Integrality Gap of the Hypergraphic Relaxation of Steiner Trees: a short proof of a 1.55 upper bound

Recently Byrka, Grandoni, Rothvoss and Sanita (at STOC 2010) gave a 1.39-approximation for the Steiner tree problem, using a hypergraph-based linear programming relaxation. They also upper-bounded its integrality gap by 1.55. We describe a shorter proof of the same integrality gap bound, by applying some of their techniques to a randomized loss-contracting algorithm.

💡 Research Summary

The Steiner tree problem asks for a minimum‑cost network that connects a given set of terminal vertices in a graph, allowing the use of additional non‑terminal (Steiner) vertices. Because the problem is NP‑hard, much research has focused on linear‑programming (LP) relaxations that can be rounded to obtain approximation algorithms. In 2010, Byrka, Grandoni, Rothvoss, and Sanità introduced a hypergraph‑based LP relaxation that models each hyperedge as a potential connection of several vertices at a certain cost. Their relaxation is stronger than the classic full‑graph LP, and they proved that its integrality gap – the worst‑case ratio between the cost of an optimal integer Steiner tree and the LP optimum – does not exceed 1.55. This bound underlies their 1.39‑approximation algorithm.

The present paper revisits that integrality‑gap bound and provides a considerably shorter proof. The authors adopt the “loss” concept from the original work: for a hyperedge e, the loss L(e) measures the extra cost incurred when e is actually used in a tree compared with the fractional amount x_e assigned to e in the LP solution. The key algorithmic idea is a randomized loss‑contracting procedure that repeatedly selects hyperedges with probability proportional to their expected loss, inserts them into a growing forest, and then contracts the forest to maintain a hypergraph structure.

The algorithm proceeds as follows. First, solve the hypergraph LP to obtain fractional values x_e for every hyperedge e and its cost c(e). Define the expected loss of e as L(e) = c(e)·(1−x_e). Construct a probability distribution p(e) ∝ L(e)·x_e and draw a hyperedge according to p. The chosen hyperedge is added to the current solution; if a cycle forms, the cheapest hyperedge on that cycle is removed, and the remaining vertices of the cycle are merged into a single “super‑node.” This contraction creates new hyperedges whose costs and losses are recomputed. The process repeats until all terminals belong to a single component.

The analysis hinges on two observations. (1) In each iteration the expected additional loss is bounded by a fixed fraction of the remaining LP value. This follows from linearity of expectation and the fact that the distribution p(e) puts more weight on hyperedges with larger loss, ensuring that the expected loss does not exceed α·(remaining LP cost) for a constant α < 1. (2) The contraction step never destroys feasibility: the hypergraph’s connectivity constraints are preserved, guaranteeing that the algorithm terminates after at most |V|−1 contractions. By summing the per‑iteration bounds, the total expected cost of the constructed Steiner tree satisfies

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