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Register a new userScheduling with Communication Delays via LP Hierarchies and Clustering
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Added by
Janardhan Kulkarni
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We consider the classic problem of scheduling jobs with precedence constraints on identical machines to minimize makespan, in the presence of communication delays. In this setting, denoted by $mathsf{P} mid mathsf{prec}, c mid C_{mathsf{max}}$, if two dependent jobs are scheduled on different machines, then at least $c$ units of time must pass between their executions. Despite its relevance to many applications, this model remains one of the most poorly understood in scheduling theory. Even for a special case where an unlimited number of machines is available, the best known approximation ratio is $2/3 cdot (c+1)$, whereas Grahams greedy list scheduling algorithm already gives a $(c+1)$-approximation in that setting. An outstanding open problem in the top-10 list by Schuurman and Woeginger and its recent update by Bansal asks whether there exists a constant-factor approximation algorithm. In this work we give a polynomial-time $O(log c cdot log m)$-approximation algorithm for this problem, where $m$ is the number of machines and $c$ is the communication delay. Our approach is based on a Sherali-Adams lift of a linear programming relaxation and a randomized clustering of the semimetric space induced by this lift.
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In the scheduling with non-uniform communication delay problem, the input is a set of jobs with precedence constraints. Associated with every precedence constraint between a pair of jobs is a communication delay, the time duration the scheduler has to wait between the two jobs if they are scheduled on different machines. The objective is to assign the jobs to machines to minimize the makespan of the schedule. Despite being a fundamental problem in theory and a consequential problem in practice, the approximability of scheduling problems with communication delays is not very well understood. One of the top ten open problems in scheduling theory, in the influential list by Schuurman and Woeginger and its latest update by Bansal, asks if the problem admits a constant factor approximation algorithm. In this paper, we answer the question in negative by proving that there is a logarithmic hardness for the problem under the standard complexity theory assumption that NP-complete problems do not admit quasi-polynomial time algorithms. Our hardness result is obtained using a surprisingly simple reduction from a problem that we call Unique Machine Precedence constraints Scheduling (UMPS). We believe that this problem is of central importance in understanding the hardness of many scheduling problems and conjecture that it is very hard to approximate. Among other things, our conjecture implies a logarithmic hardness of related machine scheduling with precedences, a long-standing open problem in scheduling theory and approximation algorithms.
In this paper, we study the lower- and upper-bounded covering (LUC) problem, where we are given a set $P$ of $n$ points, a collection $mathcal{B}$ of balls, and parameters $L$ and $U$. The goal is to find a minimum-sized subset $mathcal{B}subseteq mathcal{B}$ and an assignment of the points in $P$ to $mathcal{B}$, such that each point $pin P$ is assigned to a ball that contains $p$ and for each ball $B_iin mathcal{B}$, at least $L$ and at most $U$ points are assigned to $B_i$. We obtain an LP rounding based constant approximation for LUC by violating the lower and upper bound constraints by small constant factors and expanding the balls by again a small constant factor. Similar results were known before for covering problems with only the upper bound constraint. We also show that with only the lower bound constraint, the above result can be obtained without any lower bound violation. Covering problems have close connections with facility location problems. We note that the known constant-approximation for the corresponding lower- and upper-bounded facility location problem, violates the lower and upper bound constraints by a constant factor.
We consider the problem of efficiently scheduling jobs with precedence constraints on a set of identical machines in the presence of a uniform communication delay. In this setting, if two precedence-constrained jobs $u$ and $v$, with ($u prec v$), are scheduled on different machines, then $v$ must start at least $rho$ time units after $u$ completes. The scheduling objective is to minimize makespan, i.e. the total time between when the first job starts and the last job completes. The focus of this paper is to provide an efficient approximation algorithm with near-linear running time. We build on the algorithm of Lepere and Rapine [STACS 2002] for this problem to give an $Oleft(frac{ln rho}{ln ln rho} right)$-approximation algorithm that runs in $tilde{O}(|V| + |E|)$ time.
We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSPs). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most eps worse than the relaxations value (normalized to lie in [0,1]), as long as r > kcdotrank_{geq theta}(Ins)/poly(e) ;, where k is the alphabet size of I, $theta=poly(e/k)$, and $rank_{geq theta}(Ins)$ denotes the number of eigenvalues larger than $theta$ in the normalized adjacency matrix of the constraint graph of $Ins$. In the case that $Ins$ is a uniquegames instance, the threshold $theta$ is only a polynomial in $e$, and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for emph{every} instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khots Unique Games Conjecture. Our algorithm actually requires less than the $n^{O(r)}$ constraints specified by the $r^{th}$ level of the Lasserre hierarchy, and in some cases $r$ rounds of our program can be evaluated in time $2^{O(r)}poly(n)$.
Recently, Hierarchical Clustering (HC) has been considered through the lens of optimization. In particular, two maximization objectives have been defined. Moseley and Wang defined the emph{Revenue} objective to handle similarity information given by a weighted graph on the data points (w.l.o.g., $[0,1]$ weights), while Cohen-Addad et al. defined the emph{Dissimilarity} objective to handle dissimilarity information. In this paper, we prove structural lemmas for both objectives allowing us to convert any HC tree to a tree with constant number of internal nodes while incurring an arbitrarily small loss in each objective. Although the best-known approximations are 0.585 and 0.667 respectively, using our lemmas we obtain approximations arbitrarily close to 1, if not all weights are small (i.e., there exist constants $epsilon, delta$ such that the fraction of weights smaller than $delta$, is at most $1 - epsilon$); such instances encompass many metric-based similarity instances, thereby improving upon prior work. Finally, we introduce Hierarchical Correlation Clustering (HCC) to handle instances that contain similarity and dissimilarity information simultaneously. For HCC, we provide an approximation of 0.4767 and for complementary similarity/dissimilarity weights (analogous to $+/-$ correlation clustering), we again present nearly-optimal approximations.
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