34 research outputs found

### Constant Amortized Time Enumeration of Eulerian trails

In this paper, we consider enumeration problems for edge-distinct and
vertex-distinct Eulerian trails. Here, two Eulerian trails are
\emph{edge-distinct} if the edge sequences are not identical, and they are
\emph{vertex-distinct} if the vertex sequences are not identical. As the main
result, we propose optimal enumeration algorithms for both problems, that is,
these algorithm runs in $\mathcal{O}(N)$ total time, where $N$ is the number of
solutions. Our algorithms are based on the reverse search technique introduced
by [Avis and Fukuda, DAM 1996], and the push out amortization technique
introduced by [Uno, WADS 2015]

### Linear-Delay Enumeration for Minimal Steiner Problems

Kimelfeld and Sagiv [Kimelfeld and Sagiv, PODS 2006], [Kimelfeld and Sagiv,
Inf. Syst. 2008] pointed out the problem of enumerating $K$-fragments is of
great importance in a keyword search on data graphs. In a graph-theoretic term,
the problem corresponds to enumerating minimal Steiner trees in (directed)
graphs. In this paper, we propose a linear-delay and polynomial-space algorithm
for enumerating all minimal Steiner trees, improving on a previous result in
[Kimelfeld and Sagiv, Inf. Syst. 2008]. Our enumeration algorithm can be
extended to other Steiner problems, such as minimal Steiner forests, minimal
terminal Steiner trees, and minimal directed Steiner trees. As another variant
of the minimal Steiner tree enumeration problem, we study the problem of
enumerating minimal induced Steiner subgraphs. We propose a polynomial-delay
and exponential-space enumeration algorithm of minimal induced Steiner
subgraphs on claw-free graphs. Contrary to these tractable results, we show
that the problem of enumerating minimal group Steiner trees is at least as hard
as the minimal transversal enumeration problem on hypergraphs

### Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids

Finding a maximum cardinality common independent set in two matroids (also
known as Matroid Intersection) is a classical combinatorial optimization
problem, which generalizes several well-known problems, such as finding a
maximum bipartite matching, a maximum colorful forest, and an arborescence in
directed graphs. Enumerating all maximal common independent sets in two (or
more) matroids is a classical enumeration problem. In this paper, we address an
``intersection'' of these problems: Given two matroids and a threshold $\tau$,
the goal is to enumerate all maximal common independent sets in the matroids
with cardinality at least $\tau$. We show that this problem can be solved in
polynomial delay and polynomial space. We also discuss how to enumerate all
maximal common independent sets of two matroids in non-increasing order of
their cardinalities

### On Maximal Cliques with Connectivity Constraints in Directed Graphs

Finding communities in the form of cohesive subgraphs is a fundamental problem in network analysis. In domains that model networks as undirected graphs, communities are generally associated with dense subgraphs, and many community models have been proposed.
Maximal cliques are arguably the most widely studied among such models, with early works dating back to the \u2760s, and a continuous stream of research up to the present. In domains that model networks as directed graphs, several approaches for community detection have been proposed, but there seems to be no clear model of cohesive subgraph, i.e., of what a community should look like. We extend the fundamental model of clique to directed graphs, adding the natural constraint of strong connectivity within the clique. We characterize the problem by giving a tight bound for the number of such cliques in a graph, and highlighting useful structural properties. We then exploit these properties to produce the first algorithm with polynomial delay for enumerating maximal strongly connected cliques

### Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids

Finding a maximum cardinality common independent set in two matroids (also known as Matroid Intersection) is a classical combinatorial optimization problem, which generalizes several well-known problems, such as finding a maximum bipartite matching, a maximum colorful forest, and an arborescence in directed graphs. Enumerating all maximal common independent sets in two (or more) matroids is a classical enumeration problem. In this paper, we address an "intersection" of these problems: Given two matroids and a threshold ?, the goal is to enumerate all maximal common independent sets in the matroids with cardinality at least ?. We show that this problem can be solved in polynomial delay and polynomial space. We also discuss how to enumerate all maximal common independent sets of two matroids in non-increasing order of their cardinalities

### Efficient Enumeration Algorithm for Dominating Sets in Bounded Degenerate Graphs (Foundations and Applications of Algorithms and Computation)

Dominating sets are fundamental graph structures. However, enumeration of dominating sets has not received much attention. This study aims to propose an efficient enumeration algorithms for bounded degenerate graphs. The algorithm enumerates all the dominating sets for k-degenerate graphs in O(k) time per solution using O(n+m) space. Since planar graphs have a constant degeneracy, this algorithm can enumerate all such sets for planar graphs in constant time per solution

### Efficient Enumeration of Dominating Sets for Sparse Graphs

A dominating set D of a graph G is a set of vertices such that any vertex in G is in D or its neighbor is in D. Enumeration of minimal dominating sets in a graph is one of central problems in enumeration study since enumeration of minimal dominating sets corresponds to enumeration of minimal hypergraph transversal. However, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a k-degenerate graph in O(k) time per solution using O(n + m) space, where n and m are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, H-minor free graphs with some fixed H. The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine