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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

1Universit`

a di Buenos Aires

2Universit`

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¨

• tschel, 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 yC for each clique C ∈ C such that

C:v∈C yC ≥ w(v) for every

vertex v ∈ V and

C∈C yC 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 yC for each clique C ∈ C such that

C:v∈C yC ≥ w(v) for every

vertex v ∈ V and

C∈C yC is minimum.

In the unweighted case (MCC), we ask for

C:v∈C yC ≥ 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¨

• tschel, 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(n3)). 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(n5.5)). 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(n3) 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 Cs 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 Cs, and if v has two neighbors s1, s2 in S, either v ∈ Cs1 or v ∈ Cs1.

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 xvs will be true if and only if v ∈ Cs. We use also an auxiliary variable y. If v has only one neighbor s in S, we add (xvs ∨ y) ∧ (xvs ∨ ¬y). If v has two neighbors s1, s2 in S, we add (xvs1 ∨ xvs2). If v and w share a neighbor s in S and are nonadjacent, we add (¬xvs ∨ ¬xws). 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 xvs will be true if and only if v ∈ Cs. We use also an auxiliary variable y. If v has only one neighbor s in S, we add (xvs ∨ y) ∧ (xvs ∨ ¬y). If v has two neighbors s1, s2 in S, we add (xvs1 ∨ xvs2). If v and w share a neighbor s in S and are nonadjacent, we add (¬xvs ∨ ¬xws). 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,

• r, 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(n3).

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

• f 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

• f 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

• f 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

• f 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

• f 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

Each class of the partition of the extremities defines a clique

• f the composed graph, and is called a partition-clique.

A strip satisfies property Π when the graph obtained by adding, for each extremity, a vertex complete to it, satisfies property Π. The composition of line (claw-free, quasi-line) strips is a line (claw-free, quasi-line) graph. The composition of perfect strips is not necessary a perfect graph.

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

Strip decomposition theorems for claw-free graphs

Existence Theorem (Chudnovsky and Seymour, 2005) G claw-free, α(G) ≥ 4:

• 1. either G is the composition of fuzzy linear interval

strips

• 2. or G is a fuzzy circular interval graph.

Algorithmic Theorem (Faenza, Oriolo and Stauffer, 2010) G claw-free, α(G) ≥ 4:

• 1. either G is the composition of distance simplicial

strips

• 2. or G is distance claw-free.

There is evidence that both algorithmically and polyhedrally these two classes behave different.

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

MWSS and MWCC on strip composed (perfect) graphs

We want to find a MWCC of G, exploiting the strip decomposition. A similar approach exists for the MWSS on strip composed graphs, using the following theorem.

Theorem (Oriolo, Pietropaoli and Stauffer, 2008)

Let G(V , E) be the composition of strips Hi = (G i, Ai) i = 1 . . . , k w.r.t. a partition P. Suppose that for each i = 1, . . . , k we can compute a MWSS of Hi in time O(pi(|V |)). Then the MWSS problem on G can be solved in time O(k

i=1 pi(|V |) + match(|V |)), where match(|V |) is the time

required to solve the matching problem on a graph with V

• vertices. If pi(|V |) is polynomial for each i, then the MWSS can

be solved on G in polynomial time.

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

MWSS of strip composed graphs

They replace every strip with a weighted simple line strip (a gadget), and obtain a line graph G ′ as the composition of the gadgets, such that αw(G) = f (αw(G ′)) and they know f . In order to find a MWSS of G ′ they find a maximum weight matching in the root graph of G ′. The vertices in the MWSS S of G ′ represent a guideline to choose a suitable stable set in each strip that forms a MWSS

• f G.

They use it for the strip decomposable claw-free graphs and solve the case of distance claw-free graphs separately.

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

Extending it to the MWCC on strip composed graphs

We borrow the MWSS idea, but... The original gadgets do not preserve perfection. We introduced four different gadgets depending on the parity of the strips (non-trivial theorems to prove the reduction and the perfection). If we want to have a primal algorithm to solve MWCC, we have to solve first MWCC on strips (instead of MWSS) in

• rder to compute the weight of the gadgets vertices.

Even knowing how to solve MWCC on strips and on the line graph G ′, sometimes it is not trivial to deduce a MWCC of G from one of G ′. Some cliques of G ′ do not translate straightforward into a clique of G. We have to deal with seven different cases.

