first solves the Configuration LP and based on its solution constructs a solution of an intermediate problem, called huge N-fold integer programming. This solution is further reduced in size by a series of steps, until its encoding length is polynomial in the parameters. Then, we show that huge N-fold IP is in NP, which implies that there is a polynomial reduction back to our scheduling problem, yielding a kernel.
Our technique is highly novel in the context of kernelization, and our structural theorem about the Configuration LP is of independent interest. Moreover, we show a polynomial kernel for huge N-fold IP conditional on whether the so-called separation subproblem can be solved in polynomial time. Considering that integer programming does not admit polynomial kernels except for quite restricted cases, our "conditional kernel" provides new insight
Precedence-constrained scheduling problems parameterized by partial order width
Negatively answering a question posed by Mnich and Wiese (Math. Program.
154(1-2):533-562), we show that P2|prec,|, the
problem of finding a non-preemptive minimum-makespan schedule for
precedence-constrained jobs of lengths 1 and 2 on two parallel identical
machines, is W[2]-hard parameterized by the width of the partial order giving
the precedence constraints. To this end, we show that Shuffle Product, the
problem of deciding whether a given word can be obtained by interleaving the
letters of other given words, is W[2]-hard parameterized by , thus
additionally answering a question posed by Rizzi and Vialette (CSR 2013).
Finally, refining a geometric algorithm due to Servakh (Diskretn. Anal. Issled.
Oper. 7(1):75-82), we show that the more general Resource-Constrained Project
Scheduling problem is fixed-parameter tractable parameterized by the partial
order width combined with the maximum allowed difference between the earliest
possible and factual starting time of a job.Comment: 14 pages plus appendi
How the structure of precedence constraints may change the complexity class of scheduling problems
This survey aims at demonstrating that the structure of precedence
constraints plays a tremendous role on the complexity of scheduling problems.
Indeed many problems can be NP-hard when considering general precedence
constraints, while they become polynomially solvable for particular precedence
constraints. We also show that there still are many very exciting challenges in
this research area
A Multivariate Complexity Analysis of the Material Consumption Scheduling Problem
The NP-hard MATERIAL CONSUMPTION SCHEDULING Problem and closely related
problems have been thoroughly studied since the 1980's. Roughly speaking, the
problem deals with minimizing the makespan when scheduling jobs that consume
non-renewable resources. We focus on the single-machine case without
preemption: from time to time, the resources of the machine are (partially)
replenished, thus allowing for meeting a necessary pre-condition for processing
further jobs, each of which having individual resource demands. We initiate a
systematic exploration of the parameterized (exact) complexity landscape of the
problem, providing parameterized tractability as well as intractability
results. Doing so, we mainly investigate how parameters related to the resource
supplies influence the computational solvability. Thereby, we get a deepened
understanding of the algorithmic complexity of this fundamental scheduling
problem.Comment: Accepted for publication in The Thirty-Fifth AAAI Conference on
Artificial Intelligence (AAAI-21
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