Eight-Fifth Approximation for TSP Paths

We prove the approximation ratio 8/5 for the metric $\{s,t\}$-path-TSP problem, and more generally for shortest connected $T$-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides $\{s,t\}$-tour out of those constructed from a family \Fscr_{>0} of trees having a convex combination dominated by an optimal solution $x^*$ of the fractional relaxation. They give the approximation guarantee $\frac{\sqrt{5}+1}{2}$ for such an $\{s,t\}$-tour, which is the first improvement after the 5/3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13/8-approximation of shortest connected $T$-joins, for $|T|\ge 4$. The ratio 8/5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing $x^*/2$ in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in \Fscr_{>0} into an $\{s,t\}$-path (or more generally, into a $T$-join) and its complement, which induces a decomposition of $x^*$. This decomposition can be refined and then efficiently used to complete $x^*/2$ without using linear programming or particular properties of $T$, but by adding to each cut deficient for $x^*/2$ an individually tailored explicitly given vector, inherent in $x^*$. A simple example shows that the Best of Many Christofides algorithm may not find a shorter $\{s,t\}$-tour than 3/2 times the incidentally common optima of the problem and of its fractional relaxation.Comment: 15 pages, corrected typos in citations, minor change

The Salesman's Improved Tours for Fundamental Classes

Finding the exact integrality gap $\alpha$ for the LP relaxation of the metric Travelling Salesman Problem (TSP) has been an open problem for over thirty years, with little progress made. It is known that $4/3 \leq \alpha \leq 3/2$, and a famous conjecture states $\alpha = 4/3$. For this problem, essentially two "fundamental" classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most $\rho$ for all instances of metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simplicity of such classes, no apparent effort has been deployed for improving the integrality gap bounds for them. In this paper we take a natural first step in this endeavour, and consider the $1/2$-integer points of one such class. We successfully improve the upper bound for the integrality gap from $3/2$ to $10/7$ for a superclass of these points, as well as prove a lower bound of $4/3$ for the superclass. Our methods involve innovative applications of tools from combinatorial optimization which have the potential to be more broadly applied

Layers and Matroids for the Traveling Salesman's Paths

Gottschalk and Vygen proved that every solution of the subtour elimination linear program for traveling salesman paths is a convex combination of more and more restrictive "generalized Gao-trees". We give a short proof of this fact, as a layered convex combination of bases of a sequence of increasingly restrictive matroids. A strongly polynomial, combinatorial algorithm follows for finding this convex combination, which is a new tool offering polyhedral insight, already instrumental in recent results for the $s-t$ path TSP

Improving on Best-of-Many-Christofides for TT-tours

The $T$-tour problem is a natural generalization of TSP and Path TSP. Given a graph $G=(V,E)$, edge cost $c: E \to \mathbb{R}_{\ge 0}$, and an even cardinality set $T\subseteq V$, we want to compute a minimum-cost $T$-join connecting all vertices of $G$ (and possibly containing parallel edges). In this paper we give an $\frac{11}{7}$-approximation for the $T$-tour problem and show that the integrality ratio of the standard LP relaxation is at most $\frac{11}{7}$. Despite much progress for the special case Path TSP, for general $T$-tours this is the first improvement on Seb\H{o}'s analysis of the Best-of-Many-Christofides algorithm (Seb\H{o} [2013])

Cosmological Constraints from Galaxy Clustering and the Mass-to-Number Ratio of Galaxy Clusters: Marginalizing over the Physics of Galaxy Formation

Many approaches to obtaining cosmological constraints rely on the connection between galaxies and dark matter. However, the distribution of galaxies is dependent on their formation and evolution as well as the cosmological model, and galaxy formation is still not a well-constrained process. Thus, methods that probe cosmology using galaxies as a tracer for dark matter must be able to accurately estimate the cosmological parameters without knowing the details of galaxy formation a priori. We apply this reasoning to the method of obtaining $\Omega_m$ and $\sigma_8$ from galaxy clustering combined with the mass-to-number ratio of galaxy clusters. To test the sensitivity of this method to variations due to galaxy formation, we consider several different models applied to the same cosmological dark matter simulation. The cosmological parameters are then estimated using the observables in each model, marginalizing over the parameters of the Halo Occupation Distribution (HOD). We find that for models where the galaxies can be well represented by a parameterized HOD, this method can successfully extract the desired cosmological parameters for a wide range of galaxy formation prescriptions.Comment: 10 pages, 7 figures, Submitted to Ap