Prize-collecting Steiner Forest Problem Research Articles

An Approximation Algorithm for the Generalized Prize-Collecting Steiner Forest Problem with Submodular Penalties

An Approximation Algorithm for the Generalized Prize-Collecting Steiner Forest Problem with Submodular Penalties

Hunting multiple bumps in graphs

Bump hunting is an important approach to the extraction of insights from Euclidean datasets. Recently, it has been explored for graph datasets for the first time, and a single bump is hunted in an unweighted graph in this exploration. Here, we extend this exploration by hunting multiple bumps in a weighted graph. Given a weighted graph and a set of query nodes exhibiting a property of interest, our objective is to find k non-overlapping and connected subgraphs, i.e., bumps, in which the discrepancy between the numbers of query and non-query nodes is maximized and the sum of edge costs is minimized simultaneously. We prove that our extended bump hunting problem can be transformed to a recently formulated Prize-Collecting Steiner Forest Problem (PCSFP). We further prove that PCSFP is NP-hard even in trees. Then, we propose a fast approximation algorithm for solving PCSFP in trees. Based on this algorithm, we improve the state-of-the-art approximation algorithm for solving PCSFP in graphs, and prove that the solutions of our improvement are always better than or equal to those of the state-of-the-art algorithm. Moreover, we adapt the existing bump hunting algorithms for solving our extended bump hunting problem. We evaluate our methodology via real datasets, and show that 1) our improvement scales well to large graphs, while producing solutions that dominate those of the state-of-the-art algorithm; and 2) our adaptation of an existing bump hunting algorithm can also produce solutions that are better than those of the state-of-the-art algorithm in some cases.

A Primal-Dual Algorithm for the Generalized Prize-Collecting Steiner Forest Problem

In this paper, we consider the generalized prize-collecting Steiner forest problem, extending the prize-collecting Steiner forest problem. In this problem, we are given a connected graph \(G = (V, E)\) and a set of vertex sets \({\mathcal {V}} = \{V_1, V_2,\cdots , V_{l}\}\). Every edge in E has a nonnegative cost, and every vertex set in \({\mathcal {V}}\) has a nonnegative penalty cost. For a given edge set \(F\subseteq E\), vertex set \(V_i\in {\mathcal {V}}\) is said to be connected by edge set F if \(V_i\) is in a connected component of the F-spanned subgraph. The objective is to find such an edge set F such that the total edge cost in F and the penalty cost of the vertex sets not connected by F is minimized. Our main contribution is to give a 3-approximation algorithm for this problem via the primal-dual method.

Simultaneous Reconstruction of Multiple Signaling Pathways via the Prize-Collecting Steiner Forest Problem

Signaling and regulatory networks are essential for cells to control processes such as growth, differentiation, and response to stimuli. Although many "omic" data sources are available to probe signaling pathways, these data are typically sparse and noisy. Thus, it has been difficult to use these data to discover the cause of the diseases and to propose new therapeutic strategies. We overcome these problems and use "omic" data to reconstruct simultaneously multiple pathways that are altered in a particular condition by solving the prize-collecting Steiner forest problem. To evaluate this approach, we use the well-characterized yeast pheromone response. We then apply the method to human glioblastoma data, searching for a forest of trees, each of which is rooted in a different cell-surface receptor. This approach discovers both overlapping and independent signaling pathways that are enriched in functionally and clinically relevant proteins, which could provide the basis for new therapeutic strategies. Although the algorithm was not provided with any information about the phosphorylation status of receptors, it identifies a small set of clinically relevant receptors among hundreds present in the interactome.

Prize-collecting steiner network problems

In the Steiner Network problem, we are given a graph G with edge-costs and connectivity requirements r uv between node pairs u,v . The goal is to find a minimum-cost subgraph H of G that contains r uv edge-disjoint paths for all u,v ∈ V . In Prize-Collecting Steiner Network problems, we do not need to satisfy all requirements, but are given a penalty function for violating the connectivity requirements, and the goal is to find a subgraph H that minimizes the cost plus the penalty. The case when r uv ∈ {0,1} is the classic Prize-Collecting Steiner Forest problem. In this article, we present a novel linear programming relaxation for the Prize-Collecting Steiner Network problem, and by rounding it, obtain the first constant-factor approximation algorithm for submodular and monotone nondecreasing penalty functions. In particular, our setting includes all-or-nothing penalty functions, which charge the penalty even if the connectivity requirement is slightly violated; this resolves an open question posed by Nagarajan et al. [2008]. We further generalize our results for element-connectivity and node-connectivity.

Elementary approximation algorithms for prize collecting Steiner tree problems

This paper deals with approximation algorithms for the prize collecting generalized Steiner forest problem, defined as follows. The input is an undirected graph G = ( V , E ) , a collection T = { T 1 , … , T k } , each a subset of V of size at least 2, a weight function w : E → R + , and a penalty function p : T → R + . The goal is to find a forest F that minimizes the cost of the edges of F plus the penalties paid for subsets T i whose vertices are not all connected by F. Our main result is a ( 3 − 4 n ) -approximation for the prize collecting generalized Steiner forest problem, where n ⩾ 2 is the number of vertices in the graph. This obviously implies the same approximation for the special case called the prize collecting Steiner forest problem (all subsets T i are of size 2). The approximation algorithm is obtained by applying the local ratio method, and is much simpler than the best known combinatorial algorithm for this problem. Our approach gives a ( 2 − 1 n − 1 ) -approximation for the prize collecting Steiner tree problem (all subsets T i are of size 2 and there is some root vertex r that belongs to all of them). This latter algorithm is in fact the local ratio version of the primal-dual algorithm of Goemans and Williamson [M.X. Goemans, D.P. Williamson, A general approximation technique for constrained forest problems, SIAM Journal on Computing 24 (2) (April 1995) 296–317]. Another special case of our main algorithm is Bar-Yehuda's local ratio ( 2 − 2 n ) -approximation for the generalized Steiner forest problem (all the penalties are infinity) [R. Bar-Yehuda, One for the price of two: a unified approach for approximating covering problems, Algorithmica 27 (2) (June 2000) 131–144]. Thus, an important contribution of this paper is in providing a natural generalization of the framework presented by Goemans and Williamson, and later by Bar-Yehuda.