Stable Sets and Graphs with no Even Holes
SStable Sets and Graphs with no Even Holes
Michele Conforti , Bert Gerards , Kanstantsin Pashkovich December 6, 2018
Abstract
We develop decomposition/composition tools for efficiently solving maximum weightstable sets problems as well as for describing them as polynomially sized linear programs(using “compact systems”). Some of these are well-known but need some extra work toyield polynomial “decomposition schemes”.We apply the tools to graphs with no even hole and no cap. A hole is a chordless cycleof length greater than three and a cap is a hole together with an additional node that isadjacent to two adjacent nodes of the hole and that has no other neighbors on the hole.
A vast literature about efficiently solvable cases of the stable set problem focuses on “perfectgraphs”. Based on the ellipsoid method, Gr¨otschel, Lov´asz, and Schrijver [13] have developeda polynomial-time algorithm that computes a stable set of maximum weight in a perfectgraph. Perfect graphs have no odd holes. (A hole is a chordless cycle of length greater thanthree.) It is conceivable that the stable set problem is polynomially solvable for all graphswithout odd holes, and this may even extend to graphs with all holes having the same parity,so either all even or all odd. To our knowledge the case that all holes are odd has not receivedmuch attention and in this paper we take a first step in exploring this topic by considering“cap-free” graphs with no even holes. A cap is a hole together with an additional node thatis adjacent to two adjacent nodes of the hole and that has no other neighbors on the hole.
Theorem 1.1.
The stable set problem for cap-free graphs with no even holes is polynomiallysolvable.
The stable set polytope of a graph is the convex hull of the characteristic vectors of thestable sets of the graph. Linear descriptions of stable set polytopes require in the worstcase exponentially many inequalities and arbitrarily large coefficients (in minimum integerform), even for cap-free graphs with no even hole. However, for those graphs we can tamethe descriptions by allowing some extra variables. An extended formulation for a polytope P in R n is a system of inequalities Ax + By ≤ d such that P = { x ∈ R n : ∃ y [ Ax + By ≤ d ] } . An extended formulation for P is compact if its encoding has polynomial size in n . Dipartimento di Matematica, Universit`a degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy( [email protected] ). Supported by “Progetto di Eccellenza 2008–2009” of “Fondazione Cassa diRisparmio di Padova e Rovigo”. Centrum Wiskunde & Informatica, Amsterdam, The Netherlands ( [email protected] ). D´epartement de Math´ematique, Universit´e Libre de Bruxelles, Boulevard du Triomphe, B-1050 Brussels,Belgium. ( [email protected] ). Research partly done at: Dipartimento di Matematica,Universit`a degli Studi di Padova, Italy. a r X i v : . [ m a t h . C O ] J a n heorem 1.2. Stable set polytopes of cap-free graphs without even hole admit compact ex-tended formulations.
We develop decomposition/composition tools for solving maximum weight stable sets prob-lems. The working of such tool is that when a graph is decomposable into smaller partsaccording to the tool’s specifications, then that can be used to efficiently construct a solutionfor the whole from solutions for the parts. Some of these tools are well-known but needsome extra work to make them suitable as a component in polynomial-time algorithms. Wedevelop similar mechanisms for combining polynomially sized linear programs for stable setproblems on parts of a decomposition into such linear program for the whole.We apply these results to cap-free graphs with no even holes. Conforti, Cornu´ejols, Kapoor,and Vuˇskovi´c [6, 7, 8] give a decomposition theorem for graphs with no even hole and usethat to find even holes in polynomial time [9]. The following theorem is a simplified variantof the main result in [6].
Theorem 1.3 ([6, Theorem 4.1]) . Every cap-free graph with a triangle either admits an amal-gam decomposition or a clique cutset decomposition (both defined in Section 2) or containsa node adjacent to all other nodes.
So cap-free graphs with no even holes can be built from triangle-free graphs with no evenholes. Conforti, Cornu´ejols, Kapoor, and Vuˇskovi´c [7] prove that triangle-free graphs with noeven holes can be further decomposed into as simple graphs as “fans” and the 1-skeleton ofthe three dimensional cube. This is Theorem 2.18; as that result is a bit technical, we explainits details later, in Section 2.2. As we will see, all decompositions coming up in Theorems 1.3and 2.18 fall in our framework and thus, taking all together, we get Theorems 1.1 and 1.2.
