Prize-collecting Steiner Tree Research Articles

Transcriptional changes are tightly coupled to chromatin reorganization during cellular aging.

Human life expectancy is constantly increasing and aging has become a major risk factor for many diseases, although the underlying gene regulatory mechanisms are still unclear. Using transcriptomic and chromosomal conformation capture (Hi-C) data from human skin fibroblasts from individuals across different age groups, we identified a tight coupling between the changes in co-regulation and co-localization of genes. We obtained transcription factors, cofactors, and chromatin regulators that could drive the cellular aging process by developing a time-course prize-collecting Steiner tree algorithm. In particular, by combining RNA-Seq data from different age groups and protein-protein interaction data we determined the key transcription regulators and gene regulatory changes at different life stage transitions. We then mapped these transcription regulators to the 3D reorganization of chromatin in young and old skin fibroblasts. Collectively, we identified key transcription regulators whose target genes are spatially rearranged and correlate with changes in their expression, thereby providing potential targets for reverting cellular aging.

Genomic landscape in Saudi patients with hepatocellular carcinoma using whole-genome sequencing: a pilot study

Background and aimsHepatocellular carcinoma (HCC) is the third most prevalent cancer in Saudi Arabia. HCC poses a significant clinical challenge due to the presence of resistance among certain patients to the standard therapeutic agent sorafenib. This study aims to unravel the genomic characteristics of HCC patients in Saudi Arabia, investigate the genetic makeup of tumors in both sorafenib-sensitive and sorafenib-resistant patients, and analyze the functional implications of genomic abnormalities observed in these individuals. The resistance displayed by some HCC patients toward sorafenib underscores the need for alternative treatment approaches to effectively combat this formidable disease burden.MethodsWhole-genome sequencing (WGS) was performed on 16 HCC samples and targeted sequencing was performed on seven additional tumors. We identified and validated somatic and germline genetic aberrations. Employing a prize-collecting Steiner tree algorithm, we identified important altered genetic modules and potential biomarkers for each patient. Furthermore, we analyzed non-synonymous germline and somatic mutations, specifically in patients who underwent sorafenib treatment.ResultsOut of the 13 patients who received sorafenib, three exhibited sorafenib sensitivity, while the others showed resistance to the drug. Notably, 3 out of 16 individuals carried cancer-predisposing mutations. Additionally, 8 out of 16 patients displayed non-synonymous somatic alterations in genes associated with cancer. In the targeted-sequencing samples, rare non-synonymous variants were observed across all seven cases. The study also revealed the presence of specific somatic aberrations, including TP53, PIK3CA, APOB, CTNNB1, DPYD, LRP1B, MYC, and NFE2L2, which were identified in two patients. Among the 42 genes linked to sorafenib treatment, 4 out of 10 resistant patients carried somatic non-synonymous variants. Furthermore, when analyzing the 5,000 genes most relevant to the 42 genes, 7 out of 10 resistant individuals exhibited rare non-synonymous germline variants. Interestingly, none of the three sorafenib-sensitive patients displayed any concerning variants in those genes.ConclusionOur findings indicate that most of the HCC patients possess cancer-related genetic variants, and the altered pathways in these patients exhibit similarities. Notably, resistant patients exhibit a higher frequency of aberrations in sorafenib-related genes than do sensitive patients. Specifically, 4 out of 10 resistant individuals demonstrated 13 somatic mutations, whereas none of the three sensitive patients exhibited any. Similarly, 7 out of 10 resistant patients possessed 30 germline mutations, while none were observed in the sensitive group (two-sided Fisher’s exact test; somatic: p=0.50, germline: 0.07). These results contribute to our understanding of the genetic landscape of HCC and highlight potential therapeutic targets that could aid in overcoming treatment resistance.

Risk-aware temporal cascade reconstruction to detect asymptomatic cases.

