Combinatorial Optimization (hybrid meeting)

Combinatorial Optimization deals with optimization problems defined on combinatorial structures such as graphs and networks. Motivated by diverse practical problem setups, the topic has developed into a rich mathematical discipline with many connections to other fields of Mathematics (such as, e.g., Combinatorics, Convex Optimization and Geometry, and Real Algebraic Geometry). It also has strong ties to Theoretical Computer Science and Operations Research. A series of Oberwolfach Workshops have been crucial for establishing and developing the field. The workshop we report about was a particularly exciting event - due to the depth of results that were presented, the spectrum of developments that became apparent from the talks, the breadth of the connections to other mathematical fields that were explored, and last but not least because for many of the particiants it was the first opportunity to exchange ideas and to collaborate during an on-site workshop since almost two years

Combinatorial Optimization

Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry

Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices

We address the computation of the degrees of minors of a noncommutative symbolic matrix of form $$A[c] := \sum_{k=1}^m A_k t^{c_k} x_k,$$ where $A_k$ are matrices over a field $\mathbb{K}$, $x_i$ are noncommutative variables, $c_k$ are integer weights, and $t$ is a commuting variable specifying the degree. This problem extends noncommutative Edmonds' problem (Ivanyos et al. 2017), and can formulate various combinatorial optimization problems. Extending the study by Hirai 2018, and Hirai, Ikeda 2022, we provide novel duality theorems and polyhedral characterization for the maximum degrees of minors of $A[c]$ of all sizes, and develop a strongly polynomial-time algorithm for computing them. This algorithm is viewed as a unified algebraization of the classical Hungarian method for bipartite matching and the weight-splitting algorithm for linear matroid intersection. As applications, we provide polynomial-time algorithms for weighted fractional linear matroid matching and linear optimization over rank-2 Brascamp-Lieb polytopes

Round and Bipartize for Vertex Cover Approximation

The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair (?, S), consisting of a graph with an odd cycle transversal. If S is a stable set, we prove a tight approximation ratio of 1 + 1/?, where 2? -1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph ?? : = ?/S and satisfies ? ? [2,?], with ? = ? corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1+1/?) (1 - ?) + 2 ?, where ? ? [0,1] is a natural parameter measuring the quality of the set S. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph ??, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ? and ?, which are ? = 2 and ? = 1 - 4/n, recover the integrality gap of 2 - 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph

A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs

The theory of n-fold integer programming has been recently emerging as an important tool in parameterized complexity. The input to an n-fold integer program (IP) consists of parameter A, dimension n, and numerical data of binary encoding length L. It was known for some time that such programs can be solved in polynomial time using O(n^{g(A)}L) arithmetic operations where g is an exponential function of the parameter. In 2013 it was shown that it can be solved in fixed-parameter tractable time using O(f(A)n^3L) arithmetic operations for a single-exponential function f. This, and a faster algorithm for a special case of combinatorial n-fold IP, have led to several very recent breakthroughs in the parameterized complexity of scheduling, stringology, and computational social choice. In 2015 it was shown that it can be solved in strongly polynomial time using O(n^{g(A)}) arithmetic operations. Here we establish a result which subsumes all three of the above results by showing that n-fold IP can be solved in strongly polynomial fixed-parameter tractable time using O(f(A)n^6 log n) arithmetic operations. In fact, our results are much more general, briefly outlined as follows. - There is a strongly polynomial algorithm for integer linear programming (ILP) whenever a so-called Graver-best oracle is realizable for it. - Graver-best oracles for the large classes of multi-stage stochastic and tree-fold ILPs can be realized in fixed-parameter tractable time. Together with the previous oracle algorithm, this newly shows two large classes of ILP to be strongly polynomial; in contrast, only few classes of ILP were previously known to be strongly polynomial. - We show that ILP is fixed-parameter tractable parameterized by the largest coefficient |A |_infty and the primal or dual treedepth of A, and that this parameterization cannot be relaxed, signifying substantial progress in understanding the parameterized complexity of ILP

Integer programs with bounded subdeterminants and two nonzeros per row

We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our approach is the first polynomial-time algorithm for the weighted stable set problem on graphs that do not contain more than $k$ vertex-disjoint odd cycles, where $k$ is any constant. Previously, polynomial-time algorithms were only known for $k=0$ (bipartite graphs) and for $k=1$. We observe that integer linear programs defined by coefficient matrices with bounded subdeterminants and two nonzeros per column can be also solved in strongly polynomial-time, using a reduction to $b$-matching

Round and bipartize for vertex cover approximation

The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair (G, S), consisting of a graph with an odd cycle transversal. If S is a stable set, we prove a tight approximation ratio of 1 + 1/ρ, where 2ρ − 1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph G̃:= G/S and satisfies ρ ∈ [2, ∞], with ρ = ∞ corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1 + 1/ρ) (1 − α) + 2α, where α ∈ [0, 1] is a natural parameter measuring the quality of the set S. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph G̃, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ρ and α, which are ρ = 2 and α = 1 − 4/n, recover the integrality gap of 2 − 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph