Approximability of Connected Factors

Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by Tutte's reduction to the matching problem. By the same reduction, it is easy to find a minimal or maximal d-factor of a graph. However, if we require that the d-factor is connected, these problems become NP-hard - finding a minimal connected 2-factor is just the traveling salesman problem (TSP). Given a complete graph with edge weights that satisfy the triangle inequality, we consider the problem of finding a minimal connected $d$-factor. We give a 3-approximation for all $d$ and improve this to an (r+1)-approximation for even d, where r is the approximation ratio of the TSP. This yields a 2.5-approximation for even d. The same algorithm yields an (r+1)-approximation for the directed version of the problem, where r is the approximation ratio of the asymmetric TSP. We also show that none of these minimization problems can be approximated better than the corresponding TSP. Finally, for the decision problem of deciding whether a given graph contains a connected d-factor, we extend known hardness results.Comment: To appear in the proceedings of WAOA 201

Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming

Here we study the NP-complete $K$-SAT problem. Although the worst-case complexity of NP-complete problems is conjectured to be exponential, there exist parametrized random ensembles of problems where solutions can typically be found in polynomial time for suitable ranges of the parameter. In fact, random $K$-SAT, with $\alpha=M/N$ as control parameter, can be solved quickly for small enough values of $\alpha$. It shows a phase transition between a satisfiable phase and an unsatisfiable phase. For branch and bound algorithms, which operate in the space of feasible Boolean configurations, the empirically hardest problems are located only close to this phase transition. Here we study $K$-SAT ($K=3,4$) and the related optimization problem MAX-SAT by a linear programming approach, which is widely used for practical problems and allows for polynomial run time. In contrast to branch and bound it operates outside the space of feasible configurations. On the other hand, finding a solution within polynomial time is not guaranteed. We investigated several variants like including artificial objective functions, so called cutting-plane approaches, and a mapping to the NP-complete vertex-cover problem. We observed several easy-hard transitions, from where the problems are typically solvable (in polynomial time) using the given algorithms, respectively, to where they are not solvable in polynomial time. For the related vertex-cover problem on random graphs these easy-hard transitions can be identified with structural properties of the graphs, like percolation transitions. For the present random $K$-SAT problem we have investigated numerous structural properties also exhibiting clear transitions, but they appear not be correlated to the here observed easy-hard transitions. This renders the behaviour of random $K$-SAT more complex than, e.g., the vertex-cover problem.Comment: 11 pages, 5 figure

The Maximum Traveling Salesman Problem with Submodular Rewards

In this paper, we look at the problem of finding the tour of maximum reward on an undirected graph where the reward is a submodular function, that has a curvature of $\kappa$, of the edges in the tour. This problem is known to be NP-hard. We analyze two simple algorithms for finding an approximate solution. Both algorithms require $O(|V|^3)$ oracle calls to the submodular function. The approximation factors are shown to be $\frac{1}{2+\kappa}$ and $\max\set{\frac{2}{3(2+\kappa)},2/3(1-\kappa)}$, respectively; so the second method has better bounds for low values of $\kappa$. We also look at how these algorithms perform for a directed graph and investigate a method to consider edge costs in addition to rewards. The problem has direct applications in monitoring an environment using autonomous mobile sensors where the sensing reward depends on the path taken. We provide simulation results to empirically evaluate the performance of the algorithms.Comment: Extended version of ACC 2013 submission (including p-system greedy bound with curvature

On Semidefinite Programming Relaxations of the Travelling Salesman Problem (Replaced by DP 2008-96)

AMS classification: 90C22, 20Cxx, 70-08traveling salesman problem;semidefinite programming;quadratic as- signment problem