This paper studies the problem of detecting asymptomatic cases in a temporal contact network in which multiple outbreaks have occurred. We show that the key to detecting asymptomatic cases well is taking into account both individual risk and the likelihood of disease-flow along edges. We consider both aspects by formulating the asymptomatic case detection problem as a directed prize-collecting Steiner tree (Directed PCST) problem. We present an approximation-preserving reduction from this problem to the directed Steiner tree problem and obtain scalable algorithms for the Directed PCST problem on instances with more than 1.5M edges obtained from both synthetic and fine-grained hospital data. On synthetic data, we demonstrate that our detection methods significantly outperform various baselines (with a gain of 3.6 times ). We apply our method to the infectious disease prediction task by using an additional feature set that captures exposure to detected asymptomatic cases and show that our method outperforms all baselines. We further use our method to detect infection sources (“patient zero”) of outbreaks that outperform baselines. We also demonstrate that the solutions returned by our approach are clinically meaningful by presenting case studies.

Prioritization of cancer driver gene with prize-collecting steiner tree by introducing an edge weighted strategy in the personalized gene interaction network

BackgroundCancer is a heterogeneous disease in which tumor genes cooperate as well as adapt and evolve to the changing conditions for individual patients. It is a meaningful task to discover the personalized cancer driver genes that can provide diagnosis and target drug for individual patients. However, most of existing methods mainly ranks potential personalized cancer driver genes by considering the patient-specific nodes information on the gene/protein interaction network. These methods ignore the personalized edge weight information in gene interaction network, leading to false positive results.ResultsIn this work, we presented a novel algorithm (called PDGPCS) to predict the Personalized cancer Driver Genes based on the Prize-Collecting Steiner tree model by considering the personalized edge weight information. PDGPCS first constructs the personalized weighted gene interaction network by integrating the personalized gene expression data and prior known gene/protein interaction network knowledge. Then the gene mutation data and pathway data are integrated to quantify the impact of each mutant gene on every dysregulated pathway with the prize-collecting Steiner tree model. Finally, according to the mutant gene’s aggregated impact score on all dysregulated pathways, the mutant genes are ranked for prioritizing the personalized cancer driver genes. Experimental results on four TCGA cancer datasets show that PDGPCS has better performance than other personalized driver gene prediction methods. In addition, we verified that the personalized edge weight of gene interaction network can improve the prediction performance.ConclusionsPDGPCS can more accurately identify the personalized driver genes and takes a step further toward personalized medicine and treatment. The source code of PDGPCS can be freely downloaded from https://github.com/NWPU-903PR/PDGPCS.

An approximation algorithm for the group prize-collecting Steiner tree problem with submodular penalties

An approximation algorithm for the group prize-collecting Steiner tree problem with submodular penalties

Context dependent isoform specific PI3K inhibition confers drug resistance in hepatocellular carcinoma cells

BackgroundTargeted therapies for Primary liver cancer (HCC) is limited to the multi-kinase inhibitors, and not fully effective due to the resistance to these agents because of the heterogeneous molecular nature of HCC developed during chronic liver disease stages and cirrhosis. Although combinatorial therapy can increase the efficiency of targeted therapies through synergistic activities, isoform specific effects of the inhibitors are usually ignored. This study concentrated on PI3K/Akt/mTOR pathway and the differential combinatory bioactivities of isoform specific PI3K-α inhibitor (PIK-75) or PI3K-β inhibitor (TGX-221) with Sorafenib dependent on PTEN context.MethodsThe bioactivities of inhibitors on PTEN adequate Huh7 and deficient Mahlavu cells were investigated with real time cell growth, cell cycle and cell migration assays. Differentially expressed genes from RNA-Seq were identified by edgeR tool. Systems level network analysis of treatment specific pathways were performed with Prize Collecting Steiner Tree (PCST) on human interactome and enriched networks were visualized with Cytoscape platform.ResultsOur data from combinatory treatment of Sorafenib and PIK-75 and TGX-221 showed opposite effects; while PIK-75 displays synergistic effects on Huh7 cells leading to apoptotic cell death, Sorafenib with TGX-221 display antagonistic effects and significantly promotes cell growth in PTEN deficient Mahlavu cells. Signaling pathways were reconstructed and analyzed in-depth from RNA-Seq data to understand mechanism of differential synergistic or antagonistic effects of PI3K-α (PIK-75) and PI3K-β (TGX-221) inhibitors with Sorafenib. PCST allowed as to identify AOX1 and AGER as targets in PI3K/Akt/mTOR pathway for this combinatory effect. The siRNA knockdown of AOX1 and AGER significantly reduced cell proliferation in HCC cells.ConclusionsSimultaneously constructed and analyzed differentially expressed cellular networks presented in this study, revealed distinct consequences of isoform specific PI3K inhibition in PTEN adequate and deficient liver cancer cells. We demonstrated the importance of context dependent and isoform specific PI3K/Akt/mTOR signaling inhibition in drug resistance during combination therapies. (https://github.com/cansyl/Isoform-spesific-PI3K-inhibitor-analysis).

