Parallel algorithms for generalized clique transversal problems

Parallel algorithms for generalized clique transversal problems Elias Dahlhaus 

Paul D. Manuel y

Mirka Miller y

Abstract The K` - clique transversal problem is to locate a minimum collection of cliques of size ` in a graph G such that every maximal clique of size  ` in G contains at least one member of the collection. We give an NC algorithm to solve this problem on strongly chordal graphs. Keywords: transversal,

balanced graphs, strongly chordal graphs, clique transversal,

K` - clique transversal.

k-fold clique

1 Introduction A 0 ? 1 matrix is balanced if it does not contain as a submatrix, an edge - vertex incidence matrix of an odd cycle. A 0 ? 1 matrix is totally balanced if it does not contain as a submatrix, an edge - vertex incidence matrix of any cycle. A hypergraph H is an ordered pair (V; E ) where V is a set of vertices and E is a family of subsets of V . The members of E are called hyperedges of H . Let V = fv1; v2; : : :; vn g and E = fE1; E2; : : :; Emg. Let A(H ) denote the hyperedge - vertex incidence matrix of a hypergraph H . A hypergraph H is balanced (respectively totally balanced) if A(H ) is balanced (respectively totally balanced). A clique hypergraph H (V; E ) of a graph G(V; E ) is a hypergraph whose hyperedges are the maximal cliques of G. A graph is said to be balanced if its clique hypergraph is balanced. A graph is said to be chordal if it contains no induced cycle of length 4 or greater. A chordal graph is said to be strongly chordal if every cycle on six or more vertices contains a chord joining two vertices with an odd distance in the cycle. A k-fold clique transversal of a graph G is a set S of vertices such that every maximal clique of G has at least k vertices of S . This problem has been shown to be in the polynomial class for balanced graphs [DMM95]. Corneil and Fonlupt [CF89] have introduced the Cij -cover problem which is to nd a minimum family of cliques of size j such that every clique of size i of G contains at least one member of the family. The Cij -cover problem has been studied in [CCM86, CCM87, CF89, MRP92, DEMM94]. We study a similar concept,  y

Basser Department of Computer Science, University of Sydney, NSW, Australia 2006. Department of Computer Science, University of Newcastle, Newcastle, Australia 2308.

1

called K` - clique transversal problem. It is usual to denote a clique of size r by Kr . A K` - clique transversal of a graph G is a collection of cliques of size ` such that every maximal clique of size greater than or equal to ` in G contains at least one member of the collection. A K` - clique transversal problem is to locate a K` - clique transversal

with the minimum cardinality. A clique transversal of a graph is a set of vertices which meet all the maximal cliques. A A clique transversal is a K1-clique transversal and it has been widely studied [AST91, Tuz90, Man93, CFT93].

Note that the K` - clique transversal problem is dierent from the `-fold clique transversal problem. In Figure 1, K21 = f1; 2g, K22 = f4; 5g is a minimum K2 - clique transversal of

1

5 3

2

4 Figure 1:

the graph. But f1; 2; 4; 5g is not a minimum 2-fold clique transversal whereas f1; 3; 4g is a minimum 2-fold clique transversal of the graph. It is interesting to explore whether a minimum K` - clique transversal can be eciently extracted from a minimum `-fold clique transversal of a graph. The notion of a K` - clique transversal can be extended to hypergraphs as follows: for a hypergraph H (V; E ), let = = fS1; S2; : : :; Sr g be a family of subsets of V such that each Si is a subset of some hyperedge of E . A = - transversal problem is to nd a minimum subfamily =0 of = such that a hyperedge of H contains a member of =0 whenever it contains a member of =. This problem reduces to the transversal problem when = = V and each Si is a singleton set consisting of a vertex of V . The K` - clique transversal problem becomes a particular case of = - transversal problem when hyperedges are the maximal cliques and = is the family of all cliques of size `. We will later use the following result, due to Dahlhaus and Damaschke:

Theorem 1.1 [DD94] The transversal problem can be solved on totally balanced hypergraphs in O(log3 n) time with O(n + m) processors.

This result can be restated as

Theorem 1.2 Consider the integer programming problem 9 rj=1 xj > = ; subject to M x  1 >

minimize

2

(1)

where M is a totally balanced matrix and x = (x1; x2; : : :; xr ) is such that xi = 0 or 1. Here 1 stands for the all-one vector and vectors will be considered columnwise. The integer programming problem (1) can be solved in O(log3 n) time with O(n + m) processors.

In this paper, we study the K` - clique transversal and = - transversal problems. We give an NC - algorithm to solve = - transversal problem on totally balanced hypergraphs (strongly chordal graphs) using Theorem 1.2. The main result of this paper is that the K` - clique transversal problem on strongly chordal graphs is solvable in polylogarithmic time with polynomial number of processors.

2 = - transversal problem Given H (V; E ) and a family = = fS1; S2; : : :; Sr g of subsets of V , consider the incidence matrix B (H; =) = (b(i; j )) with m rows representing the edges E1; E2; : : :; Em of E and r columns representing the sets S1 ; S2; : : :; Sr with ( Ej b(i; j ) = 10 ifif SSi  6 E i

j

The = - transversal problem can be formulated as:

9 rj=1 xj > = ; subject to Bx  1 >

minimize

(2)

where B = B (H; =) is the incidence matrix and x = (x1; x2; : : :; xr ) is such that xi = 0 or 1. Here again, 1 stands for the all-one vector and vectors will be considered columnwise.

v1

v2






S1

vn

E1 E. 2

E1 E. 2

Em

Em

..

S2






Sr

..

(a): Matrix A(H ) in ?-free form

(b): Matrix B (H; ) =

Figure 2: In this section, we show that the = - transversal problem is in NC - class for totally balanced hypergraphs. It is enough to prove that B (H; =) is totally balanced whenever H (V; E ) is totally balanced. 3

Let H (V; E ) be a hypergraph with the vertex set V = fv1; v2; : : :; vn g. Let = = fS1; S2; : : :; Sr g be a family of ordered subsets of V such that the vertices of Si , i = 1; 2; : : :; r, are sorted by its indices. For example, (v2 ; v5; v7; v8) is sorted, whereas (v2; v7; v5; v8) is not. The ordered sets Si are sorted by lexicographic ordering `