Notation. A transversal of a collection A of disjoint nonempty sets is a set W ⊆ ∪A with | W ∩ X | = 1 for each X ∈ A .Let G = ( V, E ) be a graph. If
X, Y ⊆ V are disjoint and some node in X has a neighborin Y , then X and Y are adjacent . If each node in X is adjacent to all nodes in Y , then X is fully adjacent to Y . The set of nodes outside X that are adjacent to X is denotedby N G ( X ). The subgraph of G induced by X ⊆ V is G X . Moreover, G − X = G V \ X and B G ( X ) = N G ( V \ X ). If x ∈ V , we write G − x for G − { x } , N G ( x ) for N G ( { x } ), “ x is fullyadjacent to Y ” for “ { x } is fully adjacent to Y ”, etc.Let A be a collection of disjoint nonempty sets in G . Let G A denote the graph with asnodes the members of A and as edges the adjacent pairs in A . We call any graph isomorphicto G A a pattern of A . We call A a region of G , if each adjacent pair in A is fully adjacent;in that case, G W is a pattern of A in G , for every transversal W of A .If X is fully adjacent to N G ( X ) and X (cid:54) = ∅ , we call X a group of G . A partition of a set A into groups of G is a grouping of A in G .The collection of stable sets in a graph G = ( V, E ) is denoted by S [ G ]. The stabilitynumber α ( G ) of G is the size of the largest stable set in G . If X ⊆ V , then α ( X ) = α ( G X ).If w = ( w v : v ∈ V ) ∈ R V , then w X = ( w v : v ∈ X ) and w ( X ) = Σ v ∈ X w v . Given a graph G = ( V, E ) and a weighting ( w v : v ∈ V ), we consider the following problem:2ind in G , a stable set S , that maximizes w ( S ). ( Stable set problem )Suppose we are given a triple ( V , U, V ) of disjoint sets with union V and a grouping U of U in G V ∪ U such that V and V nonadjacent, | V ∪ U | > |U | and | V | >
0. We call such( V , U, V ) with U a node cutset separation of G . (Recall from Section 1, that U is a groupingmeans that each X ∈ U is fully adjacent to N G ( X ) ∩ V and that each pair in U is eithernonadjacent or fully adjacent.)Define for each stable set S (cid:48) in V ∪ U the value: correct ( S (cid:48) ) = max { w ( S (cid:48)(cid:48) ) : S (cid:48)(cid:48) ⊆ V , S (cid:48) ∪ S (cid:48)(cid:48) ∈ S [ G ] } . Then the maximum w ( S ) of a stable set S in G is equal to the maximum in the followingproblem.Find in G V ∪ U , a stable set S (cid:48) , that maximizes w ( S (cid:48) ) + correct ( S (cid:48) ). ( Master )Fix a transversal (cid:98) U of U . It is immaterial which particular transversal is chosen; our actualobject of interest is the graph G V ∪ (cid:98) U and (up to graph isomorphism) that does not dependon the choice of (cid:98) U —that is what “ U is a grouping of U in G V ∪ U ” means. We define for each B ⊆ U , the set (cid:98) B = { (cid:98) X : X ∈ B} and we define the map hom U : S [ G V ∪ U ] → S [ G (cid:98) U ] by: hom U ( S (cid:48) ) = { (cid:98) X : X ∈ U , S (cid:48) ∩ X (cid:54) = ∅} for each stable set S (cid:48) in V ∪ U .Now take any stable set S (cid:48) in V ∪ U . Then, for any set S (cid:48)(cid:48) in V , it is straightforwardto see that S (cid:48) ∪ S (cid:48)(cid:48) is a stable set in G if and only if hom U ( S (cid:48) ) ∪ S (cid:48)(cid:48) is a stable set in G V ∪ (cid:98) U .This means that correct ( S (cid:48) ) = service (cid:98) U ( hom U ( S (cid:48) )) , where the function T (cid:55)→ service (cid:98) U ( T ) on S [ (cid:98) U ] is given by the values w ( V ∩ S T ) of the stablesets S T determined by:Find in G V ∪ (cid:98) U , for each stable set T in (cid:98) U ,a stable set S T with (cid:98) U ∩ S T = T , that maximizes w ( V ∩ S T ). ( Servant )The discussion above implies the following results.max S ∈ S [ G ] w ( S ) = max S (cid:48) ∈ S [ G V ∪ U ] w ( S (cid:48) ) + correct ( S (cid:48) ) , (1)where correct = service (cid:98) U ◦ hom U and service (cid:98) U ( T ) = max S T ∈ S [ G V ∪ (cid:98) U ] , (cid:98) U∩ S T = T w ( V ∩ S T ) . (2) Fact 2.1.
