Ordering problems approximated: single-processor scheduling and interval graph completion

Ravi, R.; Agrawal, Ajit; Klein, Philip N.

doi:10.1007/3-540-54233-7_180

Cited by 42 publications

(34 citation statements)

References 10 publications

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“…For example, Klein et al [1995] used the flow results to design approximation algorithms for 2-CNF satisfiability. Agarwal et al [1993] and Ravi et al [1991] gave approximation algorithms for minimizing fill when solving sparse linear systems of equations, and register allocation. Garg et al [1996] used their flow results to find an approximation algorithm for the minimum multicut problem.…”

Section: Our Max-flow Min-cut Resultsmentioning

confidence: 99%

“…The main difference is that we have a DAG and are worried about node cuts instead of edge cuts. Ravi et al [1991] describe an O(log 2 n) times optimal approximation algorithm for this problem. The algorithm combines techniques of Sections 3.5 and 3.8 along with some other ideas to obtain approximately optimal register cost.…”

Section: Minimum Cut Linear Arrangement One Of the Most Famous Np-hardmentioning

confidence: 99%

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Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Leighton

Rao

1999

J. ACM

697

666

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Abstract. In this paper, we establish max-flow min-cut theorems for several important classes of multicommodity flow problems. In particular, we show that for any n-node multicommodity flow problem with uniform demands, the max-flow for the problem is within an O(log n) factor of the upper bound implied by the min-cut. The result (which is existentially optimal) establishes an important analogue of the famous 1-commodity max-flow min-cut theorem for problems with multiple commodities. The result also has substantial applications to the field of approximation algorithms. For example, we use the flow result to design the first polynomial-time (polylog n-times-optimal) approximation algorithms for well-known NP-hard optimization problems such as graph partitioning, min-cut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. Applications of the flow results to path routing problems, network reconfiguration, communication in distributed networks, scientific computing and rapidly mixing Markov chains are also described in the paper.

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Section: Our Max-flow Min-cut Resultsmentioning

confidence: 99%

Section: Minimum Cut Linear Arrangement One Of the Most Famous Np-hardmentioning

confidence: 99%

Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Leighton

Rao

1999

J. ACM

697

666

View full text Add to dashboard Cite

show abstract

“…It is known that the Register Sufficiency problem (also known as One-Shot Black Pebbling) admits a O(log 2 n) approximation algorithm (Ravi, Agrawal, & Klein, 1991). We observe that by plugging in the improved approximation algorithm for direct vertex separator (Agarwal, Charikar, Makarychev, & Makarychev, 2005) into the algorithm by Ravi et al (1991), one can improve this to an O( √ log n log n) approximation algorithm.…”

Section: Previous Workmentioning

confidence: 93%

“…We observe that by plugging in the improved approximation algorithm for direct vertex separator (Agarwal, Charikar, Makarychev, & Makarychev, 2005) into the algorithm by Ravi et al (1991), one can improve this to an O( √ log n log n) approximation algorithm.…”

Section: Previous Workmentioning

confidence: 99%

Inapproximability of Treewidth and Related Problems

Austrin

Pitassi

et al. 2014

jair

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Graphical models, such as Bayesian Networks and Markov networks play an important role in artificial intelligence and machine learning. Inference is a central problem to be solved on these networks. This, and other problems on these graph models are often known to be hard to solve in general, but tractable on graphs with bounded Treewidth. Therefore, finding or approximating the Treewidth of a graph is a fundamental problem related to inference in graphical models. In this paper, we study the approximability of a number of graph problems: Treewidth and Pathwidth of graphs, Minimum Fill-In, OneShot Black (and Black-White) pebbling costs of directed acyclic graphs, and a variety of different graph layout problems such as Minimum Cut Linear Arrangement and Interval Graph Completion. We show that, assuming the recently introduced Small Set Expansion Conjecture, all of these problems are NP-hard to approximate to within any constant factor in polynomial time.

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“…The minimum linear arrangement problem (or MinLA) was orignally formulated by Harper [9]. It has a lot of applications, including VLSI design [1], graph drawing [17], modeling of nervous activity in the cortex [15], single machine job scheduling [2] [22], and etc.…”

Section: Definition 4 (Minimum Linear Arrangement Problem [9])mentioning

confidence: 99%

Overlapping Matrix Pattern Visualization: A Hypergraph Approach

Jin

Xiang

Fuhry

et al. 2008

2008 Eighth IEEE International Conference on Data Mining

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In this work, we study a visual data mining problem: Given a set of discovered overlapping submatrices of interest, how can we order the rows and columns of the data matrix to best display these submatrices and their relationships? We find this problem can be converted to the hypergraph ordering problem, which generalizes the traditional minimal linear arrangement (or graph ordering) problem and then we are able to prove the NP-hardness of this problem. We propose a novel iterative algorithm which utilize the existing graph ordering algorithm to solve the optimal visualization problem. This algorithm can always converge to a local minimum. The detailed experimental evaluation using a set of publicly available transactional datasets demonstrates the effectiveness and efficiency of the proposed algorithm.

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Ordering problems approximated: single-processor scheduling and interval graph completion

Cited by 42 publications

References 10 publications

Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Inapproximability of Treewidth and Related Problems

Overlapping Matrix Pattern Visualization: A Hypergraph Approach

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