- Email: [email protected]
Clique Covers in Claw-Free Berge Graphs
Clique Covers in Claw-Free Berge Graphs
Gianpaolo Oriolo Dipartimento di Informatica, Sistemi e Produzione, Universita di Roma \Tor Vergata", via di Tor Vergata 110, 00133 Rome, Italy
Abstract We investigate properties of weighted clique covers in claw-free Berge graphs. We discuss a representation of weighted clique covers based on structural properties of these graphs. This leads to sketch a primal-dual algorithm for the minimum weighted clique cover problem for the case where there are no subgraphs hysomorphic to the Hajos subgraph. Key words: Claw-free Graphs, Primal Dual Algorithm, Matching Polytope.
We consider the problem of nding minimum weighted clique covers in clawfree perfect graphs. Since strong perfect graph conjecture holds for claw-free graphs, Chvatal and Sbihi (2]) proposed to call this class of graphs, claw-free Berge graphs. An e cient algorithm for solving the minimum weighted clique cover problem in claw-free Berge graphs was given by Hsu and Nemahauser (3]). This algorithm is based on Minty's algorithm (4]) for solving the maximum weighted stable set problem on claw-free graphs (Minty's algorithm was recently revised by (?])). In fact, it is well known that, if G is perfect, then the maximum weight of a stable set is equal to the minimum weight of a clique cover moreover, by complementarity slackness, a stable set S and a clique cover y are optimal if and only if the stable set meets all the cliques in the support of y - dened as the set of cliques K such that yK > 0 - and the cover y is tight for any vertex in the set S . Therefore, the algorithm in (3]) starts from a maximum weighted stable set S of G (found by Minty's algorithm) and builds a clique cover y such that S and y satisfy complementarity slackness. Vice versa, as we will explain in the following, it would be nice to have a primal-dual algorithm for solving (cuncurrently) both problems. Email address: [email protected] (Gianpaolo Oriolo).
Preprint submitted to Elsevier Science
24 March 2001
To see that, we look at our problem from a dierent perspective. Our interest in it was mainly sparkled by Ben Rebea's conjecture (1]) on the stable set polytope of quasi-line graphs. Quasi-line graphs are a relevant subclass of clawfree graphs dened as follows. A graph is quasi-line if, for any vertex there exist two cliques 1 and 2 such that ( ) = 1 2, that is, the neighborood of any vertex may be covered by two cliques. Trivially, if is quasi-line, then it is claw-free. Vice versa, we have the following lemma. v
C
C
N v
C
C
G
Lemma 0.1 If is claw free and Berge, then it is quasi-line. G
Basically, Ben Rebea conjectured (1], 7]) that the stable set polytope of a quasi-line graph may be described by trivial inequalities and another family of inequalities - called odd sets of cliques inequalities in (7]) - which are the extension of the odd set (of vertices) inequalities for the matching polytope. Observe that quasi-line are a superclass of line grahs, and the decription of the stable set polytope of a line graph follows from the decription of the matching polytope. A possible line of attack to the conjecture would be that of extending the primal-dual approach used by Edmonds in (5]). In this case, one should extend Edmonds' operation of shrinking of odd sets, namely dening an operation of shrinking of odd sets of cliques. In order to perform shrinking operations, it seems crucial to assume a nested structure of the shrunk objects. - A family of sets is nested (or laminar) if for each pair ( ) of elements of either \ = or or . - This is possible in the matching case, where one may assume, without loss of generality, any dual solution to hold this property. F
S
T
S T
S
T
T
F
S
When dealing with stable sets, even assuming perfection, things seem more involved: consider for instance the weighted graph whose vertex set is given by f 1 2 3 1 2 3g, respectively with weights f1 3 2 2 3 1g, and edge set f 1 2 2 3 1 2 2 3 1 1 2 1 2 2 3 2 3 3g. It is easy to see that the only optimal dual solution (i.e. minimum weighted clique cover) gives to each maximal clique weight one (while there are three dierent stable sets). How is it possible to look at this solution as a nested one? u u u v v v
