LDFS-Based Certifying Algorithm for the Minimum Path Cover Problem on Cocomparability Graphs

Abstract

International audienceFor graph $G(V,E)$, a minimum path cover (MPC) is a minimum cardinality set of vertex disjoint paths that cover $V$ (i.e., every vertex of $G$ is in exactly one path in the cover). This problem is a natural generalization of the Hamiltonian path problem. Cocomparability graphs (the complements of graphs that have an acyclic transitive orientation of their edge sets) are a well studied subfamily of perfect graphs that includes many popular families of graphs such as interval, permutation, and cographs. Furthermore, for every cocomparability graph $G$ and acyclic transitive orientation of the edges of $\overline{G}$ there is a corresponding poset $P_G$; it is easy to see that an MPC of $G$ is a linear extension of $P_G$ that minimizes the bump number of $P_G$. Although there are directly graph-theoretical MPC algorithms (i.e., algorithms that do not rely on poset formulations) for various subfamilies of cocomparability graphs, notably interval graphs, until now all MPC algorithms for cocomparability graphs themselves have been based on the bump number algorithms for posets. In this paper we present the first directly graph-theoretical MPC algorithm for cocomparability graphs; this algorithm is based on two consecutive graph searches followed by a certifying algorithm. Surprisingly, except for a lexicographic depth first search (LDFS) preprocessing step, this algorithm is identical to the corresponding algorithm for interval graphs. The running time of the algorithm is $O({\rm min}(n^2, n + {\rm mloglogn}))$, with the nonlinearity coming from LDFS

A Linear-Time Algorithm for Maximum-Cardinality Matching on Cocomparability Graphs

“…So far a similar phenomenon of extending an interval graph algorithm to cocomparability graphs by using an LDFS preprocessing step has also been observed for the Longest Path problem [39], the Minimum Path Cover problem [7], and the Maximum Independent Set problem [8]. Our results for the RMM algorithm, adding to the previous results [7,8,39], provide evidence that cocomparability graphs present an "interval graph structure" when they are considered with an LDFS preprocessing step. This insight is of independent interest and might lead to new and more efficient combinatorial algorithms.…”

“…LDFS + ) to π. Our RMM algorithm (see Algorithm 3) will then take this LDFS ordering σ as input, together with the graph G. It is important to note here that, starting from an umbrella-free ordering π, the LDFS vertex ordering σ = LDFS + (G, π) remains umbrella-free [7]. That is, σ satisfies both the conditions of Definition 1 and Definition 3, and thus σ is simultaneously an LDFS ordering and an umbrella-free ordering.…”

Vertex Ordering Characterizations of Graphs of Bounded Asteroidal Number

“…In particular, a simple LDFS + of a COCOMP ORDER yields such an order. This property turned out to be immensely useful in recent algorithmic results on cocomparability graphs [5,7,24]. It again would seem useful if this carried over to AT-free ORDERs.…”

A New Graph Parameter to Measure Linearity

“…We call the triple a, b, c as described in Theorem 2 above a bad LexBFS triple, and vertex d a private neighbour of b with respect to c. Given a graph class G, a vertex ordering characterization (or VOC) of G is a total ordering on the vertices with specific properties, and ∀G, G ∈ G if and only if G admits a total ordering that satisfies said properties. VOCs have led to a number of efficient algorithms, and are often the basis of various graph recognition algorithms, see for instance [2,6,10,16,18]. We recall here some vertex ordering characterizations of the graph classes we consider.…”

“…Graph searching is a mechanism to traverse the graph one vertex at a time, in a specific manner. A very promising area of research is based on graph searching and the notion of multi-sweep algorithms [1,2,6,8,14,15]. A multi-sweep algorithm is an algorithm that computes a number of orderings where each ordering σ i>1 uses the previous ordering σ i−1 to break ties using specified tie breaking rules.…”