Suppose we are given:(1) a solution ( S T : T ∈ S [ (cid:98) U ]) of the servant,(2) the function service (cid:98) U : T (cid:55)→ w ( V ∩ S T ) on S [ (cid:98) U ] ,
3) a solution S (cid:48) of the master with correct = service (cid:98) U ◦ hom U ,(4) S = S (cid:48) ∪ ( V ∩ S hom U ( S (cid:48) ) ) .Then S is a solution of the stable set problem on G with w . Fact 2.1 and the underlying formula (1) are seminal to the approach in this paper. It saysthat we can solve the stable set problem on G as follows: first, solve all the problems listedin the servant and substitute the results in the objective function of the master; after that,solve the master. We call this master/servant decomposition , and use that term freely at alllevels: for instance, we will call G V ∪ U , G V ∪ (cid:98) U a master/servant decomposition of G .Singletons are groups. The advantage of using larger groups is that that takes out replicationsin the list of problems making up the servant. For us that saving is crucial: The decompo-sitions in Theorem 1.3 come from node cutset separations with unbounded α ( U ) but with α ( (cid:98) U ) = 1; and that is a big difference: the servant comprises as many problems as there arestable sets in (cid:98) U , and that number lies between 2 α ( (cid:98) U ) and (cid:0) | (cid:98) U| α ( (cid:98) U ) (cid:1) α ( (cid:98) U ) . Our algorithms will allcome with an a priori bound α bound on α ( (cid:98) U ), but allow α ( U ) to be arbitrary high.Observe that for any node v ∈ U , the union of all groups in U that contain v is a group.So the inclusion-wise maximal groups in U form a grouping of U , we denote this grouping by U coarse . It is clearly “better than the rest”: the pattern G (cid:98) U coarse is a proper induced subgraphsof the pattern G (cid:98) U for any other U . Incidentally, note that it is not hard to find the maximalgroup in U containing a particular node v : starting with X = U , keep removing nodes from X that have not the same neighbors outside the current X as v until this is no longer possible.Then, X has become a group. As all groups in U containing v will have stayed in X duringthe procedure, X is the maximal group in U that contains v . Further decomposing the servant—rooted graphs.
Our approach will be to not onlyapply master/servant decomposition to the stable set problem on G but also to the problemsthat appear in the master and the servant. This is not without issues, both for the masteras the servant. We first consider the servant.The servant is a stable set problem on a rooted graph ; which in general is formulated as:Given a graph G = ( V, E ), a root Z ⊆ V , and a weighting ( w v : v ∈ V ):Find in G , for each stable set R in Z ,a stable set S R with Z ∩ S R = R , that maximizes w ( S R ).( Stable set problem with root )The servant is special in that it has weighting 0 on the root (because w ( S R ) = w ( S R \ Z ) if w is identical to 0 on Z ). We call a pair ( G, Z ) with Z ⊆ V a rooted graph. We call V \ Z the area of ( G, Z ) and of the stable set problem on (
G, Z ); if the area is empty we call (
G, Z ),and the stable set problem, trivial . The servant is never trivial.The stable set problem on a rooted graph is a list of stable set problems. That theservant has this “multiple-problem” feature becomes an issue when we further decomposethese subgraphs of the servant as if they were totally unrelated. We easily run into exponentialexplosion then; even if the root has as few as two stable sets, which is always the case when U is nonempty. That means that we can hardly iterate decomposing “on the servant side”.4e ran into this issue of the “multiple-problem” aspect of the servant when we wantedto use amalgam decompositions in Theorem 1.3 to design a polynomial time algorithm forthe stable set problem of cap-free graphs with no even holes. A standard way to addressthe issue is to avoid servant-graphs that can be further decomposed, for instance by taking V inclusion-wise minimal. Clique cutsets with V inclusion-wise minimal have that propertyand can be found efficiently (Whitesides [15]). But for the amalgam decompositions used inTheorem 1.3, this will not work. Cornu´ejols and Cunningham [10] gave a polynomial-timealgorithm to find an amalgam separation with minimal servant, but as illustrated by Figure 1,that does not guarantee that the amalgam blocks will have no amalgams. So, what then?Just forbidding to decompose “on the servant side” and ignore occasions that arise limits theapplicability of the approach too much—at least for our purposes.There is a way out: master/servant decomposition extends easily to rooted graphs ( G, Z )with Z ⊆ V ∪ U : in the master, just replace the graph G V ∪ U by the rooted graph ( G V ∪ U , Z ),but keep the same objective function S (cid:55)→ w ( S ) + correct ( S ) with correct = service (cid:98) U ◦ hom U ,where (cid:98) U is the same transversal of U and service (cid:98) U comes from the same servant, with thesame graph, the same root and the same weighting as before.Find in G V ∪ U , for each stable set R in Z ,a stable set S (cid:48) R with Z ∩ S (cid:48) R = R , that maximizes w ( S (cid:48) R ) + correct ( S (cid:48) R ).( Master with root )In terms of rooted graphs, (1) reads as:max S R ∈ S [ G ] , Z ∩ S R = R w ( S R ) = max S (cid:48) R ∈ S [ G V ∪ U ] , Z ∩ S (cid:48) R = R w ( S (cid:48) R ) + correct ( S (cid:48) R ) . (3)with the function correct = service (cid:98) U ◦ hom U is as in (2). Moreover: Fact 2.2.