v v v v u u u u u v u v u v u v u v
In some sense, we believe that the lack of a primal-dual algorithm (or even a primal algorithm) for the minimum weighted clique cover problem is related to the lack of an answer of this question. We oer a rst answer to the question. Briey, this goes as follows. If is a clique of a quasi-line graph , then the strong neighborood of - dened as the set of vertices not in which are adjacent to any vertex of - may be covered by two cliques 1 and 2 (actually, this is an another denition of quasi-line graphs). For = 1 2, let = . We say that the pair f 1 2g is C
G
C
C
C
j
C
C
K
j
C
C
2
j
K K
generated by C . Analogously, we say that a family K of (maximal) cliques is generated by a set of cliques C = fC1 : : : C g if: K
=
=1::p
p
1
Ki
K2 i
i
where, for each , the pair ( i
1
Ki
K 2) is generated by C . i
i
Then, it is possible to show, that, without loss of generality, we may always assume to deal with clique covers whose support - is generated by a nested family of cliques. y
Nested set sof cliques seem a powerful tool. For instance, they allow to prove the following theorem. An Hajos graph is a graph with vertex set f 1 2 3 1 2 3g and edge set f 1 2 2 3 3 1 1 1 1 2 2 2 2 3 3 2 3 1 g we call Hajos-free a graph without subgraph hysomorphic to the Hajos graph. Also, if is a weighted clique cover of , we denote by ( ) the subgraph induced by the vertices of for which is tight. u u u v
v v
u u u u u u v u v u v u v u v u v u
y
G
G
G y
y
Theorem 0.2 If is a claw-free Hajos-free Berge graph, then there exists G
an optimal pair (S y ) such that S is a stable set of maximum size in G(y).
As one would expect, on the base of the previous theorem, it is possible to sketch a primal-dual algorithm for the minimum weighted clique cover (and maximum stable set) problem on claw-free Hajos-free Berge graphs. At each iteration of the algorithm, we have to solve of a maximum cardinality stable set problem, moreover there is no need of performing any shrinking operation. We point out that, for nding a stable set of maximum cardinality on a clawfree graph, we may again use Minty's algorithm but there are more simple algorithms (by Sbihi 9] and, in particular, by Lovasz and Plummer 6]). Observe that, in general, Theorem 0.2 does not hold for claw-free Berge graph for instance, consider the Hajos graph and give weight 3 to any vertex and 1 to any vertex. It is easy to see that there exists only a minimum weighted clique cover (giving weight 1 to each maximal clique) and that this cover is tight for any vertex of nevertheless no stable of maximum weight has size three. ui
vi
y
G
S
Of course, the problem of extending this approach to the general claw-free Berge graph naturally arises. It seems that its extension is possible by doing some shrinking operations of cliques in Hajos subgraphs and hopefully nested structure would be useful again. 3
References 1] A. Ben Rebea, Etudes des stables dans le graphs quasi-adjoints, Ph.D. Thesis, Universit e de Grenoble, 1981. 2] V. Chv atal and N. Sbihi, Recognizing claw-free Berge graphs, Journal of Combinatorial Theory Series B, 44, 1988,154-176. 3] W.L. Hsu and L.G. Nemhauser, Algorithms for Maximum Weight Cliques, Minimum weighted clique covers and minimum colorings of claw-free prefect graphs, Topics on Perfect Graphs, C. Berge, V. Chv atal eds, 1984. 4] G.J. Minty, On maximal independent sets of vertices in claw-free graphs, Journal of Combinatorial Theory Series B, 28, 1980, 284-304. 5] J. Edmonds, Maximum Matching and a polyhedron with 0-1 vertices, J. Res. Nat. Bur. Standards Sect. B 69B, 1965, 125-130. 6] L. Lov asz and M.D. Plummer, Matching theory, Annals of Discrete Mathematics 29, 1986. 7] G. Oriolo, On the Stable Set Polytope of Quasi-Line Graphs, working paper. 8] K.R. Parthasarathy and G Ravindra, The strong perfect graph conjecture is true for claw-free graphs Journal of Combinatorial Theory Series B, 21, 1976, 212-223. 9] N. Sbihi, Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile, Discrete Mathematics, 29, 1980, 53-76.
