Approximability of the Multiple Stack TSP

STSP seeks a pair of pickup and delivery tours in two distinct networks, where the two tours are related by LIFO contraints. We address here the problem approximability. We notably establish that asymmetric MaxSTSP and MinSTSP12 are APX, and propose a heuristic that yields to a 1/2, 3/4 and 3/2 standard approximation for respectively Max2STSP, Max2STSP12 and Min2STSP12

Approximation Algorithms for the Asymmetric Traveling Salesman Problem : Describing two recent methods

The paper provides a description of the two recent approximation algorithms for the Asymmetric Traveling Salesman Problem, giving the intuitive description of the works of Feige-Singh[1] and Asadpour et.al\ [2].\newline [1] improves the previous $O(\log n)$ approximation algorithm, by improving the constant from 0.84 to 0.66 and modifying the work of Kaplan et. al\ [3] and also shows an efficient reduction from ATSPP to ATSP. Combining both the results, they finally establish an approximation ratio of $\left(\frac{4}{3}+\epsilon \right)\log n$ for ATSPP,\ considering a small $\epsilon>0$,\ improving the work of Chekuri and Pal.[4]\newline Asadpour et.al, in their seminal work\ [2], gives an $O\left(\frac{\log n}{\log \log n}\right)$ randomized algorithm for the ATSP, by symmetrizing and modifying the solution of the Held-Karp relaxation problem and then proving an exponential family distribution for probabilistically constructing a maximum entropy spanning tree from a spanning tree polytope and then finally defining the thin-ness property and transforming a thin spanning tree into an Eulerian walk.\ The optimization methods used in\ [2] are quite elegant and the approximation ratio could further be improved, by manipulating the thin-ness of the cuts.Comment: 12 page

On improving the approximation ratio of the r-shortest common superstring problem

The Shortest Common Superstring problem (SCS) consists, for a set of strings S = {s_1,...,s_n}, in finding a minimum length string that contains all s_i, 1<= i <= n, as substrings. While a 2+11/30 approximation ratio algorithm has recently been published, the general objective is now to break the conceptual lower bound barrier of 2. This paper is a step ahead in this direction. Here we focus on a particular instance of the SCS problem, meaning the r-SCS problem, which requires all input strings to be of the same length, r. Golonev et al. proved an approximation ratio which is better than the general one for r<= 6. Here we extend their approach and improve their approximation ratio, which is now better than the general one for r<= 7, and less than or equal to 2 up to r = 6

Balanced Combinations of Solutions in Multi-Objective Optimization

For every list of integers x_1, ..., x_m there is some j such that x_1 + ... + x_j - x_{j+1} - ... - x_m \approx 0. So the list can be nearly balanced and for this we only need one alternation between addition and subtraction. But what if the x_i are k-dimensional integer vectors? Using results from topological degree theory we show that balancing is still possible, now with k alternations. This result is useful in multi-objective optimization, as it allows a polynomial-time computable balance of two alternatives with conflicting costs. The application to two multi-objective optimization problems yields the following results: - A randomized 1/2-approximation for multi-objective maximum asymmetric traveling salesman, which improves and simplifies the best known approximation for this problem. - A deterministic 1/2-approximation for multi-objective maximum weighted satisfiability

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