The area of ( G, Z ) is the disjoint union of the area of the master and the servant. This is crucial to our approach. The stable set problem on G is the same as the stable setproblem on the rooted graph ( G, ∅ ). Starting from the rooted graph perspective, we reduce astable set problem on a rooted graph into one stable set problem on the rooted master-graphand one stable set problem on the rooted servant-graph. Regardless how often we repeat thismaster/servant decomposition for rooted graphs, by (2.2), the collective area of the list ofproblems constructed does not grow. We formalize this by “decomposition lists”.Suppose ( G, Z ) occurs in an left/right ordered list L of rooted graphs. Then a master/servantdecomposition of L along node cutset separation ( V , U, V ) with U of ( G, Z ) is a list obtainedfrom L by replacing ( G, Z ) by the rooted master graph ( G V ∪ U , Z ) (in the same position)and then inserting the rooted servant graph ( G V ∪ (cid:98) U , (cid:98) U ) anywhere further down the list (soto the right of where ( G V ∪ U , Z ) is).A master/servant decomposition-list of ( G, Z ) is either (( G , Z )), or (recursively) definedas a master/servant decomposition of a master/servant decomposition list of ( G, Z ). Lemma 2.3.
A master/servant decomposition list of a rooted graph with n nodes containsat most n non-trivial rooted graphs and at most n trivial rooted graphs. roof. Imagine a sequence of master/servant decompositions that leads from the rooted graphto the list. Trivial rooted graphs can not be decomposed, so for this analysis we ignore them.We visualize each non-trivial rooted graph encountered in the sequence as an area-root pairwith nonempty area and root of size at most n −
1. These area-root pairs behave as follows.Either we split an area into two disjoint nonempty areas and assign each of them with a root,each with at most n − n , we can apply at most n − n − n −
1, the“root-reduction” operation cannot be iterated more than n − Further decomposing the master—templates.
