Combinatorial generation: greedy approaches and symmetry (Kombinatorische Generierung: Greedy-Ansätze und Symmetrie)
Abstract: We frequently encounter different combinatorial objects in mathematics and computer science classes. Bitstrings, permutations, binary trees, combinations, spanning trees, and poset chains are just a few examples. Combinatorial objects are at the forefront of questions in algorithmic design, where we want to program a computer to perform some task on them. People design algorithms for efficiently counting objects according to some parameters, sampling objects uniformly at random, optimizing given some objective function or generating all the desired combinatorial objects exactly once. This thesis focuses on combinatorial generation, the last and most foundational task above. Since the advent of the modern computer, the area has gained much traction in both practical and academic computer science. From the practical side, they are an essential building block of many algorithms. We can see its applied relevance in the multiple efficient generation modules available in the standard libraries of many popular programming languages. On the academic side, researchers have designed generation algorithms for many objects with various desirable properties. The two most prominent examples are the binary reflected Gray code algorithm for generating bitstrings and the Steinhaus-Johnson-Trotter algorithm for permutations. Both these algorithms have surprising and desirable properties: - They have simple greedy interpretations. - They present a high degree of symmetry. - One can efficiently implement them. - They have compact encodings. Our main results are the development of combinatorial generation algorithms and constructions for more structured combinatorial objects with the exciting properties of greedy interpretations and symmetry. As a byproduct, we obtain efficient algorithms for many combinatorial objects and compact ways of encoding their combinatorial structure. Our first set of results is the design of greedy algorithms for elimination trees. The elimination trees of a graph describe the ways of searching in the given graph. They repeatedly appear in computer science and mathematics and generalize fascinating combinatorial objects like permutations, binary trees, and partial permutations. We apply the recent Hartung-Hoang-Mütze-Williams combinatorial generation framework to elimination trees. We prove that all elimination trees for a chordal graph can be generated by tree rotations using a simple greedy algorithm. Consequently, we obtain efficient and simple generation algorithms for binary trees, partial permutations, and other objects. Our second set of results is a new greedy framework for objects that can be binary encoded. Our method relies on a simple and versatile algorithm for computing a Hamilton path on the skeleton of any 0/1-polytope. Surprisingly, we can efficiently implement this algorithm whenever a variant of the classical linear optimization problem can be efficiently implemented. Consequently, we establish an exciting connection between combinatorial generation and optimization. As concrete results of our framework, we obtain efficient algorithms for generating bases and independent sets in a matroid, spanning trees, forests, matchings, and maximum matchings in a general graph, among other objects. As another consequence of our framework, we present the first improvement in the vertex enumeration problem on 0/1-polytopes in 25 years. In this thesis's second part, we explore the notion of symmetry within listings of combinatorial objects. Inspired by a conjecture of Knuth about the middle levels of the hypercube, we introduce the notion of Hamilton compression, a way of quantifying the symmetry in a listing of combinatorial objects. We investigate the Hamilton compression in five settings: bitstrings, combinations, permutations, middle levels, and abelian groups. In several cases, we determine their Hamilton compression precisely, and in other cases, we provide close lower and upper bounds. The listings we construct have a much higher compression than several classical results from the literature and provide a solution to Knuth's conjecture on the middle levels.
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