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We give the first constant-factor approximation algorithm for quasi-bipartite instances of DIRECTED STEINER TREE on graphs that exclude fixed minors. In particular, for K r -minorfree graphs our approximation guarantee is O(r • log r) and, further, for planar graphs our approximation guarantee is 20.Our algorithm uses the primal-dual scheme. We employ a more involved method of determining when to buy an edge while raising dual variables since, as we show, the natural primal-dual scheme fails to raise enough dual value to pay for the purchased solution. As a consequence, we also demonstrate integrality gap upper bounds on the standard cut-based linear programming relaxation for the DIRECTED STEINER TREE instances we consider.

In this paper, we present improved approximation algorithms for the (unsplittable) Capacitated Vehicle Routing Problem (CVRP) in general metrics. In CVRP, introduced by Dantzig and Ramser (1959), we are given a set of points (clients) V together with a depot r in a metric space, with each v ∈ V having a demand d v > 0, and a vehicle of bounded capacity Q. The goal is to find a minimum cost collection of tours for the vehicle, each starting and ending at the depot, such that each client is visited at least once and the total demands of the clients in each tour is at most Q. In the unsplittable variant we study, the demand of a node must be served entirely by one tour. We present two approximation algorithms for unsplittable CVRP: a combinatorial (α + 1.75)-approximation, where α is the approximation factor for the Traveling Salesman Problem, and an approximation algorithm based on LP rounding with approximation guarantee α + ln(2) + δ ≈ 3.194 + δ in n O(1/δ) time. Both approximations can further be improved by a small amount when combined with recent work by Blauth, Traub, and Vygen (2021), who obtained an (α + 2 • (1 − ε))-approximation for unsplittable CVRP for some constant ε depending on α (ε > 1/3000 for α = 1.5).

k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.93, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm yields a 2^{O((klog k)/{epsilon})}dn-time (1+epsilon)-approximation for Euclidean k-center, where d is the dimension. In this work, we give a faster algorithm for small dimensions: roughly speaking an O^*(2^{O((1/epsilon)^{O(d)} k^{1-1/d} log k)})-time (1+epsilon)-approximation. In particular, the running time is roughly O^*(2^{O((1/epsilon)^{O(1)}sqrt{k}log k)}) in the plane. We complement our algorithmic result with a matching hardness lower bound. We also consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a 2^{O(klog k)}n^2 time 3-approximation for NUkC, and a 2^{O((klog k)/epsilon)}dn time (1+\epsilon)-approximation for Euclidean NUkC. The latter time bound matches the bound for k-center.