Suppose the master-graph G V ∪ U has anode cutset separation ( V (cid:48) , U (cid:48) , V (cid:48) ) with grouping U (cid:48) and Z ⊆ V (cid:48) ∪ U (cid:48) and we want to usethat for a master/servant decomposition of the master. Since the factor service (cid:98) U of the extraterm can be virtually anything (see Fact 2.9), we only do that if the separation fits U , whichmeans that:(i) U ⊆ V (cid:48) ∪ U (cid:48) or U ⊆ V (cid:48) ∪ U (cid:48) .(ii) if U meets V (cid:48) and X ∈ U meets U (cid:48) , then X ∩ U (cid:48) is the union of members of U (cid:48) .When the separation does fit U , we decompose the master as follows: The master-of-the-master has graph G V (cid:48) ∪ U (cid:48) , root Z , and objective function:( service (cid:98) U (cid:48) ◦ hom U (cid:48) )( S (cid:48) R ) + w ( S (cid:48) R ) + (cid:40) ( service (cid:98) U ◦ hom U )( S (cid:48) R ) if U does not meet V (cid:48) ,0 if U does meet V (cid:48) ,and the servant-of-the-master has graph G V (cid:48) ∪ (cid:99) U (cid:48) , root (cid:98) U (cid:48) , and objective function: w ( V (cid:48) ∩ S T ) + (cid:40) U does not meet V (cid:48) ,( service (cid:98) U ◦ hom U )( S (cid:48) R ) if U does meet V (cid:48) . If we next also decompose the master-of-the-master and servant-of-the-master and keep re-peating that, we will accumulate more and more extra terms of the form service (cid:98) A ◦ hom A .This leads to templates : triples ( G, Z,
Ω) where (
G, Z ) is a rooted graph and Ω a collectionof regions in G . Also note that the presence of these extra terms has no effect on how rootedgraphs are decomposed; the extra terms only prevent some separations to lead to decompo-sitions. This means that (2.2) and Lemma 2.3 will still apply.Suppose we are given a collection Ω of regions in G and for each of regions A ∈
Ω: a pat-tern with node set (cid:98) A , a graph isomorphism X → (cid:98) X A between G A and that pattern, and areal-valued function σ (cid:98) A on S [ (cid:98) A ]. Define, for B ⊆ A , the set (cid:98) B A = { (cid:98) X A : X ∈ B} , and for S ∈ S [ G ], the set hom A ( S ) = { (cid:98) X A : X ∈ A , S (cid:48) ∩ X (cid:54) = ∅} . We consider the following problem.Find in G , for each stable set R in Z , 6 stable set S R with Z ∩ S R = R , that maximizes w ( S R ) + (cid:88) A∈ Ω ( σ (cid:98) A ◦ hom A )( S R ) . ( Stable set problem on a template )Suppose also that our node cutset separation ( V , U, V ) with U and Z ⊆ V ∪ U fits eachregion in Ω. Then the maxima in the stable set problem on a template are the same as themaxima in the following problem.Find in G V ∪ U , for each stable set R in Z ,a stable set S (cid:48) R with Z ∩ S (cid:48) R = R , that maximizes w ( S (cid:48) R ) + (cid:88) A∈ Ω ,V ∩ ( ∪A )= ∅ ( σ (cid:98) A ◦ hom A )( S (cid:48) R ) + correct ( S (cid:48) R ),( Master for a template )where correct = service (cid:98) U ◦ hom U and T (cid:55)→ service (cid:98) U ( T ) on S [ (cid:98) U ] is given by the maxima in:Find in G V ∪ (cid:98) U , for each stable set T in (cid:98) U ,a stable set S T with S T ∩ Z = T , that maximizes w ( V ∩ S T ) + (cid:88) A∈ Ω ,V ∩ ( ∪A ) (cid:54) = ∅ ( σ (cid:98) A ◦ hom A )( S T ) . ( Servant for a template )Using templates, formula (1) extends to:max S R ∈ S [ G ] ,Z ∩ S R = R w ( S R ) + (cid:88) A∈ Ω ( σ (cid:98) A ◦ hom A )( S R ) == max S (cid:48) R ∈ S [ G V ∪ U ] ,Z ∩ S (cid:48) R = R w ( S (cid:48) R ) + (cid:88) A∈ Ω ,V ∩ ( ∪A )= ∅ ( σ (cid:98) A ◦ hom A )( S (cid:48) R ) + correct ( S (cid:48) R ) , (4)where correct = service (cid:98) U ◦ hom U andmax S R ∈ S [ G ] , Z ∩ S R = R service (cid:98) U ( S T ) = w ( V ∩ S T ) + (cid:88) A∈ Ω ,V ∩ ( ∪A ) (cid:54) = ∅ ( σ (cid:98) A ◦ hom A )( S T ) . (5)Also Fact 2.1 generalizes to templates. Fact 2.4.
Suppose we are given:(1) a solution ( S T : T ∈ S [ (cid:98) U ]) of the servant for a template,(2) the function service (cid:98) U : T (cid:55)→ w ( V ∩ S T ) + (cid:80) A∈ Ω , V ∩ ( ∪A ) (cid:54) = ∅ ( σ (cid:98) A ◦ hom A )( S T ) on S [ (cid:98) U ] ,(3) a solution ( S (cid:48) R : R ∈ S [ Z ]) of the master for a template with correct = service (cid:98) U ◦ hom U ,(4) S R = S (cid:48) R ∪ ( V ∩ S hom U ( S (cid:48) R ) for each R ∈ S [ Z ] .Then ( S R : R ∈ S [ Z ]) is a solution of the stable set problem on a template. And we have the following.
