We consider the problem of augmenting an n-vertex graph embedded in a metric space, by inserting one additional edge in order to minimize the diameter of the resulting graph. We present exact algorithms for the cases when (i) the input graph is a path, running in O(n ^3 n) time, and (ii) the input graph is a tree, running in O(n^2 n) time. We also present an algorithm that computes a (1+ )-approximation in O(n + 1 ^3) time, for paths in R^d, where d is a constant.
The Diameter-Optimal @math -Augmentation Problem for edge-weighted graphs, and many of its variants, have been shown to be NP-hard @cite_16 , or even @math -hard @cite_6 @cite_18 . Because of this, several special classes of graphs have been considered. Chung and Gary @cite_13 and @cite_9 considered paths and cycles with unit edge weights and gave upper and lower bounds on the diameter that can be achieved. Ishii @cite_10 gave a constant factor approximation algorithm (approximating both @math and the diameter) for the case when the input graph is outerplanar. Erd o @cite_19 investigated upper and lower bounds for the case when the augmented graph must be triangle-free.
@cite_18 @cite_10 @cite_9 @cite_6 @cite_19 @cite_16 @cite_13
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Given an undirected graph and an integer , we consider the problem of augmenting G by a minimum set of new edges so that the diameter becomes at most D. It is known that no constant factor approximation algorithms to this problem with an arbitrary graph G can be obtained unless , while the problem with only a few graph classes such as forests is approximable within a constant factor. In this article, we give the first constant factor approximation algorithm to the problem with an outerplanar graph G. We also show that if the target diameter D is even, then the case where G is a partial 2-tree is also approximable within a constant. par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . Let G=(V,E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of [SIAM J. Comput., 28 (1999), pp. 1167--1181], we describe an @math -time algorithm APASP2 for computing all distances in G with an additive one-sided error of at most 2. Algorithm APASP2 is simple, easy to implement, and faster than the fastest known matrix-multiplication algorithm. Furthermore, for every even k>2, we describe an @math -time algorithm APASPk for computing all distances in G with an additive one-sided error of at most k. We also give an @math -time algorithm @math for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in @math time. We say that a weighted graph F=(V,E') k-emulates an unweighted graph G=(V,E) if for every @math we have @math . We show that every unweighted graph on n vertices has a 2-emulator with @math edges and a 4-emulator with @math edges. These results are asymptotically tight. Finally, we show that any weighted undirected graph on n vertices has a 3-spanner with @math edges and that such a 3-spanner can be built in @math time. We also describe an @math -time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3. We present the best known algorithms for approximating the minimum-size undirected @math -edge connected spanning subgraph. For simple graphs our approximation ratio is @math . The more precise version of this bound requires @math , and for all such @math it improves the long-standing performance ratio of Cheriyan and Thurimella [SIAM J. Comput., 30 (2000), pp. 528-560], @math . The improvement comes in two steps. First we show that for simple @math -edge connected graphs, any laminar family of degree @math sets is smaller than the general bound ( @math versus @math ). This immediately implies that iterated rounding improves the performance ratio of Cheriyan and Thurimella. The second step carefully chooses good edges for rounding. For multigraphs our approximation ratio is @math . This improves the previous ratio @math [H. N. Gabow, M. X. Goemans, E. Tardos, and D. P. Williamson, Networks, 53 (2009), pp. 345-357]. It is of interest since it is known that for some constant @math , an approximation ratio @math implies @math . Our approximation ratio extends to the minimum-size Steiner network problem, where @math denotes the average vertex demand. The algorithm exploits rounding properties of the first two linear programs in iterated rounding.