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Gianpaolo Oriolo Dipartimento di Informatica, Sistemi e Produzione, Universita di Roma \Tor Vergata", via di Tor Vergata 110, 00133 Rome, Italy
Abstract We investigate properties of weighted clique covers in claw-free Berge graphs. We discuss a representation of weighted clique covers based on structural properties of these graphs. This leads to sketch a primal-dual algorithm for the minimum weighted clique cover problem for the case where there are no subgraphs hysomorphic to the Hajos subgraph. Key words: Claw-free Graphs, Primal Dual Algorithm, Matching Polytope.
We consider the problem of nding minimum weighted clique covers in clawfree perfect graphs. Since strong perfect graph conjecture holds for claw-free graphs, Chvatal and Sbihi (2]) proposed to call this class of graphs, claw-free Berge graphs. An e cient algorithm for solving the minimum weighted clique cover problem in claw-free Berge graphs was given by Hsu and Nemahauser (3]). This algorithm is based on Minty's algorithm (4]) for solving the maximum weighted stable set problem on claw-free graphs (Minty's algorithm was recently revised by (?])). In fact, it is well known that, if G is perfect, then the maximum weight of a stable set is equal to the minimum weight of a clique cover moreover, by complementarity slackness, a stable set S and a clique cover y are optimal if and only if the stable set meets all the cliques in the support of y - dened as the set of cliques K such that yK > 0 - and the cover y is tight for any vertex in the set S . Therefore, the algorithm in (3]) starts from a maximum weighted stable set S of G (found by Minty's algorithm) and builds a clique cover y such that S and y satisfy complementarity slackness. Vice versa, as we will explain in the following, it would be nice to have a primal-dual algorithm for solving (cuncurrently) both problems. Email address: [email protected] (Gianpaolo Oriolo).
Preprint submitted to Elsevier Science
24 March 2001
To see that, we look at our problem from a dierent perspective. Our interest in it was mainly sparkled by Ben Rebea's conjecture (1]) on the stable set polytope of quasi-line graphs. Quasi-line graphs are a relevant subclass of clawfree graphs dened as follows. A graph is quasi-line if, for any vertex there exist two cliques 1 and 2 such that ( ) = 1 2, that is, the neighborood of any vertex may be covered by two cliques. Trivially, if is quasi-line, then it is claw-free. Vice versa, we have the following lemma. v
C
C
N v
C
C
G
Lemma 0.1 If is claw free and Berge, then it is quasi-line. G
Basically, Ben Rebea conjectured (1], 7]) that the stable set polytope of a quasi-line graph may be described by trivial inequalities and another family of inequalities - called odd sets of cliques inequalities in (7]) - which are the extension of the odd set (of vertices) inequalities for the matching polytope. Observe that quasi-line are a superclass of line grahs, and the decription of the stable set polytope of a line graph follows from the decription of the matching polytope. A possible line of attack to the conjecture would be that of extending the primal-dual approach used by Edmonds in (5]). In this case, one should extend Edmonds' operation of shrinking of odd sets, namely dening an operation of shrinking of odd sets of cliques. In order to perform shrinking operations, it seems crucial to assume a nested structure of the shrunk objects. - A family of sets is nested (or laminar) if for each pair ( ) of elements of either \ = or or . - This is possible in the matching case, where one may assume, without loss of generality, any dual solution to hold this property. F
S
T
S T
S
T
T
F
S
When dealing with stable sets, even assuming perfection, things seem more involved: consider for instance the weighted graph whose vertex set is given by f 1 2 3 1 2 3g, respectively with weights f1 3 2 2 3 1g, and edge set f 1 2 2 3 1 2 2 3 1 1 2 1 2 2 3 2 3 3g. It is easy to see that the only optimal dual solution (i.e. minimum weighted clique cover) gives to each maximal clique weight one (while there are three dierent stable sets). How is it possible to look at this solution as a nested one? u u u v v v
v v v v u u u u u v u v u v u v u v
In some sense, we believe that the lack of a primal-dual algorithm (or even a primal algorithm) for the minimum weighted clique cover problem is related to the lack of an answer of this question. We oer a rst answer to the question. Briey, this goes as follows. If is a clique of a quasi-line graph , then the strong neighborood of - dened as the set of vertices not in which are adjacent to any vertex of - may be covered by two cliques 1 and 2 (actually, this is an another denition of quasi-line graphs). For = 1 2, let = . We say that the pair f 1 2g is C
G
C
C
C
j
C
C
K
j
C
C
2
j
K K
generated by C . Analogously, we say that a family K of (maximal) cliques is generated by a set of cliques C = fC1 : : : C g if: K
=
=1::p
p
1
Ki
K2 i
i
where, for each , the pair ( i
1
Ki
K 2) is generated by C . i
i
Then, it is possible to show, that, without loss of generality, we may always assume to deal with clique covers whose support - is generated by a nested family of cliques. y
Nested set sof cliques seem a powerful tool. For instance, they allow to prove the following theorem. An Hajos graph is a graph with vertex set f 1 2 3 1 2 3g and edge set f 1 2 2 3 3 1 1 1 1 2 2 2 2 3 3 2 3 1 g we call Hajos-free a graph without subgraph hysomorphic to the Hajos graph. Also, if is a weighted clique cover of , we denote by ( ) the subgraph induced by the vertices of for which is tight. u u u v
v v
u u u u u u v u v u v u v u v u v u
y
G
G
G y
y
Theorem 0.2 If is a claw-free Hajos-free Berge graph, then there exists G
an optimal pair (S y ) such that S is a stable set of maximum size in G(y).
As one would expect, on the base of the previous theorem, it is possible to sketch a primal-dual algorithm for the minimum weighted clique cover (and maximum stable set) problem on claw-free Hajos-free Berge graphs. At each iteration of the algorithm, we have to solve of a maximum cardinality stable set problem, moreover there is no need of performing any shrinking operation. We point out that, for nding a stable set of maximum cardinality on a clawfree graph, we may again use Minty's algorithm but there are more simple algorithms (by Sbihi 9] and, in particular, by Lovasz and Plummer 6]). Observe that, in general, Theorem 0.2 does not hold for claw-free Berge graph for instance, consider the Hajos graph and give weight 3 to any vertex and 1 to any vertex. It is easy to see that there exists only a minimum weighted clique cover (giving weight 1 to each maximal clique) and that this cover is tight for any vertex of nevertheless no stable of maximum weight has size three. ui
vi
y
G
S
Of course, the problem of extending this approach to the general claw-free Berge graph naturally arises. It seems that its extension is possible by doing some shrinking operations of cliques in Hajos subgraphs and hopefully nested structure would be useful again. 3
References 1] A. Ben Rebea, Etudes des stables dans le graphs quasi-adjoints, Ph.D. Thesis, Universit e de Grenoble, 1981. 2] V. Chv atal and N. Sbihi, Recognizing claw-free Berge graphs, Journal of Combinatorial Theory Series B, 44, 1988,154-176. 3] W.L. Hsu and L.G. Nemhauser, Algorithms for Maximum Weight Cliques, Minimum weighted clique covers and minimum colorings of claw-free prefect graphs, Topics on Perfect Graphs, C. Berge, V. Chv atal eds, 1984. 4] G.J. Minty, On maximal independent sets of vertices in claw-free graphs, Journal of Combinatorial Theory Series B, 28, 1980, 284-304. 5] J. Edmonds, Maximum Matching and a polyhedron with 0-1 vertices, J. Res. Nat. Bur. Standards Sect. B 69B, 1965, 125-130. 6] L. Lov asz and M.D. Plummer, Matching theory, Annals of Discrete Mathematics 29, 1986. 7] G. Oriolo, On the Stable Set Polytope of Quasi-Line Graphs, working paper. 8] K.R. Parthasarathy and G Ravindra, The strong perfect graph conjecture is true for claw-free graphs Journal of Combinatorial Theory Series B, 21, 1976, 212-223. 9] N. Sbihi, Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile, Discrete Mathematics, 29, 1980, 53-76.
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