Submodular Minimization Under Congruency Constraints

Submodular function minimization (SFM) is a fundamental and efficiently solvable problem class in combinatorial optimization with a multitude of applications in various fields. Surprisingly, there is only very little known about constraint types under which SFM remains efficiently solvable. The arguably most relevant non-trivial constraint class for which polynomial SFM algorithms are known are parity constraints, i.e., optimizing only over sets of odd (or even) cardinality. Parity constraints capture classical combinatorial optimization problems like the odd-cut problem, and they are a key tool in a recent technique to efficiently solve integer programs with a constraint matrix whose subdeterminants are bounded by two in absolute value. We show that efficient SFM is possible even for a significantly larger class than parity constraints, by introducing a new approach that combines techniques from Combinatorial Optimization, Combinatorics, and Number Theory. In particular, we can show that efficient SFM is possible over all sets (of any given lattice) of cardinality r mod m, as long as m is a constant prime power. This covers generalizations of the odd-cut problem with open complexity status, and with relevance in the context of integer programming with higher subdeterminants. To obtain our results, we establish a connection between the correctness of a natural algorithm, and the inexistence of set systems with specific combinatorial properties. We introduce a general technique to disprove the existence of such set systems, which allows for obtaining extensions of our results beyond the above-mentioned setting. These extensions settle two open questions raised by Geelen and Kapadia [Combinatorica, 2017] in the context of computing the girth and cogirth of certain types of binary matroids

Covering Vectors by Spaces in Perturbed Graphic Matroids and Their Duals

Perturbed graphic matroids are binary matroids that can be obtained from a graphic matroid by adding a noise of small rank. More precisely, an r-rank perturbed graphic matroid M is a binary matroid that can be represented in the form I +P, where I is the incidence matrix of some graph and P is a binary matrix of rank at most r. Such matroids naturally appear in a number of theoretical and applied settings. The main motivation behind our work is an attempt to understand which parameterized algorithms for various problems on graphs could be lifted to perturbed graphic matroids. We study the parameterized complexity of a natural generalization (for matroids) of the following fundamental problems on graphs: Steiner Tree and Multiway Cut. In this generalization, called the Space Cover problem, we are given a binary matroid M with a ground set E, a set of terminals T subseteq E, and a non-negative integer k. The task is to decide whether T can be spanned by a subset of E T of size at most k. We prove that on graphic matroid perturbations, for every fixed r, Space Cover is fixed-parameter tractable parameterized by k. On the other hand, the problem becomes W[1]-hard when parameterized by r+k+|T| and it is NP-complete for r <= 2 and |T|<= 2. On cographic matroids, that are the duals of graphic matroids, Space Cover generalizes another fundamental and well-studied problem, namely Multiway Cut. We show that on the duals of perturbed graphic matroids the Space Cover problem is fixed-parameter tractable parameterized by r+k

On the Complexity of Recovering Incidence Matrices

The incidence matrix of a graph is a fundamental object naturally appearing in many applications, involving graphs such as social networks, communication networks, or transportation networks. Often, the data collected about the incidence relations can have some slight noise. In this paper, we initiate the study of the computational complexity of recovering incidence matrices of graphs from a binary matrix: given a binary matrix M which can be written as the superposition of two binary matrices L and S, where S is the incidence matrix of a graph from a specified graph class, and L is a matrix (i) of small rank or, (ii) of small (Hamming) weight. Further, identify all those graphs whose incidence matrices form part of such a superposition. Here, L represents the noise in the input matrix M. Another motivation for this problem comes from the Matroid Minors project of Geelen, Gerards and Whittle, where perturbed graphic and co-graphic matroids play a prominent role. There, it is expected that a perturbed binary matroid (or its dual) is presented as L+S where L is a low rank matrix and S is the incidence matrix of a graph. Here, we address the complexity of constructing such a decomposition. When L is of small rank, we show that the problem is NP-complete, but it can be decided in time (mn)^O(r), where m,n are dimensions of M and r is an upper-bound on the rank of L. When L is of small weight, then the problem is solvable in polynomial time (mn)^O(1). Furthermore, in many applications it is desirable to have the list of all possible solutions for further analysis. We show that our algorithms naturally extend to enumeration algorithms for the above two problems with delay (mn)^O(r) and (mn)^O(1), respectively, between consecutive outputs

Blocking optimal kk-arborescences

Given a digraph $D=(V,A)$ and a positive integer $k$, an arc set $F\subseteq A$ is called a \textbf{$k$-arborescence} if it is the disjoint union of $k$ spanning arborescences. The problem of finding a minimum cost $k$-arborescence is known to be polynomial-time solvable using matroid intersection. In this paper we study the following problem: find a minimum cardinality subset of arcs that contains at least one arc from every minimum cost $k$-arborescence. For $k=1$, the problem was solved in [A. Bern\'ath, G. Pap , Blocking optimal arborescences, IPCO 2013]. In this paper we give an algorithm for general $k$ that has polynomial running time if $k$ is fixed

Odd Paths, Cycles and TT-joins: Connections and Algorithms

Minimizing the weight of an edge set satisfying parity constraints is a challenging branch of combinatorial optimization as witnessed by the binary hypergraph chapter of Alexander Schrijver's book ``Combinatorial Optimization'' (Chapter 80). This area contains relevant graph theory problems including open cases of the Max Cut problem, or some multiflow problems. We clarify the interconnections of some problems and establish three levels of difficulties. On the one hand, we prove that the Shortest Odd Path problem in an undirected graph without cycles of negative total weight and several related problems are NP-hard, settling a long-standing open question asked by Lov\'asz (Open Problem 27 in Schrijver's book ``Combinatorial Optimization''. On the other hand, we provide a polynomial-time algorithm to the closely related and well-studied Minimum-weight Odd $\{s,t\}$-Join problem for non-negative weights, whose complexity, however, was not known; more generally, we solve the Minimum-weight Odd $T$-Join problem in FPT time when parameterized by $|T|$. If negative weights are also allowed, then finding a minimum-weight odd $\{s,t\}$-join is equivalent to the Minimum-weight Odd $T$-Join problem for arbitrary weights, whose complexity is only conjectured to be polynomially solvable. The analogous problems for digraphs are also considered.Comment: 24 pages, 2 figure