Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

The D irected S teiner N etwork (DSN) problem takes as input a directed graph G =( V , E ) with non-negative edge-weights and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N⊆ G for which there is an s\rightarrow t path for each ( s , t )∈ D. It is known that this problem is notoriously hard, as there is no k 1/4− o (1) -approximation algorithm under Gap-ETH, even when parametrizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k . For the bi -DSNP lanar problem, the aim is to compute a solution N⊆ G whose cost is at most that of an optimum planar solution in a bidirected graph G , i.e., for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k . We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) no PAS exists for any generalization of bi-DSNP lanar , under standard complexity assumptions. The techniques we use also imply a polynomial-sized approximate kernelization scheme (PSAKS). Additionally, we study several generalizations of bi -DSNP lanar and obtain upper and lower bounds on obtainable runtimes parameterized by k . One important special case of DSN is the S trongly C onnected S teiner S ubgraph (SCSS) problem, for which the solution network N⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter k [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for k no parameterized (2 − ε)-approximation algorithm exists under Gap-ETH. Additionally, we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k .

New concept of connection in bidirected graphs

In bidirected graph an edge has a direction at each end. We introduce a new definition of connection in a bidirected graph. We prove some properties of this definition and we establish a relationship to connection and imbalance in the corresponding signed graph. The main result gives a sufficient condition for a signed graph to have a Biconnected biorientation.

Describing the Local Structure of Sequence Graphs

Analysis of genetic variation using graph structures is an emerging paradigm of genomics. However, defining genetic sites on sequence graphs remains an open problem. Paten’s invention of the ultra-bubble and snarl, special subgraphs of sequence graphs which can identified with efficient algorithms, represents important first step to segregating graphs into genetic sites. We extend the theory of ultrabubbles to a special subclass where every detail of the ultrabubble can be described in a series and parallel arrangement of genetic sites. We furthermore introduce the concept of bundle structures, which allows us to recognize the graph motifs created by additional combinations of variation in the graph, including but not limited to runs of abutting single nucleotide variants. We demonstrate linear-time identification of bundles in a bidirected graph. These two advances build on initial work on ultrabubbles in bidirected graphs, and define a more granular concept of genetic site.