The Weighted Tree Augmentation Problem (WTAP) is a fundamental well-studied problem in the field of network design. Given an undirected tree @math , an additional set of edges @math disjoint from @math called , and a cost vector @math , WTAP asks to find a minimum-cost set @math with the property that @math is @math -edge connected. The special case where @math for all @math is called the Tree Augmentation Problem (TAP). Both problems are known to be NP-hard. For the class of bounded cost vectors, we present a first improved approximation algorithm for WTAP since more than three decades. Concretely, for any @math and @math we present an LP based @math -approximation for WTAP restricted to cost vectors @math in @math for @math . For the special case of TAP we improve this factor to @math . Our results rely on a new LP, that significantly differs from existing LPs achieving improved bounds for TAP. We round a fractional solution in two phases. The first phase uses the fractional solution to decompose the tree and its fractional solution into so-called @math -simple pairs losing only an @math -factor in the objective function. We then show how to use the additional constraints in our LP combined with the @math -simple structure to round a fractional solution in each part of the decomposition.
The techniques used to achieve the latter improved algorithms for TAP are combinatorial in nature, and seem very hard to modify for an improved approximation of WTAP. In an effort to improve the approximation factor for WTAP, several algorithm with approximation factors better than @math have recently been developed for TAP, that are based on continuous relaxations of the problem. Along these lines, Kortsarz and Nutov @cite_8 recently showed an LP-based @math -approximation algorithm. Our @math -approximation for TAP improves on that factor for LP-based algorithms. Cheriyan and Gao @cite_12 presented an approximation with respect to a semidefinite program (SDP) obtained from Lasserre tightening of an LP relaxation with factor @math . A strong point of both papers compared to our result is that the mathematical program is used in the analysis, while the algorithms themselves are combinatorial.
@cite_12 @cite_8
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In Part II, we study the unweighted tree augmentation problem (TAP) via the Lasserre (sum of squares) system. 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 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 (Integer Programming and Combinatorial Optimization, LNCS, volume 6655, Springer, pp 301–314, 2011). In the Tree Augmentation Problem (TAP) the goal is to augment a tree T by a minimum size edge set F from a given edge set E such that T+F is 2-edge-connected. The best approximation ratio known for TAP is 1.5. In the more general Weighted TAP problem, F should be of minimum weight. Weighted TAP admits several 2-approximation algorithms w.r.t. the standard cut-LP relaxation. The problem is equivalent to the problem of covering a laminar set family. Laminar set families play an important role in the design of approximation algorithms for connectivity network design problems. In fact, Weighted TAP is the simplest connectivity network design problem for which a ratio better than 2 is not known. Improving this "natural" ratio is a major open problem, which may have implications on many other network design problems. It seems that achieving this goal requires finding an LP-relaxation with integrality gap better than 2, which is an old open problem even for TAP. In this paper we introduce two different LP-relaxations, and for each of them give a simple algorithm that computes a feasible solution for TAP of size at most 7 4 times the optimal LP value. This gives some hope to break the ratio 2 for the weighted case.