How can the authors get a 1/4 approximation for the MTO problem on stars?

As the MTO problem on stars is equivalent to the Max-DICut problem, an 0.874-approximation for the problem can be obtained using the semidefinite programming based approximation algorithms [2, 9].

How many pairs of vertices can be satisfied?

It is easy to use the method of conditional expectations to obtain a deterministic linear time algorithm that produces an orientation of a star that satisfies at least 1/4 of the pairs.

What is the objective function for the p-p program?

3. For each pair p = (a, b) ∈ P and every tree edge e = (u, v) ∈ T , where u < v: if the path from a to b in T uses e in the direction from u to v, introduce a constraint y(p) ≤ x(e), and if it uses the edge in the direction from v to u, introduce a constraint y(p) ≤ 1− x(e).4. Maximize the objective function ∑p∈P y(p).

What is the way to predict the direction of the signal flow?

The authors complemented this network by additional 596 (kinase-substrate) PPIs from [10] for which the direction of signal flow is known (from the kinase to the substrate), represented as undirected edges in the constructed network.

How did the algorithm work in practice?

The algorithm was shown to yield very tight approximation ratios in practice, and attained remarkable accuracy in predicting edge directions on a real protein network.

How many pairs of P1 are satisfied?

If the total number of satisfied pairs in P1 is less than 18n, then for some i there are less than 9ni satisfied pairs (out of the ones associated with it).

How can the authors extend the results to unrooted trees?

All the results can be extended to unrooted, bounded-diameter trees by rooting them at a “central” vertex so that their depth is bounded by roughly half the diameter.

How many pairs of vertices are satisfied by the MTO algorithm?

Approximating MTO on CaterpillarsRecall that a caterpillar is a graph in which all vertices are on a central path or at most one edge away from it.

how do the authors force a path between sti and t t i?

to force the consistency of truth association in Ts and Tt, the authors will force a directed path from sti to t t i or from s f i to t f i by adding 3ni pairs (s t i, t t i) and 3ni pairs (s f i , t f i ) to P1.

How do the authors direct the edges between v and Ti?

To direct the edges between v and Ti, construct an instance of MTO on a star T ′ as follows: the root is v, and there are l leaves v1, . . . , vl.

What is the proof of the MTO on stars?

In this section the authors describe an approximation algorithm for MTO on stars that will also serve as a building block in their approximation algorithms for general trees.

Open AccessBook Chapter10.1007/978-3-540-87361-7_19

An Algorithm for Orienting Graphs Based on Cause-Effect Pairs and Its Applications to Orienting Protein Networks

Q: How many pairs of vertices were generated in the algorithm?

the algorithm attains very high approximation ratios in practice reaching up to 0.9 and higher ratios on instances with 500 or more pairs.

Q: How many vertices were generated by the random tree?

The random trees contained 1,000 vertices and were generated by iteratively adding a vertex, and connecting it to one of the already existing vertices uniformly at random.

Q: What is the goal of the graph orientation problem?

The goal is to direct the edges of the graph, assigning a single direction to each edge, so that a maximum number of pairs admit a directed path from the cause to the effect.

Q: How many pairs of proteins were in the data set?

The data set contained 24,457 pairs of a knocked out gene (cause) and an affected gene (effect), out of which 14,295 are pairs of proteins from the network.

Q: What is the way to predict the direction of the signal flow?

The authors complemented this network by additional 596 (kinase-substrate) PPIs from [10] for which the direction of signal flow is known (from the kinase to the substrate), represented as undirected edges in the constructed network.

Q: How did the algorithm work in practice?

The algorithm was shown to yield very tight approximation ratios in practice, and attained remarkable accuracy in predicting edge directions on a real protein network.

Q: How many pairs of P1 are satisfied?

If the total number of satisfied pairs in P1 is less than 18n, then for some i there are less than 9ni satisfied pairs (out of the ones associated with it).

Alexander Medvedovsky, +3 more

- 15 Sep 2008

- pp 222-232

TL;DR: This work considers a graph orientation problem arising in the study of biological networks, and provides an O(logn) approximation algorithm for the general case that achieves very tight approximation ratios in practice and is able to infer edge directions with high accuracy on both simulated and real network data.

Abstract: We consider a graph orientation problem arising in the study of biological networks. Given an undirected graph and a list of ordered source-target pairs, the goal is to orient the graph so that a maximum number of pairs will admit a directed path from the source to the target. We show that the problem is NP-hard and hard to approximate to within a constant ratio. We then study restrictions of the problem to various graph classes, and provide an O(logn) approximation algorithm for the general case. We show that this algorithm achieves very tight approximation ratios in practice and is able to infer edge directions with high accuracy on both simulated and real network data.

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Most frequently asked questions

1. How many pairs of vertices were generated in the algorithm?

the algorithm attains very high approximation ratios in practice reaching up to 0.9 and higher ratios on instances with 500 or more pairs.

2. How many vertices were generated by the random tree?

The random trees contained 1,000 vertices and were generated by iteratively adding a vertex, and connecting it to one of the already existing vertices uniformly at random.

3. What is the goal of the graph orientation problem?

The goal is to direct the edges of the graph, assigning a single direction to each edge, so that a maximum number of pairs admit a directed path from the cause to the effect.

