Related papers: Branching with a pre-specified finite list of $k$-…
A permutation is $k$-coverable if it can be partitioned into $k$ monotone subsequences. Barber conjectured that, for any given permutation, if every subsequence of length $k+2 \choose 2$ is $k$-coverable then the permutation itself is…
The main result of the paper is motivated by the following two, apparently unrelated graph optimization problems: (A) as an extension of Edmonds' disjoint branchings theorem, characterize digraphs comprising $k$ disjoint branchings $B_i$…
We prove that for each fixed $m \ge 2$, there are only finitely many disjoint covering systems with minimum modulus at least $3$ in which precisely one modulus is repeated, namely the largest modulus, and it occurs exactly $m$ times.
Spanning trees of complete bipartite graphs exhibit a rich interaction between degree sequences and graph structure. In this paper, we obtain lower bounds on the number of isomorphism classes of spanning trees in $K_{a,b}, 2 \leq a \leq b$…
Let $t\ge 1$ be a given integer. Let ${\cal F}$ be a family of subsets of $[m]=\{1,2,\ldots,m\}$. Assume that for every pair of disjoint sets $S,T\subset [m]$ with $|S|=|T|=k$, there do not exist $2t$ sets in ${\cal F}$ where $t$ subsets of…
As a generalization of matching preclusion number of a graph, we provide the (strong) integer $k$-matching preclusion number, abbreviated as $MP^{k}$ number ($SMP^{k}$ number), which is the minimum number of edges (vertices and edges) whose…
Cutting planes and branching are two of the most important algorithms for solving mixed-integer linear programs. For both algorithms, disjunctions play an important role, being used both as branching candidates and as the foundation for…
In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure and the lattice-free closure for packing sets as a function of the integrality gap. We also provide a similar lower bound on the split rank of…
Coverage problems are central in optimization and have a wide range of applications in data mining and machine learning. While several distributed algorithms have been developed for coverage problems, the existing methods suffer from…
In this paper we give a generalization of the well known split cuts of Cook, Kannan and Schrijver to cuts which are based on multi-term disjunctions. They will be called k-disjunctive cuts. The starting point is the question what kind of…
We investigate the list packing number of a graph, the least $k$ such that there are always $k$ disjoint proper list-colourings whenever we have lists all of size $k$ associated to the vertices. We are curious how the behaviour of the list…
A functional $k$-PIR code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information symbols. Any linear combination of the $s$ information symbols can be recovered by $k$ disjoint subsets…
In the context of algorithm theory, various studies have been conducted on spanning trees with desirable properties. In this paper, we consider the \textsc{Minimum Cover Spanning Tree} problem (MCST for short). Given a graph $G$ and a…
Recently, many studies have been devoted to finding diverse solutions in classical combinatorial problems, such as Vertex Cover (Baste et al., IJCAI'20), Matching (Fomin et al., ISAAC'20) and Spanning Tree (Hanaka et al., AAAI'21). We…
We consider $k$-graphs on $n$ vertices, that is, $\mathcal{F}\subset \binom{[n]}{k}$. A $k$-graph $\mathcal{F}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. In the present paper we prove that for $k\geq…
We study the problem of covering the maximum number of vertices in a graph by a collection of vertex-disjoint stars, each with a number of satellites in a given interval $[k, \ell]$, where $1 \le k < \ell$ and $\ell$ can be infinity. This…
We investigate the theoretical complexity of branch-and-bound (BB) and cutting plane (CP) algorithms for mixed-integer optimization. In particular, we study the relative efficiency of BB and CP, when both are based on the same family of…
A digraph is {\bf \( k \)-linked} if for arbitary two disjoint vertex sets \(\{s_1, \ldots, s_k\}\) and \(\{t_1, \ldots, t_k\}\), there exist vertex-disjoint directed paths \(P_1, \ldots, P_k\) {such that \(P_i\) is a directed path from…
Let p be a rational prime and let F be a number field. Then, for each i>0, there is a short exact localization sequence for K_{2i}(F). If p is odd or F is nonexceptional, we find necessary and sufficient conditions for this exact sequence…
Let $G$ be a graph with an even number of vertices. The matching preclusion number of $G$, denoted by $mp(G)$, is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching. We introduced a $0$-$1$…