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.
WTAP can also be interpreted as a problem of covering a laminar family with point-to-point links (see e.g. @cite_8 ). This observation implies that the problem of augmenting a @math -edge connected graph to a @math -edge connected graph can be reduced to WTAP, whenever @math is odd, due to the laminar structure of the family of minimum cuts, in this case.
@cite_8
2052687843 2065259354 2149283591 2963972775
In the survivable network design problem (SNDP), the goal is to find a minimum-cost spanning subgraph satisfying certain connectivity requirements. We study the vertex-connectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertex-disjoint paths connecting them. We give the first strong lower bound on the approximability of SNDP, showing that the problem admits no efficient @math ratio approximation for any fixed @math , unless @math . We show hardness of approximation results for some important special cases of SNDP, and we exhibit the first lower bound on the approximability of the related classical NP-hard problem of augmenting the connectivity of a graph using edges from a given set. In the Graph Realization Problem (GRP), one is given a graph (G ), a set of non-negative edge-weights, and an integer (d ). The goal is to find, if possible, a realization of (G ) in the Euclidian space ( R ^d ), such that the distance between any two vertices is the assigned edge weight. The problem has many applications in mathematics and computer science, but is NP-hard when the dimension (d ) is fixed. Characterizing tractable instances of GRP is a classical problem, first studied by Menger in 1931. We construct two new infinite families of GRP instances which can be solved in polynomial time. Both constructions are based on the blow-up of fixed small graphs with large expanders. Our main tool is the Connelly’s condition in Rigidity Theory, combined with an explicit construction and algebraic calculations of the rigidity (stress) matrix. As an application of our results, we describe a general framework to construct uniquely k-colorable graphs. These graphs have the extra property of being uniquely vector k-colorable. We give a deterministic explicit construction of such a family using Cayley expander graphs. Discusses time complexity analysis of the minimum vertex cover and minimum connected vertex cover problems for 3-connected graphs. A vertex cover of a graph G=(V, E) is a subset N of V such that each element of E is incident upon some element of N, where V and E are the sets of vertices and of edges of G, respectively. A connected vertex cover of a graph G is a vertex cover of G such that the subgraph G(N) induced by N of G is connected. The minimum vertex cover problem (VCP) is the problem of finding a vertex cover of minimum cardinality, and the minimum connected vertex cover problem is similarly defined. > We present a deterministic algorithm that computes the edge-connectivity of a graph in near-linear time. This is for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. Our algorithm is easily extended to find a concrete minimum edge-cut. In fact, we can construct the classic cactus representation of all minimum cuts in near-linear time. The previous fastest deterministic algorithm by Gabow from STOC '91 took O(m+λ2 n), where λ is the edge connectivity, but λ can be as big as n−1. Karger presented a randomized near-linear time Monte Carlo algorithm for the minimum cut problem at STOC’96, but the returned cut is only minimum with high probability. Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree Δ, producing a multigraph Ḡ with O(m Δ) edges, which preserves all minimum cuts of G with at least two vertices on each side. In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS’06. Normally, such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed.