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Register a new userA New Perspective on the Windows Scheduling Problem
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Added by
Tobias Jacobs
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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The Windows Scheduling Problem, also known as the Pinwheel Problem, is to schedule periodic jobs subject to their processing frequency demands. Instances are given as a set of jobs that have to be processed infinitely often such that the time interval between two consecutive executions of the same job j is no longer than the jobs given period $p_j$. The key contribution of this work is a new interpretation of the problem variant with exact periods, where the time interval between consecutive executions must be strictly $p_j$. We show that this version is equivalent to a natural combinatorial problem we call Partial Coding. Reductions in both directions can be realized in polynomial time, so that both hardness proofs and algorithms for Partial Coding transfer to Windows Scheduling. Applying this new perspective, we obtain a number of new results regarding the computational complexity of various Windows Scheduling Problem variants. We prove that even the case of one processor and unit-length jobs does not admit a pseudo-polynomial time algorithm unless SAT can be solved by a randomized method in expected quasi-polynomial time. This result also extends to the case of inexact periods, which answers a question that has remained open for more than two decades. Furthermore, we report an error found in a hardness proof previously given for the multi-machine case without machine migration, and we show that this variant reduces to the single-machine case. Finally, we prove that even with unit-length jobs the problem is co-NP-hard when jobs are allowed to migrate between machines.
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We consider time-space tradeoffs for exactly computing frequency moments and order statistics over sliding windows. Given an input of length 2n-1, the task is to output the function of each window of length n, giving n outputs in total. Computations over sliding windows are related to direct sum problems except that inputs to instances almost completely overlap. We show an average case and randomized time-space tradeoff lower bound of TS in Omega(n^2) for multi-way branching programs, and hence standard RAM and word-RAM models, to compute the number of distinct elements, F_0, in sliding windows over alphabet [n]. The same lower bound holds for computing the low-order bit of F_0 and computing any frequency moment F_k for k not equal to 1. We complement this lower bound with a TS in tilde O(n^2) deterministic RAM algorithm for exactly computing F_k in sliding windows. We show time-space separations between the complexity of sliding-window element distinctness and that of sliding-window $F_0bmod 2$ computation. In particular for alphabet [n] there is a very simple errorless sliding-window algorithm for element distinctness that runs in O(n) time on average and uses O(log{n}) space. We show that any algorithm for a single element distinctness instance can be extended to an algorithm for the sliding-window version of element distinctness with at most a polylogarithmic increase in the time-space product. Finally, we show that the sliding-window computation of order statistics such as the maximum and minimum can be computed with only a logarithmic increase in time, but that a TS in Omega(n^2) lower bound holds for sliding-window computation of order statistics such as the median, a nearly linear increase in time when space is small.
The hypergraph duality problem DUAL is defined as follows: given two simple hypergraphs $mathcal{G}$ and $mathcal{H}$, decide whether $mathcal{H}$ consists precisely of all minimal transversals of $mathcal{G}$ (in which case we say that $mathcal{G}$ is the dual of $mathcal{H}$). This problem is equivalent to deciding whether two given non-redundant monotone DNFs are dual. It is known that non-DUAL, the complementary problem to DUAL, is in $mathrm{GC}(log^2 n,mathrm{PTIME})$, where $mathrm{GC}(f(n),mathcal{C})$ denotes the complexity class of all problems that after a nondeterministic guess of $O(f(n))$ bits can be decided (checked) within complexity class $mathcal{C}$. It was conjectured that non-DUAL is in $mathrm{GC}(log^2 n,mathrm{LOGSPACE})$. In this paper we prove this conjecture and actually place the non-DUAL problem into the complexity class $mathrm{GC}(log^2 n,mathrm{TC}^0)$ which is a subclass of $mathrm{GC}(log^2 n,mathrm{LOGSPACE})$. We here refer to the logtime-uniform version of $mathrm{TC}^0$, which corresponds to $mathrm{FO(COUNT)}$, i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for non-DUAL that requires to guess $O(log^2 n)$ bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in $mathrm{FO(COUNT)}$. From this result, by the well known inclusion $mathrm{TC}^0subseteqmathrm{LOGSPACE}$, it follows that DUAL belongs also to $mathrm{DSPACE}[log^2 n]$. Finally, by exploiting the principles on which the proposed nondeterministic algorithm is based, we devise a deterministic algorithm that, given two hypergraphs $mathcal{G}$ and $mathcal{H}$, computes in quadratic logspace a transversal of $mathcal{G}$ missing in $mathcal{H}$.
We study a family of generalizations of Edge Dominating Set on directed graphs called Directed $(p,q)$-Edge Dominating Set. In this problem an arc $(u,v)$ is said to dominate itself, as well as all arcs which are at distance at most $q$ from $v$, or at distance at most $p$ to $u$. First, we give significantly improved FPT algorithms for the two most important cases of the problem, $(0,1)$-dEDS and $(1,1)$-dEDS (that correspond
This paper introduces a multi-period inspector scheduling problem (MPISP), which is a new variant of the multi-trip vehicle routing problem with time windows (VRPTW). In the MPISP, each inspector is scheduled to perform a route in a given multi-period planning horizon. At the end of each period, each inspector is not required to return to the depot but has to stay at one of the vertices for recuperation. If the remaining time of the current period is insufficient for an inspector to travel from his/her current vertex $A$ to a certain vertex B, he/she can choose either waiting at vertex A until the start of the next period or traveling to a vertex C that is closer to vertex B. Therefore, the shortest transit time between any vertex pair is affected by the length of the period and the departure time. We first describe an approach of computing the shortest transit time between any pair of vertices with an arbitrary departure time. To solve the MPISP, we then propose several local search operators adapted from classical operators for the VRPTW and integrate them into a tabu search framework. In addition, we present a constrained knapsack model that is able to produce an upper bound for the problem. Finally, we evaluate the effectiveness of our algorithm with extensive experiments based on a set of test instances. Our computational results indicate that our approach generates high-quality solutions.
This PhD thesis summarizes research works on the design of exact algorithms that provide a worst-case (time or space) guarantee for NP-hard scheduling problems. Both theoretical and practical aspects are considered with three main results reported. The first one is about a Dynamic Programming algorithm which solves the F3Cmax problem in O*(3^n) time and space. The algorithm is easily generalized to other flowshop problems and single machine scheduling problems. The second contribution is about a search tree method called Branch & Merge which solves the 1||SumTi problem with the time complexity converging to O*(2^n) and in polynomial space. Our third contribution aims to improve the practical efficiency of exact search tree algorithms solving scheduling problems. First we realized that a better way to implement the idea of Branch & Merge is to use a technique called Memorization. By the finding of a new algorithmic paradox and the implementation of a memory cleaning strategy, the method succeeded to solve instances with 300 more jobs with respect to the state-of-the-art algorithm for the 1||SumTi problem. Then the treatment is extended to another three problems 1|ri|SumCi, 1|dtilde|SumwiCi and F2||SumCi. The results of the four problems all together show the power of the Memorization paradigm when applied on sequencing problems. We name it Branch & Memorize to promote a systematic consideration of Memorization as an essential building block in branching algorithms like Branch and Bound. The method can surely also be used to solve other problems, which are not necessarily scheduling problems.
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