An efficient algorithm for solving the homogeneous set sandwich problem

doi:10.1016/S0020-0190(00)00145-9

Information Processing Letters

Volume 77, Issue 1, 31 January 2001, Pages 17-22

https://doi.org/10.1016/S0020-0190(00)00145-9 Get rights and content

Abstract

A set $H$ of vertices of graph $G(V,E)$ is a homogeneous set if each vertex in $V\H$ is either adjacent to all vertices of $H$ or to none of the vertices in $H$ , where $V$ and $E$ are the vertex set and edge set, respectively, of graph $G$ . A graph $G_{s} (V,E_{s})$ is called a sandwich graph for the pair of graphs $G(V,E)$ and $G_{t} (V,E_{t})$ if $E_{t} ⫅E_{s} ⫅E$ . The homogeneous set sandwich problem is to determine whether there exists a sandwich graph for the pair of graphs $G$ and $G_{t}$ such that there is a homogeneous set in $G_{s}$ . In this paper, we shall propose an $O (Δn^{2})$ time algorithm for solving the homogeneous set sandwich problem, where $Δ$ is the maximum degree of $G$ . Furthermore, a surprising result comes from the bias graph which is an auxiliary graph for reducing the homogeneous set sandwich problem to the problem of finding strongly connected components. Using the bias graph, we can easily find all homogeneous sets for a sandwich graph problem with the same time complexity, i.e., $O (Δn^{2})$ .

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Cited by (9)

Complexity issues for the sandwich homogeneous set problem
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Citation Excerpt :
In particular, it is not known if one can enumerate such sets with a polynomial delay (see [13,18] for preliminary considerations on this problem).
Graph sandwich problems were introduced by Golumbic et al. (1994) in [12] for DNA physical mapping problems and can be described as follows. Given a property $Π$ of graphs and two disjoint sets of edges $E_{1}$ , $E_{2}$ with $E_{1} \subseteq E_{2}$ on a vertex set $V$ , the problem is to find a graph $G$ on $V$ with edge set $E_{s}$ having property $Π$ and such that $E_{1} \subseteq E_{s} \subseteq E_{2}$ .
In this paper, we exhibit a quasi-linear reduction between the problem of finding an independent set of size $k \geq 2$ in a graph and the problem of finding a sandwich homogeneous set of the same size $k$ . Using this reduction, we prove that a number of natural (decision and counting) problems related to sandwich homogeneous sets are hard in general. We then exploit a little further the reduction and show that finding efficient algorithms to compute small sandwich homogeneous sets would imply substantial improvement for computing triangles in graphs.
Can transitive orientation make sandwich problems easier?
2007, Discrete Mathematics
A graph $G_{s} = (V, E_{s})$ is a sandwich for a pair of graphs $G_{t} = (V, E_{t})$ and $G = (V, E)$ if $E_{t} \subseteq E_{s} \subseteq E$ . A sandwich problem asks for the existence of a sandwich graph having an expected property. In a seminal paper, Golumbic et al. [Graph sandwich problems, J. Algorithms 19 (1995) 449–473] present many results on sub-families of perfect graphs. We are especially interested in comparability (resp., co-comparability) graphs because these graphs (resp., their complements) admit one or more transitive orientations (each orientation is a partially ordered set or poset). Thus, fixing the orientations of the edges of $G_{t}$ and G restricts the number of possible sandwiches. We study whether adding an orientation can decrease the complexity of the problem. Two different types of problems should be considered depending on the transitivity of the orientation: the poset sandwich problems and the directed sandwich problems. The orientations added to both graphs G and $G_{s}$ are transitive in the first type of problem but arbitrary for the second type.
The pair completion algorithm for the homogeneous set sandwich problem
2006, Information Processing Letters
A homogeneous set is a non-trivial module of a graph, i.e., a non-empty, non-unitary, proper vertex subset such that all its elements present the same outer neighborhood. Given two graphs $G_{1} (V, E_{1})$ and $G_{2} (V, E_{2})$ , the Homogeneous Set Sandwich Problem (HSSP) asks whether there exists a graph $G_{S} (V, E_{S})$ , $E_{1} \subseteq E_{S} \subseteq E_{2}$ , which has a homogeneous set. This paper presents an algorithm that uses the concept of bias graph [S. Tang, F. Yeh, Y. Wang, An efficient algorithm for solving the homogeneous set sandwich problem, Inform. Process. Lett. 77 (2001) 17–22] to solve the problem in $O (n \min {| E_{1} |, | {\bar{E}}_{2} |} \log n)$ time, thus outperforming the other known HSSP deterministic algorithms for inputs where $\max {| E_{1} |, | {\bar{E}}_{2} |} = Ω (n \log n)$ .
Note on the Homogeneous Set Sandwich Problem
2005, Information Processing Letters
A homogeneous set is a non-trivial module of a graph, i.e., a non-unitary, proper subset H of a graph's vertices such that all vertices in H have the same neighbors outside H. Given two graphs $G_{1} (V, E_{1})$ , $G_{2} (V, E_{2})$ , the Homogeneous Set Sandwich Problem asks whether there exists a sandwich graph $G_{S} (V, E_{S})$ , $E_{1} \subseteq E_{S} \subseteq E_{2}$ , which has a homogeneous set. Recently, Tang et al. [Inform. Process. Lett. 77 (2001) 17–22] proposed an interesting $O (▵_{1} \cdot n^{2})$ algorithm for this problem, which has been considered its most efficient algorithm since. We show the incorrectness of their algorithm by presenting three counterexamples.
A note on finding all homogeneous set sandwiches
2003, Information Processing Letters
A module is a set of vertices H of a graph G=(V,E) such that each vertex of V⧹H is either adjacent to all vertices of H or to none of them. A homogeneous set is a nontrivial module. A graph G_s=(V,E_s) is a sandwich for a pair of graphs G_t=(V,E_t) and G=(V,E) if E_t⊆E_s⊆E. In a recent paper, Tang et al. [Inform. Process. Lett. 77 (2001) 17–22] described an O(Δn²) algorithm for testing the existence of a homogeneous set in sandwich graphs of G_t=(V,E_t) and G=(V,E) and then extended it to an enumerative algorithm computing all these possible homogeneous sets. In this paper, we invalidate this latter algorithm by proving there are possibly exponentially many such sets, even if we restrict our attention to strong modules. We then give a correct characterization of a homogeneous set of a sandwich graph.
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^☆: This work was supported by the National Science Council, Republic of China, under Contract NSC-89-2213-E011-049.

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An efficient algorithm for solving the homogeneous set sandwich problem☆

Abstract

Inform. Process. Lett.

J. Algorithms

Discrete Math.

Discrete Math.