Abstract: In the Covering Steiner problem, we are given an undirected graph with edgecosts, and some subsets of vertices called groups, with each group being equipped with a non-negative integer value (called its requirement); the problem is to find a minimum-cost tree which spans at least the required number of vertices from every group. The Covering Steiner problem is a common generalization of the k-MST and Group Steiner problems; indeed, when all the vertices of the graph lie in one group with a requirement of k, we get the k-MST problem, and when there are multiple groups with unit requirements, we obtain the Group Steiner problem. While many covering problems (e.g., the covering integer programs such as set cover) become easier to approximate as the requirements increase, the Covering Steiner problem

We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a collection of

We consider the problem of embedding a metric into low-dimensional Euclidean space. The classical theorems of Bourgain and of Johnson and Lindenstrauss imply that any metric on n points embeds into an O(log n)-dimensional Euclidean space with O(log n) distortion. Moreover, a simple “volume ” argument shows that this bound is nearly tight: the uniform metric on n points requires Ω(log n / log log n) dimensions to embed with logarithmic distortion. It is natural to ask whether such a volume restriction is the only hurdle to low-dimensional low-distortion embeddings. Do doubling metrics, which do not have large uniform submetrics, embed in low dimensional Euclidean spaces with small distortion? In this paper, we answer the question positively and show that any doubling metric embeds into O(log log n) dimensions with o(log n) distortion. In fact, we give a suite of embeddings with a smooth trade-off between distortion and dimension: given an n-point metric (V, d) with doubling dimension dimD, and any target dimension T in the range Ω(dimD log log n) ≤ T ≤ O(log n), we embed the metric into Euclidean space R T with O(log n � dimD /T) distortion.

We present a poly-log-competitive deterministic online algorithm for the online transportation problem on hierarchically separated trees when the online algorithm has one extra server per site. Using metric embedding results in the literature, one can then obtain a poly-log-competitive randomized online algorithm for the online transportation on an arbitrary metric space when the online algorithm has one extra server per site. 1

Based on a recent work by Abraham, Bartal and Neiman (2007), we construct a strictly fundamental cycle basis of length O(n 2) for any unweighted graph, whence proving the conjecture of Deo et al. (1982). For weighted graphs, we construct cycle bases of length O(W ·log n log log n), where W denotes the sum of the weights of the edges. This improves the upper bound that follows from the result of Elkin et al. (2005) by a logarithmic factor and, for comparison from below, some natural classes of large girth graphs are known to exhibit minimum cycle bases of length Ω(W · log n). We achieve this bound for weighted graphs by not restricting ourselves to strictly fundamental cycle bases—as it is inherent to the approach of Elkin et al.—but rather also considering weakly fundamental cycle bases in our construction. This way we profit from some nice properties of Hierarchically Well-Separated Trees that were introduced by Bartal (1998). 1

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Can any metric be embedded into a single tree with low distortion? Unfortunately, the answer to this question is “no”. Embedding Cn into a single subtree requires distortion at least n − 1 (deleting a single edge from the cycle yields a tree metric with distortion n − 1). Rabinovich and Raz (1995) proved that embedding the unit weight n-cycle Cn into a tree (which is not necessarily a subtree of Cn, and may have vertices and edges not in Cn) still requires a distortion of Ω(n). We will overcome this lower bound by embedding metrics into distributions of trees. Definition 8.1 Suppose G is a graph family. Then (X, d) ↩ D − → G means that there exists a graph H ∈ G with edge lengths such that (X, d) ↩ D − → (vH, dH). Definition 8.2 (X, d) ↩ D − → distrib(G) means that there exists a distribution π on the graph family G and an r> 0 such that: r ≤ EH←π[dH(x, y)] d(x, y) ≤ Dr It is easy to see that dπ(x, y) = EH←π[dH(x, y)] is a metric (because of linearity). 8.2 Line Metrics Assume L is the set of all line metrics. (L is equal to the set of all metrics that isometrically embed into the real line ℓ 1 p.) Theorem 8.3 Let L be the set of all line metrics. For any metric (X, d), log n (X, d) ↩−−− → distrib(L) This follows from the following (simple) result: Lemma 8.4 Let L be the set of all line metrics. Given a metric µ, µ ∈ ℓ1 ⇐ ⇒ µ ∈ distrib(L). Proof. For one direction, assume µ ∈ ℓ1. Then µ = � S ySδS, where δS are all elementary cut metrics. But elementary cut metrics are line metrics with two points, so µ is equivalent to a ySi distribution over line metrics, each one having probability mass note that line metrics are in ℓ1, and hence distributions over them, which are the same as convex combinations of them, are in ℓ1 as well. IV-1 P. For the other direction,

In this lecture, we will study a different type of graph cuts than the max-cut problem previously discussed. Let us recall that the maximum-cut problem is to find a 2-coloring of the vertices to maximize the total weight of the edges whose endpoints have different colors. We have seen a simple factor-1/2 approximation in Homework #1, and an SDP-based factor-0.878 approximation in Lecture 14. We also know that the problem is NP-hard. It is natural to wonder what we can say if the objective is instead to minimize the weight. +1 Surprisingly, the minimization version turns out to be much eas--1 ier than max-cut: by a celebrated theorem of Ford and Fulkerson [FF62], the minimum s-t cut problem can be solved efficiently using the duality between max-flow and min-cut. Thus, we can try all possible (s, t) pairs and solve this problem exactly in polynomial time. In fact, we can be a little more careful about which (s, t) pairs Cut to examine and obtain a slightly more efficient algorithm. Let’s look at a generalization of this problem.

We study approximation algorithms for generalized network design where the cost of an edge depends on the identities of the demands using it (as a monotone subadditive function). Our main result is that even a very special case of this problem cannot be approximated to within a factor 2 log1−ε |D| if D is the set of demands

We consider optimization problems for which the best known approximation algorithms are randomized algorithms: these algorithms make random choices during their execution, and it has been shown that in expectation the cost of the algorithm’s solution is at most a known constant factor more than optimal. We show how to give deterministic variants of these algorithms that have similar performance guarantees. In particular, we give conditions under which the Sample-Augment algorithms proposed by Gupta et al. [42] can be derandomized, thus obtaining the best known deterministic algorithms for a number of network design problems such as the connected facility location, virtual private network design and single sink buy-at-bulk problems. We also give deterministic variants of the “pivoting ” algorithms proposed by Ailon et al. [4] for several ranking and clustering problems. In addition to obtaining the same performance guarantees, the analysis of our algorithms is actually simpler than that of their randomized counterparts. Finally, we take a more practical approach to one of the ranking problems considered: the rank aggregation problem. We perform an extensive evaluation of several known and new algorithms for rank aggregation on web search data. We argue that there are two important classes of algorithms for rank aggregation: positional methods and comparison sort methods. We find that hybrid algorithms, that combine a positional and comparison sort approach, work especially well on our data sets.