A 2-Approximation for the k-Prize-Collecting Steiner Tree Problem

We consider the \(k\)-Prize-Collecting Steiner Tree Problem. An instance is composed of an integer k and a graph G with costs on edges and penalties on vertices. The objective is to find a tree spanning at least k vertices which minimizes the cost of the edges in the tree plus the penalties of vertices not in the tree. This is one of the most fundamental network design problems and is a common generalization of the Prize-Collecting Steiner Tree Problem and the \(k\)-Minimum Spanning Tree Problem. Our main result is a 2-approximation algorithm, which improves on the currently best known approximation factor of 3.96 and has a faster running time. The algorithm builds on a modification of the primal-dual framework of Goemans and Williamson, and reveals interesting properties that can be applied to other similar problems.

A simple rounding scheme for multistage optimization

We consider the multistage framework introduced in (Gupta et al., Eisenstat et al., both in ICALP 2014), where we are given a time horizon and a sequence of instances of a (static) combinatorial optimization problem (one for each time step), and the goal is to find a sequence of solutions (one for each time step) reaching a tradeoff between the quality of the solutions in each time step and the stability/similarity of the solutions in consecutive time steps. We first introduce a novel rounding scheme, tailored for multistage problems, that accounts for the moving cost (or stability revenue) of adjacent solutions. Using this rounding scheme, we propose improved approximation algorithms for the multistage variants of Prize-Collecting Steiner Tree and Prize-Collecting Traveling Salesman problems. Furthermore, we introduce a 2-approximation algorithm for multistage Multi-Cut on trees, and we also show that our general scheme can be extended to maximization problems, by providing a 0.75-approximation algorithm for the multistage variant of MaxSat.

Approximation algorithm with constant ratio for stochastic prize-collecting Steiner tree problem

<p style='text-indent:20px;'>Steiner tree problem is a typical NP-hard problem, which has vast application background and has been an active research topic in recent years. Stochastic optimization problem is an important branch in the field of optimization. Compared with deterministic optimization problem, it is an optimization problem with random factors, and requires the use of tools such as probability and statistics, stochastic process and stochastic analysis. In this paper, we study a two-stage finite-scenario stochastic prize-collecting Steiner tree problem, where the goal is to minimize the sum of the first stage cost, the expected second stage cost and the expected penalty cost. Our main contribution is to present a primal-dual 3-approximation algorithm for this problem.</p>

