The traveling salesman problem on cubic and subcubic graphs

We study the traveling salesman problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3-conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal value of a TSP instance and that of its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on TeX vertices a tour of length TeX exists, which also implies the 4/3-conjecture, as an upper bound, for this class of graph-TSP. Recently, Mömke and Svensson presented an algorithm that gives a 1.461-approximation for graph-TSP on general graphs and as a side result a 4/3-approximation algorithm for this problem on subcubic graphs, also settling the 4/3-conjecture for this class of graph-TSP. The algorithm by Mömke and Svensson is initially randomized but the authors remark that derandomization is trivial. We will present a different way to derandomize their algorithm which leads to a faster running time. All of the latter also works for multigraphs

The Orienteering Problem under Uncertainty Stochastic Programming and Robust Optimization compared

The Orienteering Problem (OP) is a generalization of the well-known traveling salesman problem and has many interesting applications in logistics, tourism and defense. To reflect real-life situations, we focus on an uncertain variant of the OP. Two main approaches that deal with optimization under uncertainty are stochastic programming and robust optimization. We will explore the potentialities and bottlenecks of these two approaches applied to the uncertain OP. We will compare the known robust approach for the uncertain OP (the robust orienteering problem) to the new stochastic programming counterpart (the two-stage orienteering problem). The application of both approaches will be explored in terms of their suitability in practice

Some Insects of the Hudsonian Zone in New Mexico— V.

We consider scheduling problems over scenarios where the goal is to find a single assignment of the jobs to the machines which performs well over all scenarios in an explicitly given set. Each scenario is a subset of jobs that must be executed in that scenario. The two objectives that we consider are minimizing the maximum makespan over all scenarios and minimizing the sum of the makespans of all scenarios. For both versions, we give several approximation algorithms and lower bounds on their approximability. We also consider some (easier) special cases. Combinatorial optimization problems under scenarios in general, and scheduling problems under scenarios in particular, have seen only limited research attention so far. With this paper, we make a step in this interesting research direction