Konrad Borys

Generating cut conjunctions and bridge avoiding extensions in graphs

Let G=(V,E) be an undirected graph, and let B⊆V ×V be a collection of vertex pairs. We give an incremental polynomial time algorithm to enumerate all minimal edge sets X⊆E such that every vertex pair (s,t) ∈ B is disconnected in

Generating minimal k -vertex connected spanning subgraphs

(V,E \smallsetminus X)

Enumerating spanning and connected subsets in graphs and matroids

, generalizing well-known efficient algorithms for enumerating all minimal s-t cuts, for a given pair s,t ∈ V of vertices. We also present an incremental polynomial time algorithm for enumerating all minimal subsets X⊆E such that no (s,t) ∈ B is a bridge in (V,X ∪ B). These two enumeration problems are special cases of the more general cut conjunction problem in matroids: given a matroid M on ground set S=E ∪ B, enumerate all minimal subsets X⊆E such that no element b ∈ B is spanned by

Combinatorial games modeling seki in GO

E \smallsetminus X

Generating 3-vertex connected spanning subgraphs

. Unlike the above special cases, corresponding to the cycle and cocycle matroids of the graph (V,E ∪ B), the enumeration of cut conjunctions for vectorial matroids turns out to be NP-hard.

Generating all vertices of a polyhedron is hard

We show that minimal k-vertex connected spanning subgraphs of a given graph can be generated in incremental polynomial time for any fixed k.

Generating All Vertices of a Polyhedron Is Hard

We show that enumerating all minimal spanning and connected subsets of a given matroid can be solved in incremental quasi-polynomial time. In the special case of graphical matroids, we improve this complexity bound by showing that all minimal 2-vertex connected edge subsets of a given graph can be generated in incremental polynomial time.

On Short Paths Interdiction Problems: Total and Node-Wise Limited Interdiction

Abstract The game SEKI is played on an ( m × n ) -matrix A with non-negative integer entries. Two players R (for rows) and C (for columns) alternately reduce a positive entry of A by 1 or pass. If they pass successively, the game is a draw. Otherwise, the game ends when a row or column contains only zeros, in which case R or C wins, respectively. If a zero row and column appear simultaneously, then the player who made the last move is the winner. We will also study another version of the game, called D-SEKI, in which the above case is defined as a draw. An integer non-negative matrix A is a seki or d-seki if the corresponding game results in a draw, regardless of whether R or C begins. Of particular interest are the matrices in which each player loses after every option except pass. Such a matrix is called a complete seki or a complete d-seki . For example, each matrix with entries in { 0 , 1 } that has the same sum (at least 2) in each row and column is a complete d-seki, and each such matrix with entries in { 0 , 1 , 2 } is a complete seki. The game SEKI is closely related to the seki (shared life) positions in the classical game of GO.

Enumerating Spanning and Connected Subsets in Graphs and Matroids

In this paper we present an algorithm to generate all minimal 3-vertex connected spanning subgraphs of an undirected graph with n vertices and m edges in incremental polynomial time, i.e., for every K we can generate K (or all) minimal 3-vertex connected spanning subgraphs of a given graph in O(K^2log(K)m^2+K^2m^3) time, where n and m are the number of vertices and edges of the input graph, respectively. This is an improvement over what was previously available and is the same as the best known running time for generating 2-vertex connected spanning subgraphs. Our result is obtained by applying the decomposition theory of 2-vertex connected graphs to the graphs obtained from minimal 3-vertex connected graphs by removing a single edge.