On the Exact Solution of Prize-Collecting Steiner Tree Problems

The prize-collecting Steiner tree problem (PCSTP) is a well-known generalization of the classic Steiner tree problem in graphs, with a large number of practical applications. It attracted particular interest during the 11th DIMACS Challenge in 2014, and since then, several PCSTP solvers have been introduced in the literature. Although these new solvers further, and often drastically, improved on the results of the DIMACS Challenge, many PCSTP benchmark instances have remained unsolved. The following article describes further advances in the state of the art in exact PCSTP solving. It introduces new techniques and algorithms for PCSTP, involving various new transformations (or reductions) of PCSTP instances to equivalent problems, for example, to decrease the problem size or to obtain a better integer programming formulation. Several of the new techniques and algorithms provably dominate previous approaches. Further theoretical properties of the new components, such as their complexity, are discussed. Also, new complexity results for the exact solution of PCSTP and related problems are described, which form the base of the algorithm design. Finally, the new developments also translate into a strong computational performance: the resulting exact PCSTP solver outperforms all previous approaches, both in terms of runtime and solvability. In particular, it solves several formerly intractable benchmark instances from the 11th DIMACS Challenge to optimality. Moreover, several recently introduced large-scale instances with up to 10 million edges, previously considered to be too large for any exact approach, can now be solved to optimality in less than two hours. Summary of Contribution: The prize-collecting Steiner tree problem (PCSTP) is a well-known generalization of the classic Steiner tree problem in graphs, with many practical applications. The article introduces and analyses new techniques and algorithms for PCSTP that ultimately aim for improved (practical) exact solution. The algorithmic developments are underpinned by results on theoretical aspects, such as fixed-parameter tractability of PCSTP. Computationally, we considerably push the limits of tractibility, being able to solve PCSTP instances with up to 10 million edges. The new solver, which also considerably outperforms the state of the art on smaller instances, will be made publicly available as part of the SCIP Optimization Suite.

Combinatorial Heuristics for Inventory Routing Problems

We consider the deterministic inventory routing problem over a discrete finite time horizon. Given clients on a metric, each with daily demands that must be delivered from a depot and holding costs over the planning horizon, an optimal solution selects a set of daily tours through a subset of clients to deliver all demands before they are due and minimizes the total holding and tour routing costs over the horizon. In the capacitated case, a limited number of vehicles are available, where each vehicle makes at most one trip per day. Each trip from the depot is allowed to carry a limited amount of supply to deliver. We develop fast heuristics for both cases by solving a family of prize-collecting Steiner tree instances. Computational experiments show our heuristics can find near-optimal solutions for both cases and substantially reduce the runtime compared with a pure mixed integer programming formulation approach.

Towards the Sustainable Development Goals: A Bi-objective framework for electricity access

Traditionally, the main focus of evaluation in universal electricity access problems has been cost. However, additional criteria such as increasing renewable penetration due to environmental concerns or grid penetration due to reliability concerns, have become increasingly important. We acknowledge the importance of additional criteria and propose a bi-objective framework so as to help decision makers investigate the trade-offs between potentially conflicting criteria in rural electrification. We consider two objective space based exact approaches using the Prize Collecting Steiner Tree (PCST) formulation and two metaheuristic algorithms to find Pareto solutions, and investigate their performances on real life problem instances. This study is expected to be an important decision support tool for the electrification of underdeveloped communities, having the potential of contributing to their socio-economic development.

Primal-dual based distributed approximation algorithm for Prize-collecting Steiner tree

