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Background Network 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 discussion We 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 , for every and , 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. Conclusion Our 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.

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Yannakakis [1991, 1988] showed that the matching problem does not have a small symmetric linear program. Rothvoß [2014] recently proved that any, not necessarily symmetric, linear program also has exponential size. In light of this, it is natural to ask whether the matching problem can be expressed compactly in a framework such as semidefinite programming (SDP) that is more powerful than linear programming but still allows efficient optimization. We answer this question negatively for symmetric SDPs: any symmetric SDP for the matching problem has exponential size.We also show that an O(k)-round Lasserre SDP relaxation for the asymmetric metric traveling salesperson problem yields at least as good an approximation as any symmetric SDP relaxation of size n k . The key technical ingredient underlying both these results is an upper bound on the degree needed to derive polynomial identities that hold over the space of matchings or traveling salesperson tours.

Yannakakis [1991, 1988] showed that the matching problem does not have a small symmetric linear program. Rothvoß [2014] recently proved that any, not necessarily symmetric, linear program also has exponential size. In light of this, it is natural to ask whether the matching problem can be expressed compactly in a framework such as semidefinite programming (SDP) that is more powerful than linear programming but still allows efficient optimization. We answer this question negatively for symmetric SDPs: any symmetric SDP for the matching problem has exponential size.We also show that an O(k)-round Lasserre SDP relaxation for the asymmetric metric traveling salesperson problem yields at least as good an approximation as any symmetric SDP relaxation of size n k . The key technical ingredient underlying both these results is an upper bound on the degree needed to derive polynomial identities that hold over the space of matchings or traveling salesperson tours.