Integrality Gap of the Hypergraphic Relaxation of Steiner Trees: a short proof of a 1.55 upper bound

Recently Byrka, Grandoni, Rothvoss and Sanita (at STOC 2010) gave a 1.39-approximation for the Steiner tree problem, using a hypergraph-based linear programming relaxation. They also upper-bounded its integrality gap by 1.55. We describe a shorter proof of the same integrality gap bound, by applying some of their techniques to a randomized loss-contracting algorithm

DCCast: Efficient Point to Multipoint Transfers Across Datacenters

Using multiple datacenters allows for higher availability, load balancing and reduced latency to customers of cloud services. To distribute multiple copies of data, cloud providers depend on inter-datacenter WANs that ought to be used efficiently considering their limited capacity and the ever-increasing data demands. In this paper, we focus on applications that transfer objects from one datacenter to several datacenters over dedicated inter-datacenter networks. We present DCCast, a centralized Point to Multi-Point (P2MP) algorithm that uses forwarding trees to efficiently deliver an object from a source datacenter to required destination datacenters. With low computational overhead, DCCast selects forwarding trees that minimize bandwidth usage and balance load across all links. With simulation experiments on Google's GScale network, we show that DCCast can reduce total bandwidth usage and tail Transfer Completion Times (TCT) by up to $50\%$ compared to delivering the same objects via independent point-to-point (P2P) transfers.Comment: 9th USENIX Workshop on Hot Topics in Cloud Computing, https://www.usenix.org/conference/hotcloud17/program/presentation/noormohammadpou

Travelling on Graphs with Small Highway Dimension

We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP naturally occur for various applications in logistics. It was previously shown [Feldmann et al. ICALP 2015] that these problems admit a quasi-polynomial time approximation scheme (QPTAS) on graphs of constant highway dimension. We demonstrate that a significant improvement is possible in the special case when the highway dimension is 1, for which we present a fully-polynomial time approximation scheme (FPTAS). We also prove that STP is weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015]

Approximating Activation Edge-Cover and Facility Location Problems

What approximation ratio can we achieve for the Facility Location problem if whenever a client u connects to a facility v, the opening cost of v is at most theta times the service cost of u? We show that this and many other problems are a particular case of the Activation Edge-Cover problem. Here we are given a multigraph G=(V,E), a set R subseteq V of terminals, and thresholds {t^e_u,t^e_v} for each uv-edge e in E. The goal is to find an assignment a={a_v:v in V} to the nodes minimizing sum_{v in V} a_v, such that the edge set E_a={e=uv: a_u >= t^e_u, a_v >= t^e_v} activated by a covers R. We obtain ratio 1+max_{x>=1}(ln x)/(1+x/theta)~= ln theta - ln ln theta for the problem, where theta is a problem parameter. This result is based on a simple generic algorithm for the problem of minimizing a sum of a decreasing and a sub-additive set functions, which is of independent interest. As an application, we get the same ratio for the above variant of {Facility Location}. If for each facility all service costs are identical then we show a better ratio 1+max_{k in N}(H_k-1)/(1+k/theta), where H_k=sum_{i=1}^k 1/i. For the Min-Power Edge-Cover problem we improve the ratio 1.406 of [Calinescu et al, 2019] (achieved by iterative randomized rounding) to 1.2785. For unit thresholds we improve the ratio 73/60~=1.217 of [Calinescu et al, 2019] to 1555/1347~=1.155