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

Connectivity for Matroids and Graphs.

This dissertation studies connectivity for matroids and graphs. The main results generalize Tutte\u27s Wheels and Whirls Theorem and have numerous applications. In Chapter 2, we prove two structural theorems for 3-connected matroids. An element e of a 3-connected matroid M is essential if neither the deletion M$\\$e nor the contraction M/e is 3-connected. Tutte\u27s Wheels and Whirls Theorem proves that the only 3-connected matroids in which every element is essential are the wheels and whirls. If M is not a wheel or a whirl, we prove that the essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan containing both. In particular, M must have at least two non-essential elements. In the second structural theorem, we show that if M has a fan with 2k or 2k + 1 elements for some $k \geq \ 2$, then M can be obtained by sticking together a (k + 1)-spoked wheel and a certain 3-connected minor of M. In Chapters 3 and 4, we characterize all 3-connected matroids whose set of non-essential elements has rank two. In particular, we completely determine all 3-connected matroids with exactly two non-essential elements. In Chapter 5, we derive some consequences of these results for the 3-connected binary matroids and graphs. We prove that there are exactly six classes of 3-connected binary matroids whose set of non-essential elements has rank two and we prove that there are exactly two classes of graphs, multi-dimensional wheels and twisted wheels, with exactly two non-essential edges. In Chapter 6, we use our first structural theorem to investigate the set of elements e in a 3-connected matroid M such that the simplification of M/e is 3-connected. We get best-possible lower bounds on the number of such elements thereby improving a result which was derived by Cunningham and Seymour independently. We also give some generalizations of the Wheels and Whirls Theorem and the Wheels Theorem

Local properties of graphs with large chromatic number

This thesis deals with problems concerning the local properties of graphs with large chromatic number in hereditary classes of graphs. We construct intersection graphs of axis-aligned boxes and of lines in $\mathbb{R}^3$ that have arbitrarily large girth and chromatic number. We also prove that the maximum chromatic number of a circle graph with clique number at most $\omega$ is equal to $\Theta(\omega \log \omega)$. Lastly, extending the $\chi$-boundedness of circle graphs, we prove a conjecture of Geelen that every proper vertex-minor-closed class of graphs is $\chi$-bounded

Signed Lozenge Tilings

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a new method that we call resolution of a puncture, we show that the two signs are in fact equivalent. As a consequence, we obtain the equality of determinants, up to sign, that enumerate signed perfect matchings and signed families of lattice paths of a triangular region, respectively. We also describe triangular regions, for which the signed enumerations agree with the unsigned enumerations