Fact 2.5.
The master for a template and the servant for a template are both stable setproblems on a template. roof. For the master this is obvious: each region that does not meet V —and that includesalso U —is a region in the master-graph. For the servant the situation is slightly subtle. Theregions in Ω that meet V are regions of G V ∪ U , but not of G V ∪ (cid:98) U . However, if we replace themembers of any such region by their intersection with V ∪ (cid:98) U , we do get a region of G V ∪ (cid:98) U ;for the formula in (5) that replacement has no effect.This extension of the master/servant decomposition to stable set problems on templates“closes” our model. Theorem 2.6.
Let R be a family of templates closed under master/servant decompositionand P be a subfamily of R . If there is a polynomial time that finds a fitting node cutsetseparation for any template from R\P and there is a polynomial time algorithm for thestable set problem on templates from P , then there exists a polynomial time algorithm for thestable set problem on templates from R .Proof. Suppose we are given a stable set problem on a template ( G ◦ , Z ◦ , Ω ◦ ) ∈ R . To solveit, we keep a right/left ordered list of stable set problems on templates ( G, Z,
Ω) such thatthe underlying list of rooted graphs (
G, Z ) is a master/servant decomposition list of rootedgraphs. Initially this list consists of just the single stable set problem on ( G ◦ , Z ◦ , Ω ◦ ).For any template ( G, Z,
Ω) on the list, the regions
A ∈
Ω all come with a real valuedfunction σ (cid:98) A on S [ (cid:98) A ], which is either given by an explicit listing of all function values σ (cid:98) A ( T )with T ∈ S [ (cid:98) A ] or by σ (cid:98) A = service (cid:98) A , where the function values of service (cid:98) A are the maxima ofa stable set problem with template ( G (cid:48) , (cid:98) A , Ω (cid:48) ) further down the list (so to the right of where( G, Z,
Ω) is). As soon as the solution of that stable set problem on ( G (cid:48) , (cid:98) A , Ω (cid:48) ) comes available,we store the solution by an explicit listing of the values service (cid:98) A ( T ) and remove the stableset problem on ( G (cid:48) , (cid:98) A , Ω (cid:48) ) from the list. As of that moment σ (cid:98) A is represented by the storedexplicit listing. To solve that stable set problem on ( G (cid:48) , (cid:98) A , Ω (cid:48) ), we need an explicit listing foreach σ (cid:98) A (cid:48) with A (cid:48) ∈ Ω (cid:48) . The right most problem on the list has that property. Therefore, wealways try find a solution to that right most problem.So the algorithm is to iterate the following procedure: Remove the right most stable setproblem from the list and either decompose that problem and place the master and servantin that order at the end of the list, or solve the removed problem and store its solution. Bythe given decomposition algorithm for R\P and the given optimization algorithm for P wecan carry out each iteration in polynomial time. By Lemma 2.3 we can iterate the procedureat most 2 n + n times if G ◦ has n nodes. So after at most 2 n + n iterations the list is empty.That means that we stored an explicit listing of the solution of the first problem in the list.Since throughout the entire algorithm the first problem keeps the initial root Z ◦ , that explicitlisting solves the original stable set problem on ( G ◦ , Z ◦ , Ω ◦ ). Linearized decomposition.
Instead of carrying around the nonlinear terms σ (cid:98) A ◦ hom A , wecan “linearize” them by adding suitably weighted nodes that are adjacent to nodes in ∪A .We are free in choosing which terms σ (cid:98) A ◦ hom A we eliminate in this way and when we do that.Our algorithms in this paper either not use the option at all or do it at each decompositionat once, as part of the decomposition procedure.We say that a triple [ H, γ, σ ] linearizes a real-valued function d on S [ A ], if H = ( W, F ) is a8raph with A ⊆ W , γ ∈ R W , σ ∈ R such that for each T ∈ S [ A ], the maximum value γ ( S T )of stable set in H with A ∩ S T = T is equal to d ( T ) − σ .To see the relevance of this definition, suppose that W ∩ V = U and that [ H, γ, σ ] linearizes σ (cid:98) U ◦ hom U . Consider the graph G V ∪ U ∪ H = ( V ∪ W, E ∪ F ). Define γ v = 0 if v ∈ V \ W . Fact 2.7. S + ∈ S [ G V ∪ U ∪ H ] with Z ∩ S + = R maximizes w ( V ∩ S + ) + γ ( W ∩ S + ) , then S R = V ∩ S + is a stable set in G with Z ∩ S R = R that maximizes w ( S R ) + ( σ (cid:98) U ◦ hom U )( S R ) . So we can use a triple [
H, γ, σ ] linearizing σ (cid:98) U ◦ hom U to reformulate the master problem sothat the objective function is linear. This linearization comes with the expense of addingnodes (unless W = U ).A canonical way to linearize functions on S [ A ] with A ⊆ V is to “add a record of S [ A ] to G A ”. Adding a record of S [ A ] to a graph G , means to add a clique A record (the record ) consistingof new nodes r T , one for each T ∈ S [ A ], such that each r T is fully adjacent to A \ T andnonadjacent to T and to V \ A . We call the new graph the record graph of G and A , and wedenote it by G ( A ). Records play a major role in this paper. The following fact only uses therecord graph G A ( A ) of G A and A . Fact 2.8.
