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Approximation Algorithms for Orienting Mixed Graphs
"... Abstract. Graph orientation is a fundamental problem in graph theory that has recently arisen in the study of signalingregulatory pathways in protein networks. Given a graph and a list of ordered sourcetarget vertex pairs, it calls for assigning directions to the edges of the graph so as to maximi ..."
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Abstract. Graph orientation is a fundamental problem in graph theory that has recently arisen in the study of signalingregulatory pathways in protein networks. Given a graph and a list of ordered sourcetarget vertex pairs, it calls for assigning directions to the edges of the graph so as to maximize the number of pairs that admit a directed sourcetotarget path. When the input graph is undirected, a sublogarithmic approximation is known for the problem. However, the approximability of the biologicallyrelevant variant, in which the input graph has both directed and undirected edges, was left open. Here we give the first approximation algorithm to this problem. Our algorithm provides a sublinear guarantee in the general case, and logarithmic guarantees for structured instances. Key words: proteinprotein interaction network, mixed graph, graph orientation, approximation algorithm 1
Exploiting bounded signal flow for graph orientation based on causeeffect pairs
 In Proceedings of the 1st International ICST Conference on Theory and Practice of Algorithms in (Computer) Systems (TAPAS 2011
"... Background: We consider the following problem: Given an undirected network and a set of sender–receiver pairs, direct all edges such that the maximum number of “signal flows ” defined by the pairs can be routed respecting edge directions. This problem has applications in understanding protein intera ..."
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Background: We consider the following problem: Given an undirected network and a set of sender–receiver pairs, direct all edges such that the maximum number of “signal flows ” defined by the pairs can be routed respecting edge directions. This problem has applications in understanding protein interaction based cell regulation mechanisms. Since this problem is NPhard, research so far concentrated on polynomialtime approximation algorithms and tractable special cases. Results: We take the viewpoint of parameterized algorithmics and examine several parameters related to the maximum signal flow over vertices or edges. We provide several fixedparameter tractability results, and in one case a sharp complexity dichotomy between a lineartime solvable case and a slightly more general NPhard case. We examine the value of these parameters for several realworld network instances. Conclusions: Several biologically relevant special cases of the NPhard problem can be solved to optimality. In this way, parameterized analysis yields both deeper insight into the computational complexity and practical solving strategies. Background Current technologies [1] like twohybrid screening can
Network orientation via shortest paths
 BIOINFORMATICS
, 2014
"... The graph orientation problem calls for orienting the edges of a graph so as to maximize the number of prespecified source–target vertex pairs that admit a directed path from the source to the target. Most algorithmic approaches to this problem share a common preprocessing step, in which the input ..."
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The graph orientation problem calls for orienting the edges of a graph so as to maximize the number of prespecified source–target vertex pairs that admit a directed path from the source to the target. Most algorithmic approaches to this problem share a common preprocessing step, in which the input graph is reduced to a tree by repeatedly contracting its cycles. Although this reduction is valid from an algorithmic perspective, the assignment of directions to the edges of the contracted cycles becomes arbitrary, and the connecting source–target paths may be arbitrarily long. In the context of biological networks, the connection of vertex pairs via shortest paths is highly motivated, leading to the following problem variant: given a graph and a collection of source–target vertex pairs, assign directions to the edges so as to maximize the number of pairs that are connected by a shortest (in the original graph) directed path. This problem is NPcomplete and hard to approximate to within subpolynomial factors. Here we provide a first polynomialsize integer linear program formulation for this problem, which allows its exact solution in seconds on current networks. We apply our algorithm to orient protein–protein interaction networks in yeast and compare it with two stateoftheart algorithms. We find that our algorithm outperforms previous approaches and can orient considerable parts of the network, thus revealing its structure and function. Availability and implementation: The source code is available at www.cs.tau.ac.il/*roded/shortest.zip.
Steiner Forest Orientation Problems
, 2012
"... We consider connectivity problems with orientation constraints. Given a directed graph D and a collection of ordered node pairs P let P[D] = {(u,v) ∈ P: D contains a uvpath}. In the Steiner Forest Orientation problem we are given an undirected graph G = (V,E) with edgecosts and a set P ⊆ V × V ..."
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We consider connectivity problems with orientation constraints. Given a directed graph D and a collection of ordered node pairs P let P[D] = {(u,v) ∈ P: D contains a uvpath}. In the Steiner Forest Orientation problem we are given an undirected graph G = (V,E) with edgecosts and a set P ⊆ V × V of ordered node pairs. The goal is to find a minimumcost subgraph H of G and an orientation D of H such that P[D] = P. We give a 4approximation algorithm for this problem. In the Maximum Pairs Orientation problem we are given a graph G and a multicollection of ordered node pairs P on V. The goal is to find an orientation D of G such that P[D]  is maximum. Generalizing the result of Arkin and Hassin [DAM’02] for P  = 2, we will show that for a mixed graph G (that may have both directed and undirected edges), one can decide in n O(P) time whether G has an orientation D with P[D] = P (for undirected graphs this problem admits a polynomial time algorithm for any P, but it is NPcomplete on mixed graphs). For undirected graphs, we will show that one can decide whether G admits an orientation D with P[D]  ≥ k in O(n + m) + 2 O(k·log log k) time; hence this decision problem is fixedparameter tractable, which answers an open question from Dorn et al. [AMB’11]. We also show that Maximum Pairs Orientation admits ratio O(logP/log logP), which is better than the ratio O(log n/log log n) of Gamzu et al. [WABI’10] when P  < n.
