We complement our emulator upper bound with a lower bound construction that is essentially tight (within logn factors of the upper bound) when the stretch is 2k−1 and k is either a fixed odd integer or 2. We also show constructions of fault-tolerant emulators with additive error, demonstrating that these also enjoy significantly improved tradeoffs over those available for fault-tolerant additive spanners.
Our main conceptual contribution is establishing a formal connection between LD2S and a variant of the Densest k-Subgraph (DkS) problem. Specifically, we design for both problems strong relaxations based on Linear Programming (LP) hierarchies, and show that "faithful" randomized rounding of the DkS-variant can be used to round LD2S solutions. Our notion of faithfulness intuitively means that all vertices and edges are chosen with probability proportional to their LP value, but the precise formulation is more subtle.
Unfortunately, the best algorithms known for DkS use the Sherali-Adams LP hierarchy in a non-faithful way [Bhaskara, Charikar, Chlamtáč, Feige, and Vijayaraghavan, STOC 2010]. Our main technical contribution is to overcome this shortcoming, while still matching the gap that arises in random graphs by planting a subgraph with same log-density.
The main technique we use is a reduction to a new connectivity-based network design problem that we call Constrained Connectivity. In this problem we are given a graph G=(V,E), and for every pair of vertices u,v \in V we are given a set S(u,v) ⊆ V called the safe set of the pair. The goal is to find the smallest subgraph H of G in which every pair of vertices u,v is connected by a path contained in S(u,v). We show that the IBGP problem can be reduced to the special case of Constrained Connectivity where G = K_n and safe sets are defined geometrically based on the IGP distances in the AS. We also believe that Constrained Connectivity is an interesting problem in its own right, so provide stronger hardness results (2^(log^(1-ε) n)-hardness of approximation) and integrality gaps (n^(1/3 - ε) for the general case. On the positive side, we show that Constrained Connectivity turns out to be much simpler for some interesting special cases other than iBGP: when safe sets are symmetric and hierarchical, we give a polynomial time algorithm that computes an optimal solution.
Techniques derived from metric embeddings have been considered extensively by the networking community, usually under the name of network coordinate systems. On the other hand, while the computation of distance oracles has received considerable attention in the context of web graphs and social networks, there has been little work towards similar algorithms within the networking community. In this paper, we present the first theoretical study of distance sketches derived from distance oracles in a distributed network. We first present a fast distributed algorithm for computing approximate distance sketches, based on a distributed implementation of the distance oracle scheme of [Thorup-Zwick, JACM 2005]. We also show how to modify this basic construction to achieve different tradeoffs between the number of pairs for which the distance estimate is accurate, the size of the sketches, and the time and message complexity necessary to compute them. These tradeoffs can then be combined to give an efficient construction of small sketches with provable average-case as well as worst-case performance. Our algorithms use only small-sized messages and hence are suitable for bandwidth-constrained networks, and can be used in various networking applications such as topology discovery and construction, token management, load balancing, monitoring overlays, and several other problems in distributed algorithms.
For stretch k ≥ 3 we design a simple transformation that converts every k-spanner construction with at most f(n) edges into an r-fault-tolerant k-spanner construction with at most O(r^3 log n) ⋅ f(2n/r) edges. Applying this to standard greedy spanner constructions gives r-fault tolerant k-spanners with Õ(r^2 n^(1+(2/(k+1)))) edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [STOC 2009] depends similarly on n but exponentially on r (approximately like k^r).
For the case k=2 and unit-length edges, an O(r log n)-approximation algorithm is known from recent work of Dinitz and Krauthgamer [STOC 2011], where several spanner results are obtained using a common approach of rounding a natural flow-based linear programming relaxation. Here we use a different (stronger) LP relaxation and improve the approximation ratio to O(log n), which is, notably, independent of the number of faults r. We further strengthen this bound in terms of the maximum degree by using the Lovasz Local Lemma.
Finally, we show that most of our constructions are inherently local by designing equivalent distributed algorithms in the LOCAL model of distributed computation.
A virtue of all our algorithms is that they are relatively simple. Technically, we introduce a new yet natural flow-based relaxation, and show how to approximately solve it even when its size is not polynomial. The main challenge is to design a rounding scheme that "coordinates" the choices of flow-paths between the many demand pairs while using few edges overall. We achieve this, roughly speaking, by randomization at the level of vertices.
We believe that analyzing this problem is important both in its own right and also because it arises as a subproblem in many other areas of wireless networking. We study both the complexity of the problem and also present some game theoretic results regarding capacity that is achieved by completely distributed algorithms. We also feel that this problem is intriguing since it involves both continuous aspects (i.e. choosing the transmission powers) as well as discrete aspects (i.e. which connections should be supported). Our results are:
—We show that maximizing the number of supported connections is NP-hard, even when there is no background noise. This is in contrast to the problem of determining whether or not a given set of connections is feasible since that problem can be solved via linear programming.
—We present a number of approximation algorithms for the problem. All of these approximation algorithms run in polynomial time and have an approximation ratio that is independent of the number of connections.
—We examine a completely distributed algorithm and analyze it as a game in which a connection receives a positive payoff if it is successful and a negative payoff if it is unsuccessful while transmitting with nonzero power. We show that in this game there is not necessarily a pure Nash equilibrium but if such an equilibrium does exist the corresponding price of anarchy is independent of the number of connections. We also show that a mixed Nash equilibrium corresponds to a probabilistic transmission strategy and in this case such an equilibrium always exists and has a price of anarchy that is independent of the number of connections.
—In the discounted secretary problem, there is a time-dependent "discount" factor d(t), and the benefit derived from selecting an element/secretary e at time t is d(t)⋅ v(e). For this problem with arbitrary (not necessarily decreasing) functions d(t), we show a constant-competitive algorithm when the expected optimum is known in advance. With no prior knowledge, we exhibit a lower bound of Ω(log n/log log n), and give a nearly-matching O(log n)-competitive algorithm.
—In the weighted secretary problem, up to K secretaries can be selected; when a secretary is selected (s)he must be irrevocably assigned to one of K positions, with position k having weight w(k), and assigning object/secretary e to position k has benefit w(k) ⋅ v(e). The goal is to select secretaries and assign them to positions to maximize the total benefit. We give constant-competitive algorithms for this problem.
Most of these results can also be extended to the matroid secretary case (Babaioff et al.) for a large family of matroids with a constant-factor loss, and an O(log rank) loss for general matroids. These results are based on a reduction from various matroids to partition matroids which present a unified approach to many of the upper bounds of Babaioff et al. These problems have connections to online mechanism design (see, e.g., Hajiaghayi et al.). All our algorithms are monotone, and hence lead to truthful mechanisms for the corresponding online auction problems.
We also show how to obtain sparse and low-weight spanners with slack from existing constructions of conventional spanners, and these techniques allow us to also obtain the best known results for distance oracles and distance labelings with slack. This paper complements similar results obtained in recent research on slack embeddings into normed metric spaces.
Just for fun: