MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs On minimum weight clique cover problem of claw-free perfect graphs Flavia Bonomo 1 Gianpaolo Oriolo 2 Claudia Snels 2 1 Universit` a di Buenos Aires 2 Universit` a di Roma “Tor Vergata” 16th Combinatorial Optimization Workshop, Aussois 2012
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs Maximum weighted stable set (MWSS) Given a graph G ( V , E ) with a nonnegative weight function on the vertices w , find a set of pairwise nonadjacent vertices maximizing the sum of their weight.
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs Maximum weighted stable set (MWSS) Given a graph G ( V , E ) with a nonnegative weight function on the vertices w , find a set of pairwise nonadjacent vertices maximizing the sum of their weight. In the unweighted case (MSS), we look for a maximum cardinality stable set of G .
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs Maximum weighted stable set This problem is polynomial-time solvable on perfect graphs (Gr¨ otschel, Lov´ asz, and Schrijver, 1981) but an important open question is to find a combinatorial algorithm to solve it.
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs Minimum weight clique cover (MWCC) Given a graph G ( V , E ) with a nonnegative weight function on the vertices w , find a collection of cliques C and a non negative value y C for each clique C ∈ C such that � C : v ∈ C y C ≥ w ( v ) for every vertex v ∈ V and � C ∈C y C is minimum.
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs Minimum weight clique cover (MWCC) Given a graph G ( V , E ) with a nonnegative weight function on the vertices w , find a collection of cliques C and a non negative value y C for each clique C ∈ C such that � C : v ∈ C y C ≥ w ( v ) for every vertex v ∈ V and � C ∈C y C is minimum. In the unweighted case (MCC), we ask for � C : v ∈ C y C ≥ 1 for every vertex v ∈ V .
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs Minimum weight clique cover The MWCC is the dual of the linear relaxation of the clique formulation for the MWSS. For perfect graphs, the weights of a MWSS and a MWCC are equal and, moreover, for an integer weight function w , there is a MWCC where the weight of each clique is integer. The MWCC problem is NP-hard for general graphs but it can be polynomially solved for perfect graphs using a method proposed by Gr¨ otschel, Lov´ asz, and Schrijver in 1988 (not combinatorial, it uses Lov´ asz θ -function).
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs Maximum weighted stable set on claw-free graphs For the maximum weighted stable set on claw-free graphs (even not perfect ones) several combinatorial algorithms are known (Minty 1980, Nakamura and Tamura 2001, Oriolo, Pietropaoli and Stauffer 2008, Nobili and Sassano 2011), and the more efficient until now, by Faenza, Oriolo and Stauffer (2010, O ( n 3 )). These algorithms are general, there are not special algorithms for claw-free perfect graphs. So, another interesting question is the existence of an algorithm for perfect claw-free perfect graphs that exploits the perfection.
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs MCC and MWCC on claw-free graphs Nemhauser and Hsu (1981) designed a combinatorial algorithm for claw-free perfect graphs which can solve in polynomial time the minimum clique cover (unweighted) building upon a solution to the MSS (it uses augmenting paths, and runs in O ( n 5 . 5 )). In 1982, they extended it to the weighted case (MWCC of claw-free perfect graphs), again building upon a solution to the MWSS (it uses complementary slackness).
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs Our results Our aim was to obtain a combinatorial algorithm for the minimum clique cover (unweighted and weighted) problem on claw-free perfect graphs with a primal-dual approach. By now, our results are: A simple combinatorial algorithm for solving concurrently the MCC and MSS on claw-free perfect graphs, in O ( n 3 ) time. An algorithmic theorem for the MWCC on strip composed perfect graphs, which states that we can compute a MWCC of a strip composed perfect graph if we can solve the same problem on each strip. An application of the previous theorem to strip composed claw-free perfect graphs.
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs MSS and MCC on claw-free perfect graphs Let S be a maximal (non necessarily maximum) stable set of a claw-free graph G ( V , E ). Every vertex v in V \ S has either one or two neighbors in S .
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs MSS and MCC on claw-free perfect graphs This property implies that if S is not a maximum stable set, then there is an augmenting path with respect to S .
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs MSS and MCC on claw-free perfect graphs This property implies that if S is not a maximum stable set, then there is an augmenting path with respect to S .
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs MSS and MCC on claw-free perfect graphs In an integer MCC of G of weight | S | , there is a one-to-one correspondence between cliques in the cover and vertices in S . We will call C s the clique in the cover containing vertex s . Let v in V \ S . If v has only one neighbor s in S , v should belong to C s , and if v has two neighbors s 1 , s 2 in S , either v ∈ C s 1 or v ∈ C s 1 .
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs MSS and MCC on claw-free perfect graphs We can write these conditions along to the condition that two nonadjacent vertices cannot be part of the same clique as 2-SAT instance. For each s ∈ S , v ∈ N ( s ), the variable x vs will be true if and only if v ∈ C s . We use also an auxiliary variable y . If v has only one neighbor s in S , we add ( x vs ∨ y ) ∧ ( x vs ∨ ¬ y ). If v has two neighbors s 1 , s 2 in S , we add ( x vs 1 ∨ x vs 2 ). If v and w share a neighbor s in S and are nonadjacent, we add ( ¬ x vs ∨ ¬ x ws ). The final formula will be the conjunction of all these clauses.
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs MSS and MCC on claw-free perfect graphs We can write these conditions along to the condition that two nonadjacent vertices cannot be part of the same clique as 2-SAT instance. For each s ∈ S , v ∈ N ( s ), the variable x vs will be true if and only if v ∈ C s . We use also an auxiliary variable y . If v has only one neighbor s in S , we add ( x vs ∨ y ) ∧ ( x vs ∨ ¬ y ). If v has two neighbors s 1 , s 2 in S , we add ( x vs 1 ∨ x vs 2 ). If v and w share a neighbor s in S and are nonadjacent, we add ( ¬ x vs ∨ ¬ x ws ). The final formula will be the conjunction of all these clauses.
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs MSS and MCC on claw-free perfect graphs We follow the 2-SAT algorithm by Aspvall, Plass and Tarjan (1979) and build the implication digraph corresponding to that instance. We prove that: either the instance is satisfiable and we can obtain a MCC from a truth assignment of the variables, or, if G is perfect, from the evidence of infeasibility on the implication digraph, we can find an augmenting path for S in G . In the second case, we update the stable set and corresponding formula and try again. The overall complexity is O ( n 3 ).
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs Strip composition A graph G is strip composed if G is a composition of some set of strips w.r.t. some partition P . A strip H = ( G , A ) is a graph G (not necessarily connected) with a multi-family A of either one or two designated non-empty cliques of G . The cliques in A are called the extremities of H . This generalizes line graphs, where each strip is a single vertex.
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs Strip composition A graph G is strip composed if G is a composition of some set of strips w.r.t. some partition P . A strip H = ( G , A ) is a graph G (not necessarily connected) with a multi-family A of either one or two designated non-empty cliques of G . The cliques in A are called the extremities of H . This generalizes line graphs, where each strip is a single vertex.
MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs Strip composition A graph G is strip composed if G is a composition of some set of strips w.r.t. some partition P . A strip H = ( G , A ) is a graph G (not necessarily connected) with a multi-family A of either one or two designated non-empty cliques of G . The cliques in A are called the extremities of H . This generalizes line graphs, where each strip is a single vertex.
Recommend
More recommend