The Prize-collecting Steiner tree (PCST) problem is a generalization of the Steiner tree problem that finds applications in network design, content distribution networks, and many more. There are a few centralized approximation algorithms [D. Bienstock, M. X. Goemans, D. Simchi-Levi and D. Williamson, A note on the prize collecting traveling salesman problem. Math. Program. 59 (1993) 413–420; M. X. Goemans and D. E. Williamson, A general approximation technique for constrained forest problems, SIAM J. Appl. Math. 24(2) (1995) 296–317; D. S. Johnson, M. Minkoff and S. Phillips, The prize collecting Steiner tree problem: Theory and practice, in Proc. Eleventh Annual ACM-SIAM Symp. Discrete Algorithms, SODA ’00 (2000), pp. 760–769; A. Archer, M. Hossein Bateni and M. Taghi Hajiaghayi, Improved approximation algorithms for prize-collecting Steiner tree and TSP, SIAM J. Comput. 40(2) (2011) 309–332] for solving the PCST problem. However, the problem has seen very little progress in the distributed setting; to the best of our knowledge, the only distributed algorithms proposed so far are due to Rossetti [N. G. Rossetti, A first attempt on the distributed prize-collecting Steiner tree problem, M.Sc. thesis, University of Iceland, Reykjavik (2015)]: one of them fails to guarantee a constant approximation factor while the other one is essentially centralized. In this work, first, we present a deterministic [Formula: see text] factor distributed approximation algorithm (D-PCST algorithm) that constructs a PCST for a given connected undirected graph of [Formula: see text] nodes with non-negative edge weights and non-negative prize value for each node. The D-PCST algorithm is based on the primal-dual method and uses a technique of preserving dual constraints in a distributed manner, without relying on knowledge of the global structure of the network. For an input graph [Formula: see text], the round and message complexities of the D-PCST algorithm in the CONGEST model are [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] and [Formula: see text]. Furthermore, we modify the D-PCST algorithm and show that a [Formula: see text]-approximate PCST can be deterministically computed using [Formula: see text] rounds and [Formula: see text] messages in the CONGEST model, where [Formula: see text] is the unweighted diameter of [Formula: see text]. For networks with [Formula: see text], the modified D-PCST algorithm performs better than the original one in terms of the round complexity. Both the algorithms require [Formula: see text] bits of memory in each node, where [Formula: see text] is the maximum degree of a node in the graph.

An Efficient Approach to Finding Dense Temporal Subgraphs

Dense subgraph discovery has proven useful in various applications of temporal networks. We focus on a special class of temporal networks whose nodes and edges are kept fixed, but edge weights regularly vary with timestamps. However, finding dense subgraphs in temporal networks is non-trivial, and its state of the art solution uses a filter-and-verification framework that is not scalable on large temporal networks. In this study, we propose a highly efficient approach to finding dense subgraphs in large temporal networks with $T$ T timestamps. (1) We first develop a statistics-driven approach that employs hidden statistics to identifying $k$ k time intervals, instead of $T(T+1)/2$ T ( T + 1 ) / 2 ones ( $k$ k is typically much smaller than $T$ T ), which strikes a balance between quality and efficiency. (2) After proving that the problem has no constant factor approximation algorithms, we design better heuristic algorithms to attack the problem, by connecting finding dense subgraphs with a variant of the Prize Collecting Steiner Tree problem. (3) Finally, we have conducted an extensive experimental study to verify that our approach is both effective and efficient.

A Unified PTAS for Prize Collecting TSP and Steiner Tree Problem in Doubling Metrics

We present a unified (randomized) polynomial-time approximation scheme (PTAS) for the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP) in doubling metrics. Given a metric space and a penalty function on a subset of points known as terminals, a solution is a subgraph on points in the metric space whose cost is the weight of its edges plus the penalty due to terminals not covered by the subgraph. Under our unified framework, the solution subgraph needs to be Eulerian for PCTSP, while it needs to be a tree for PCSTP. Before our work, even a QPTAS for the problems in doubling metrics is not known. Our unified PTAS is based on the previous dynamic programming frameworks proposed in Talwar (STOC 2004) and Bartal, Gottlieb, Krauthgamer (STOC 2012). However, since it is unknown which part of the optimal cost is due to edge lengths and which part is due to penalties of uncovered terminals, we need to develop new techniques to apply previous divide-and-conquer strategies and sparse instance decompositions.