4. How many pairs of proteins were in the data set?

The data set contained 24,457 pairs of a knocked out gene (cause) and an affected gene (effect), out of which 14,295 are pairs of proteins from the network.

Fig. 1. An example of the reduction from MAX-2-SAT to MTO. The input 2-SAT formula is (x1 ∨ ¬x2) ∧ (x3 ∨ ¬x2); x1 and x3 are assigned a True value, and x2 is assigned a False value.

Fig. 2. Performance on simulated data. Percents of satisfied pairs are displayed for both an optimal solution (based on IP) and the approximation algorithm’s solution. A third plot depicts the implied approximation ratio.

Citations

•Journal Article•10.1093/NAR/GKQ1207

Discovering pathways by orienting edges in protein interaction networks

Anthony Gitter, +3 more

- 01 Mar 2011

- Nucleic Acids Research

TL;DR: This work formalizes the orientation problem in weighted protein interaction graphs as an optimization problem and presents three approximation algorithms based on either weighted Boolean satisfiability solvers or probabilistic assignments that are used to identify pathways in yeast.

...read moreread less

120

•Journal Article•10.1103/PHYSREVE.84.036117

Computationally efficient measure of topological redundancy of biological and social networks

Réka Albert, +7 more

- 29 Sep 2011

- Physical Review E

TL;DR: A topological redundancy measure for labeled directed networks that is formal, computationally efficient, and applicable to a variety of directed networks such as cellular signaling, and metabolic and social interaction networks is introduced.

...read moreread less

•Journal Article•10.1089/CMB.2011.0163

Optimally orienting physical networks.

Dana Silverbush, +2 more

- 14 Oct 2011

- Journal of Computational Biology

TL;DR: This work provides a first polynomial-size ILP formulation for a network orientation problem, and finds that the algorithm achieves high accuracy and coverage in the orientation, outperforming simplified algorithmic variants that do not use information on edge directions.

...read moreread less

Journal Article•10.1016/J.MIB.2011.09.003

Network-based functional modeling of genomics, transcriptomics and metabolism in bacteria.

Lore Cloots, +2 more

- 01 Oct 2011

- Current Opinion in Microbiology

TL;DR: An overview of how integrated networks can be reconstructed from multiple omics data and how they can subsequently be used for network-based modeling of cellular function in bacteria is given.

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...

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References

Reducibility Among Combinatorial Problems.

Richard M. Karp

- 01 Jan 1972

TL;DR: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability.

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Book Chapter•10.1007/978-3-540-68279-0_8

Reducibility Among Combinatorial Problems

Richard M. Karp

- 01 Jan 2010

- Journal of Symbolic Logic

TL;DR: The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible.

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Journal Article•10.1038/NATURE01511

Mass spectrometry-based proteomics

Ruedi Aebersold, +1 more

- 13 Mar 2003

- Nature

TL;DR: The ability of mass spectrometry to identify and, increasingly, to precisely quantify thousands of proteins from complex samples can be expected to impact broadly on biology and medicine.

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Journal Article•10.1038/340245A0

A novel genetic system to detect protein-protein interactions.

Stanley Fields, +1 more

- 20 Jul 1989

- Nature

TL;DR: A novel genetic system to study protein-protein interactions between two proteins by taking advantage of the properties of the GAL4 protein of the yeast Saccharomyces cerevisiae, which may be applicable as a general method to identify proteins that interact with a known protein by the use of a simple galactose selection.

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Journal Article•10.1038/35001009

A comprehensive analysis of protein–protein interactions in Saccharomyces cerevisiae

Peter Uetz, +19 more

- 10 Feb 2000

- Nature

TL;DR: Examination of large-scale yeast two-hybrid screens reveals interactions that place functionally unclassified proteins in a biological context, interactions between proteins involved in the same biological function, and interactions that link biological functions together into larger cellular processes.

...read moreread less

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An Algorithm for Orienting Graphs Based on Cause-Effect Pairs and Its Applications to Orienting Protein Networks

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AI Agents for this Paper

Most frequently asked questions

1. How many pairs of vertices were generated in the algorithm?

2. How many vertices were generated by the random tree?

3. What is the goal of the graph orientation problem?

4. How many pairs of proteins were in the data set?

5. How can the authors get a 1/4 approximation for the MTO problem on stars?

6. How many pairs of vertices can be satisfied?

7. What is the objective function for the p-p program?

8. What is the way to predict the direction of the signal flow?

9. How did the algorithm work in practice?

10. How many pairs of P1 are satisfied?

11. How can the authors extend the results to unrooted trees?

12. How many pairs of vertices are satisfied by the MTO algorithm?

13. how do the authors force a path between sti and t t i?

14. How do the authors direct the edges between v and Ti?

15. What is the proof of the MTO on stars?

Figures

Citations

Global Analysis of Protein Phosphorylation in Yeast

Discovering pathways by orienting edges in protein interaction networks

Computationally efficient measure of topological redundancy of biological and social networks

Optimally orienting physical networks.

Network-based functional modeling of genomics, transcriptomics and metabolism in bacteria.

References

Reducibility Among Combinatorial Problems.

Reducibility Among Combinatorial Problems

Mass spectrometry-based proteomics

A novel genetic system to detect protein-protein interactions.

A comprehensive analysis of protein–protein interactions in Saccharomyces cerevisiae

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