Efficient Methods for Congruency-Constrained Optimization

In this thesis, we study discrete combinatorial optimization problems with congruency constraints and present new techniques for dealing with such constraint types. Strong motivation for studying congruency constraints comes from a long-standing open question in Integer Programming whether integer programs with constraint matrices with bounded subdeterminants are efficiently solvable. An important special case thereof are congruency-constrained integer programs min { : Tx ≤ b, ≡ r (mod m), x ∈ Z^n } with a totally unimodular constraint matrix T. Such problems have been shown to be polynomial-time solvable for m = 2, which led to an efficient algorithm for integer programs with bimodular constraint matrices, i.e., full-rank matrices whose n by n subdeterminants are bounded by two in absolute value. Whereas these advances heavily relied on existing results on well-known combinatorial minimum cut and circulation problems with parity constraints, new approaches are needed beyond the bimodular case, i.e., m > 2. We make first progress in this direction through several new techniques. In particular, we show how to efficiently decide feasibility of congruency-constrained integer programs with a totally unimodular constraint matrix for m = 3. Furthermore, for general m, our techniques also allow for identifying flat directions of infeasible problems, and deducing bounds on the proximity between solutions of the problem and its relaxation. In the second part of this thesis, we study a generalization of the aforementioned parity-constrained minimum cut problems (and therefore also a generalization of other well-known variants of minimum cut problems such as global minimum cuts and minimum s-t cuts), namely congruency-constrained minimum cuts, where we consider cuts whose number of vertices is congruent to r modulo m, for some integers r and m. We develop a new contraction technique inspired by Karger's celebrated contraction algorithm for minimum cuts, which, together with further insights, leads to a polynomial time randomized approximation scheme for congruency-constrained minimum cuts for any constant modulus m. Instead of contracting edges of the original graph, we use splitting-off techniques to create an auxiliary graph on a smaller vertex set, which is used for performing random edge contractions. This way, a well-structured distribution of candidate pairs of vertices to be contracted is obtained, where the involved pairs are generally not connected by an edge. As a byproduct, our technique reveals new structural insights into near-minimum odd cuts, and, more generally, near-minimum congruency-constrained cuts. Finally, we present results on finding a spanning tree subject to constraints on the edges in a family of cuts forming a laminar family of small width. Especially in the context of the Traveling Salesman Problem (TSP), new techniques for finding spanning trees with well-defined properties, and in particular, trees satisfying parity constraints on a chain of cuts, have been crucial in pushing a the analysis of a certain class of approximation algorithms. Our main contribution is a new dynamic programming approach where the value of a table entry does not only depend on the values of previous table entries, as it is usually the case, but also on a specific representative solution saved together with each table entry. This allows for handling a broad range of constraint types, including a relaxation of congruency constraints, or upper and lower bounds on the number of edges in each cut. Concretely, we present a quasi-polynomial time algorithm for the Minimum Chain-Constrained Spanning Tree Problem with an essentially best possible guarantee. Furthermore, we show how parity constraints as used in the context of (path) TSP and a generalization thereof can be handled, and discuss implications in this context