Approximating Node-Weighted k-MST on Planar Graphs

We study the problem of finding a minimum weight connected subgraph spanning at least k vertices on planar, node-weighted graphs. We give a (4 + ε)-approximation algorithm for this problem. We achieve this by utilizing the recent Lagrangian-multiplier preserving (LMP) primal-dual 3-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al. (SWAT’16) and adopting an approach by Chudak et al. (Math. Prog. ’04) regarding Lagrangian relaxation for the edge-weighted variant. In particular, we improve the procedure of picking additional vertices (tree merging procedure) given by Sadeghian (2013) by taking a constant number of recursive steps and utilizing the limited guessing procedure of Arora and Karakostas (Math. Prog. ’06). More generally, our approach readily gives a (4/3 ⋅ r + ε)-approximation on any graph class where the algorithm of Byrka et al. for the prize-collecting version gives an r-approximation. We argue that this can be interpreted as a generalization of an analogous result by Könemann et al. (Algorithmica ’11) for partial cover problems. Together with a lower bound construction by Mestre (STACS’08) for partial cover this implies that our bound is essentially best possible among algorithms that utilize an LMP algorithm for the Lagrangian relaxation as a black box. In addition to that, we argue by a more involved lower bound construction that even using the LMP algorithm by Byrka et al. in a non-black-box fashion could not beat the factor 4/3 ⋅ r when the tree merging step relies only on the solutions output by the LMP algorithm.

PRODIGY: personalized prioritization of driver genes.

MotivationEvolution of cancer is driven by few somatic mutations that disrupt cellular processes, causing abnormal proliferation and tumor development, whereas most somatic mutations have no impact on progression. Distinguishing those mutated genes that drive tumorigenesis in a patient is a primary goal in cancer therapy: Knowledge of these genes and the pathways on which they operate can illuminate disease mechanisms and indicate potential therapies and drug targets. Current research focuses mainly on cohort-level driver gene identification but patient-specific driver gene identification remains a challenge.MethodsWe developed a new algorithm for patient-specific ranking of driver genes. The algorithm, called PRODIGY, analyzes the expression and mutation profiles of the patient along with data on known pathways and protein–protein interactions. Prodigy quantifies the impact of each mutated gene on every deregulated pathway using the prize-collecting Steiner tree model. Mutated genes are ranked by their aggregated impact on all deregulated pathways.ResultsIn testing on five TCGA cancer cohorts spanning >2500 patients and comparison to validated driver genes, Prodigy outperformed extant methods and ranking based on network centrality measures. Our results pinpoint the pleiotropic effect of driver genes and show that Prodigy is capable of identifying even very rare drivers. Hence, Prodigy takes a step further toward personalized medicine and treatment.Availability and implementationThe Prodigy R package is available at: https://github.com/Shamir-Lab/PRODIGY.Supplementary information Supplementary data are available at Bioinformatics online.

Developing a network view of type 2 diabetes risk pathways through integration of genetic, genomic and functional data

BackgroundGenome-wide association studies (GWAS) have identified several hundred susceptibility loci for type 2 diabetes (T2D). One critical, but unresolved, issue concerns the extent to which the mechanisms through which these diverse signals influencing T2D predisposition converge on a limited set of biological processes. However, the causal variants identified by GWAS mostly fall into a non-coding sequence, complicating the task of defining the effector transcripts through which they operate.MethodsHere, we describe implementation of an analytical pipeline to address this question. First, we integrate multiple sources of genetic, genomic and biological data to assign positional candidacy scores to the genes that map to T2D GWAS signals. Second, we introduce genes with high scores as seeds within a network optimization algorithm (the asymmetric prize-collecting Steiner tree approach) which uses external, experimentally confirmed protein-protein interaction (PPI) data to generate high-confidence sub-networks. Third, we use GWAS data to test the T2D association enrichment of the “non-seed” proteins introduced into the network, as a measure of the overall functional connectivity of the network.ResultsWe find (a) non-seed proteins in the T2D protein-interaction network so generated (comprising 705 nodes) are enriched for association to T2D (p = 0.0014) but not control traits, (b) stronger T2D-enrichment for islets than other tissues when we use RNA expression data to generate tissue-specific PPI networks and (c) enhanced enrichment (p = 3.9 × 10− 5) when we combine the analysis of the islet-specific PPI network with a focus on the subset of T2D GWAS loci which act through defective insulin secretion.ConclusionsThese analyses reveal a pattern of non-random functional connectivity between candidate causal genes at T2D GWAS loci and highlight the products of genes including YWHAG, SMAD4 or CDK2 as potential contributors to T2D-relevant islet dysfunction. The approach we describe can be applied to other complex genetic and genomic datasets, facilitating integration of diverse data types into disease-associated networks.