Approximation Algorithms and Hardness Results for Shortest Path Based Graph Orientations
"... The graph orientation problem calls for orienting the edges of an undirected graph so as to maximize the number of prespecified sourcetarget vertex pairs that admit a directed path from the source to the target. Most algorithmic approaches to this problem share a common preprocessing step, in whi ..."
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The graph orientation problem calls for orienting the edges of an undirected graph so as to maximize the number of prespecified sourcetarget vertex pairs that admit a directed path from the source to the target. Most algorithmic approaches to this problem share a common preprocessing step, in which the input graph is reduced to a tree by repeatedly contracting its cycles. While this reduction is valid from an algorithmic perspective, the assignment of directions to the edges of the contracted cycles becomes arbitrary, and the connecting sourcetarget paths may be arbitrarily long. In the context of biological networks, the connection of vertex pairs via shortest paths is highly motivated, leading to the following variant: Given an undirected graph and a collection of sourcetarget vertex pairs, assign directions to the edges so as to maximize the number of pairs that are connected by a shortest (in the original graph) directed path. Here we study this variant, provide strong inapproximability results for it and propose an approximation algorithm for the problem, as well as for relaxations of it where the connecting paths need only be approximately shortest.
On the Complexity of two Problems on Orientations of Mixed Graphs
"... Abstract Interactions between biomolecules within the cell can be modeled by biological networks, i.e. graphs whose vertices are the biomolecules (proteins, genes, metabolites etc.) and whose edges represent their functional relationships. Depending on their nature, the interactions can be undirecte ..."
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Abstract Interactions between biomolecules within the cell can be modeled by biological networks, i.e. graphs whose vertices are the biomolecules (proteins, genes, metabolites etc.) and whose edges represent their functional relationships. Depending on their nature, the interactions can be undirected (e.g. proteinprotein interactions, PPIs) or directed (e.g. proteinDNA interactions, PDIs). A physical network is a network formed by both PPIs and PDIs, and is thus modeled by a mixed graph. External cellular events are transmitted into the nucleus via cascades of activation/deactivation of proteins, that correspond to paths (called signaling pathways) in the physical network from a source protein (cause) to a target protein (effect). There exists experimental methods to identify the causeeffect pairs, but such methods do not provide the signaling pathways. A key challenge is to infer such pathways based on the causeeffect informations. In terms of graph theory, this problem, called MAXIMUM GRAPH ORIENTATION (MGO), is defined as follows: given a mixed graph G and a set P of sourcetarget pairs, find an orientation of G that replaces each (undirected) edge by a single (directed) arc in such a way that there exists a directed path, from s to t, for a maximum number of pairs (s, t) ∈ P. In this work, we consider a variant of MGO, called SGO, in which we ask whether all the pairs in P can be connected by a
Inferring Host Gene Subnetworks Involved in Viral Replication
"... Systematic, genomewide lossoffunction experiments can be used to identify host factors that directly or indirectly facilitate or inhibit the replication of a virus in a host cell. We present an approach that combines an integer linear program and a diffusion kernel method to infer the pathways th ..."
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Systematic, genomewide lossoffunction experiments can be used to identify host factors that directly or indirectly facilitate or inhibit the replication of a virus in a host cell. We present an approach that combines an integer linear program and a diffusion kernel method to infer the pathways through which those host factors modulate viral replication. The inputs to the method are a set of viral phenotypes observed in singlehostgene mutants and a background network consisting of a variety of host intracellular interactions. The output is an ensemble of subnetworks that provides a consistent explanation for the measured phenotypes, predicts which unassayed host factors modulate the virus, and predicts which host factors are the most direct interfaces with the virus. We infer hostvirus interaction subnetworks using data from experiments screening the yeast genome for genes modulating the replication of two RNA viruses. Because a goldstandard network is unavailable, we assess the predicted subnetworks using both computational and qualitative analyses. We conduct a crossvalidation experiment in which we predict whether heldaside test genes have an effect on viral replication. Our approach is able to make highconfidence predictions more accurately than several baselines, and about as well as the best baseline, which does not infer mechanistic pathways. We also examine two kinds of predictions made by our method: which host factors are nearest to a direct interaction with a viral component, and which unassayed host genes are likely to be involved
On the Approximability of Reachability Preserving Network Orientations
"... We introduce a graph orientation problem arising in the study of biological networks. Given an undirected graph and a list of ordered sourcetarget vertex pairs, the goal is to orient the graph such that a maximum number of pairs admit a directed sourcetotarget path. We study the complexity and ap ..."
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We introduce a graph orientation problem arising in the study of biological networks. Given an undirected graph and a list of ordered sourcetarget vertex pairs, the goal is to orient the graph such that a maximum number of pairs admit a directed sourcetotarget path. We study the complexity and approximability of this problem. We show that the problem is NPhard even on star graphs and hard to approximate to within some constant factor. On the positive side, we provide an Ω(loglogn/logn)factor approximation algorithm for the problem on nvertex graphs. We further show that for any instance of the problem there exists an orientation of the input graph that satisfies a logarithmic fraction of all pairs and that this bound is tight up to a constant factor. Our techniques also lead to constant factor approximation algorithms for some restricted variants of the problem.