Approximating (Unweighted) Tree Augmentation via Lift-and-Project, Part I: Stemless TAP

Abstract

In Part I, we study a special case of the unweighted Tree Augmentation Problem (TAP) via the Lasserre (Sum of Squares) system. In the special case, we forbid so-called stems; these are a particular type of subtree configuration. For stemless TAP, we prove that the integrality ratio of an SDP relaxation (the Lasserre tightening of an LP relaxation) is ≤ 3 2 + , where > 0 can be any small constant. We obtain this result by designing a polynomial-time algorithm for stemless TAP that achieves an approximation guarantee of 3 2 + relative to the SDP relaxation. The algorithm is combinatorial and does not solve the SDP relaxation, but our analysis relies on the SDP relaxation. We generalize the combinatorial analysis of integral solutions from the previous literature to fractional solutions by identifying some properties of fractional solutions of the Lasserre system via the decomposition result of Karlin et al. (Integer programming and combinatoral optimization (IPCO), Lecture Notes in Computer Science, vol 6655. Springer, Berlin/Heidelberg, pp 301-314, 2011). Also, we present an example of stemless TAP such that the approximation guarantee of 3 2 is tight for the algorithm. In Part II of this paper, we extend the methods of Part I to prove the same results relative to the same SDP relaxation for TAP.

On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

“…As an example, recently Nutov [29] showed that the standard cut LP for TAP has an integrality gap of at most 28/15 while a lower bound of 3/2 was known [7]. An LP-based 5 3 + εapproximation was given by Adjiashvili [1] and then refined by Fiorini et al [12] to obtain a 3 2 + ε -approximation (see also [4,5,26]). Both results are obtained by adding a proper family of extra constraints to the standard cut LP.…”

Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

“…The first case is known in the literature as the Tree Augmentation problem and many results have been developed for it. The problem is known to be APX-hard [13,7,19], and several better-than-2 approximation algorithms have been developed using a wide range of techniques [24,11,8,20,1,12,6,21,16], being 1.393 the current best approximation ratio achieved by Cecchetto et al [5]. Tree Augmentation has also been studied in the framework of Fixed-Parameter Tractability [23,3] and in presence of general edge weights.…”

On Small-Depth Tree Augmentations

“…The optimal integral solution has cost 3. The optimal fractional solution to the odd-LP has cost at most 5 2 , since (x, y) = ( 12 , 1 2 , 1 2 , 1 2 , 1 2 , 1) is feasible. Hence the integrality gap is at least 6 5 .…”

“…While TAP is well studied in both the weighted and unweighted case [10,14,17,8,5,16,1,9,12], it is NP-hard even when the tree has diameter 4 [10] or when the set of available links form a single cycle on the leaves of the tree T [6], and is also APX-hard [15]. Weighted TAP remains one of the simplest network design problems without a better than 2-approximation in the case of general (unbounded) link costs and arbitrary depth trees, until very recently [18,19].…”