Connectivity problems on heterogeneous graphs

BackgroundNetwork connectivity problems are abundant in computational biology research, where graphs are used to represent a range of phenomena: from physical interactions between molecules to more abstract relationships such as gene co-expression. One common challenge in studying biological networks is the need to extract meaningful, small subgraphs out of large databases of potential interactions. A useful abstraction for this task turned out to be the Steiner Network problems: given a reference “database” graph, find a parsimonious subgraph that satisfies a given set of connectivity demands. While this formulation proved useful in a number of instances, the next challenge is to account for the fact that the reference graph may not be static. This can happen for instance, when studying protein measurements in single cells or at different time points, whereby different subsets of conditions can have different protein milieu.Results and discussionWe introduce the condition Steiner Network problem in which we concomitantly consider a set of distinct biological conditions. Each condition is associated with a set of connectivity demands, as well as a set of edges that are assumed to be present in that condition. The goal of this problem is to find a minimal subgraph that satisfies all the demands through paths that are present in the respective condition. We show that introducing multiple conditions as an additional factor makes this problem much harder to approximate. Specifically, we prove that for C conditions, this new problem is NP-hard to approximate to a factor of C - epsilon , for every C ge 2 and epsilon > 0, and that this bound is tight. Moving beyond the worst case, we explore a special set of instances where the reference graph grows monotonically between conditions, and show that this problem admits substantially improved approximation algorithms. We also developed an integer linear programming solver for the general problem and demonstrate its ability to reach optimality with instances from the human protein interaction network.ConclusionOur results demonstrate that in contrast to most connectivity problems studied in computational biology, accounting for multiplicity of biological conditions adds considerable complexity, which we propose to address with a new solver. Importantly, our results extend to several network connectivity problems that are commonly used in computational biology, such as Prize-Collecting Steiner Tree, and provide insight into the theoretical guarantees for their applications in a multiple condition setting.

The Fast Heuristic Algorithms and Post-Processing Techniques to Design Large and Low-Cost Communication Networks

It is challenging to design large and low-cost communication networks. In this paper, we formulate this challenge as the prize-collecting Steiner Tree Problem (PCSTP). The objective is to minimize the costs of transmission routes and the disconnected monetary or informational profits. Initially, we note that the PCSTP is MAX SNP-hard. Then, we propose some post-processing techniques to improve suboptimal solutions to PCSTP. Based on these techniques, we propose two fast heuristic algorithms: the first one is a quasilinear time heuristic algorithm that is faster and consumes less memory than other algorithms; and the second one is an improvement of a stateof-the-art polynomial time heuristic algorithm that can find high-quality solutions at a speed that is only inferior to the first one. We demonstrate the competitiveness of our heuristic algorithms by comparing them with the state-of-the-art ones on the largest existing benchmark instances (169 800 vertices and 338 551 edges). Moreover, we generate new instances that are even larger (1 000 000 vertices and 10 000 000 edges) to further demonstrate their advantages in large networks. The state-ofthe-art algorithms are too slow to find high-quality solutions for instances of this size, whereas our new heuristic algorithms can do this in around 6 to 45s on a personal computer. Ultimately, we apply our post-processing techniques to update the bestknown solution for a notoriously difficult benchmark instance to show that they can improve near-optimal solutions to PCSTP. In conclusion, we demonstrate the usefulness of our heuristic algorithms and post-processing techniques for designing large and low-cost communication networks.