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

The perfection problem

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

The perfection problem

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

The perfection problem

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

The perfection problem

New gadget strips (still line strips)

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

The perfection problem

For a strip Hi = (G i, {Ai

1, Ai 2}), the weights of the vertices of the

corresponding gadget strips will be functions of αw(G i), αw(G i \ (Ai

1)), αw(G i \ (Ai 2)) and αw(G i \ (Ai 1 ∪ Ai 2)) (these values

can be computed by solving four MWCC problems on each strip). The second gadget can be used to replace strip Hi if and only if αw(G i \ Ai

1) + αw(G i \ Ai 2) = αw(G i) + αw(G i \ (Ai 1 ∪ Ai 2)).

The third gadget can be used to replace strip Hi if and only if αw(G i \ Ai

1) + αw(G i \ Ai 2) ≥ αw(G i) + αw(G i \ (Ai 1 ∪ Ai 2)).

The fourth gadget can be used to replace strip Hi if and only if αw(G i \ Ai

1) + αw(G i \ Ai 2) ≤ αw(G i) + αw(G i \ (Ai 1 ∪ Ai 2)).

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

Parity theorems

A strip is called non-connected if there is no path joining its extremities A strip is called odd(resp. even) if its extremities are connected and every induced path from one extremity to the

• ther (with interior outside the extremity) is odd(resp. even).

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

Parity theorems

One of the main steps on the proof of correctness of the algorithm was to prove that: If Hi = (G i, {Ai

1, Ai 2}) is a non-connected strip then

αw(G i \ Ai

1) + αw(G i \ Ai 2) = αw(G i) + αw(G i \ (Ai 1 ∪ Ai 2)).

If Hi = (G i, {Ai

1, Ai 2}) is a perfect odd strip then

αw(G i \ Ai

1) + αw(G i \ Ai 2) ≥ αw(G i) + αw(G i \ (Ai 1 ∪ Ai 2)).

If Hi = (G i, {Ai

1, Ai 2}) is a perfect even strip then

αw(G i \ Ai

1) + αw(G i \ Ai 2) ≤ αw(G i) + αw(G i \ (Ai 1 ∪ Ai 2)).

So, if we choose the gadgets according to the sign of the inequality, we will always replace a non-connected/odd/even strip with a non-connected/odd/even gadget, preventing from odd holes. Odd antiholes are not line graphs, so they will not appear in G ′.

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

The “odd” cliques problem

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

The “odd” cliques problem

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

The “odd” cliques problem

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

MWCC on strip composed perfect graphs

The outline of the algorithm is the following:

• 1. Find a strip decomposition of the graph (Faenza et al.)
• 2. Compute the values of an MWCC on four suitable induced

subgraphs of each strip.

• 3. Replace in the composition each strip by a weighted gadget

(the weight of the vertices will be a function of the values computed in 2., and the gadgets will depend on some parity issues in order to preserve perfection)

• 4. Obtain a weighted perfect line graph, and solve the MWCC

using, for instance, the primal-dual algorithm for maximum weight matching by Gabow (1990).

• 5. Using this clique cover and the ability of computing a MWCC
• n a strip, reconstruct a MWCC and a MWSS of the original

graph (analyzing the different cases).

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

MWCC on strip composed perfect graphs

Theorem

Let G(V , E) be the composition of strips Hi = (G i, Ai) i = 1 . . . , k w.r.t. a partition P. Suppose that for each i = 1, . . . , k we can compute a MWCC of Hi in time O(pi(|V |)). Then the MWCC problem on G can be solved in time O(k

i=1 pi(|V |) + match(|V |)), where match(|V |) is the time

required to solve the matching problem on a graph with V

• vertices. If pi(|V |) is polynomial for each i, then the MWCC can

be solved on G in polynomial time. This result can be considered as the MWCC version of the result

• n the MWSS by Oriolo Pietropaoli and Stauffer.

MWSS and MWCC MSS and MCC on claw-free perfect graphs MWCC on strip composed perfect graphs

Conclusions and future work

We have presented:

An algorithm for the MCC in claw-free perfect graphs which builds concurrently a MSS and a MCC. An algorithmic theorem for strip composed perfect graphs which states that we can compute a MWCC of a strip composed perfect graph if we can compute a MWCC on each strip. An algorithm which builds concurrently a MWCC and a MWSS on strip composed claw-free perfect graphs, which is an application of the previous result.

Future work:

We will try to obtain now an algorithm for the MWCC on the remaining case for claw-free perfect graphs, that is, distance claw-free perfect graphs. Ideally, one that builds concurrently a MWSS and a MWCC.