2021/10/26 by Shi Li, Li, Shi, Bundit Laekhanukit +1 · 1 citation
Computer Science · Engineering · #68Q25 #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #F.2.2 #FOS: Computer and information sciences #FOS: Mathematics #Formal Methods in Verification #G.2.2 #Optimization and Control (math.OC) #VLSI and FPGA Design Techniques
paper · pdf · doi:10.48550/arxiv.2110.13350
openalex publication_date 2021/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the Directed Steiner Tree (DST) problem, we are given a directed graph G=(V,E) on n vertices with edge-costs c ∈ ℝ≥ 0E, a root vertex r ∈ V, and a set K ⊆ V ∖ \r\ of k terminals. The goal is to find a minimum-cost subgraph of G that contains a path from r to every terminal t ∈ K. DST has been a notorious problem for decades as there is a large gap between the best-known polynomial-time approximation ratio of O(kε) for any constant ε> 0, and the best quasi-polynomial-time approximation ratio of O((log2 k)/(log log k)). Towards understanding this gap, we study the integrality gap of the standard flow LP relaxation for the problem. We show that the LP has an integrality gap of Ω(n0.0418). Previously, the integrality gap of the LP is only known to be Ω((log2n)/(loglog n)) [Halperin~et~al., SODA'03 & SIAM J.~Comput.] and Ω(√(k)) [Zosin-Khuller, SODA'02] in some instance with √(k)=O((log n)/(log log n)). Our result gives the first known lower bound on the integrality gap of this standard LP that is polynomial in n, the number of vertices. Consequently, we rule out the possibility of developing a poly-logarithmic approximation algorithm for the problem based on the flow LP relaxation.