Let d be a real-valued function on S [ G A ] and let γ v = 0 if v ∈ U and γ r T = d ( T ) if T ∈ S [ U ] . Then [ G A ( A ) , γ, linearizes d if and only of d is nonnegative and inclusion-wisenon-increasing on S [ G A ] .Proof. For each T ∈ S [ U ], the stable sets in G ( A ) that meet A in T are: the set T with γ ( T ) = 0, and for each T (cid:48) ∈ S [ A ] with T (cid:48) ⊇ T , the set T ∪ { r T (cid:48) } with γ ( T ∪ { r T (cid:48) } ) = d ( T (cid:48) ).The maximum of these weights is d ( T ) if and only if d ( T ) ≥ d ( T ) ≥ d ( T (cid:48) ) for T (cid:48) ⊆ T ,as claimed.Incidentally, Fact 2.8 characterizes the functions that can turn up as solution service (cid:98) U of theservant: each nonnegative and inclusion-wise non-increasing function on S [ (cid:98) U ]. Fact 2.9.
The solution service (cid:98) U of the servant is nonnegative and inclusion-wise non-increasingon S [ (cid:98) U ] . Moreover, every nonnegative, inclusion-wise non-increasing function on the stablesets of a graph arises in this way.Proof. That service (cid:98) U is nonnegative and non-increasing is obvious. For the second statementjust take G such that G V is a record of G (cid:98) U and apply Fact 2.8.Actually, any function on S [ G A ] is the sum of a constant function, a linear function and anonnegative non-increasing function. So by by toggling in [ G A ( A ) , δ,
0] the values δ v with v ∈ A (which are 0 in Fact 2.8) and the third entry (which is also 0 in Fact 2.8), we can findlinearizations for any function on S [ A ].We say that H = ( W, F ) linearizes G U with U , if W ∩ V = U and H U = G U , and for everynonnegative and inclusion-wise non-increasing function σ (cid:98) U on S [ (cid:98) U ], there exist γ ∈ R W , σ ∈ R , such that [ H, γ, σ ] linearizes σ (cid:98) U ◦ hom U .So, instead of taking the term ( service (cid:98) U ◦ hom U )( S R ) into the objective function of the master,we can replace the master-graph by the linearized master-graph G V ∪ U ∪ H and the term9 service (cid:98) U ◦ hom U )( S R ) by γ ( W ∩ S + ). This gives the following alternative for the master.Find in G V ∪ U ∪ H , for each stable set R in Z ,a stable set S + R with Z ∩ S + R = R , that maximizes w (( V ∪ U ) ∩ S + R ) + γ ( W ∩ S + R ).( Linearized master with root )The linearized master/servant decomposition consists of this “linearized master with root”together with the original servant with servant-graph G V ∪ (cid:98) U , root (cid:98) U and objective function S T (cid:55)→ w ( V ∩ S T ).Note that as defined the linearized master may well be larger then G . If | W \ U | ≤ τ < | V | ,we speak of a τ -linearized cutset decomposition and we call ( V , U, V ) with U a τ -linearizablecutset separation and U a τ -linearizable cutset . We will use τ -linearized decomposition with τ ≤ G . Mind that the τ -linearized master need not be an induced subgraph of G , not even for τ = 1. The servant-graph, as always, is a proper induced subgraph of G .We use τ -linearized decompositions for algorithms in the same way as decribed for templatesabove. When running an algorithm using τ -linearized decomposition, we will generate aleft/right ordered list of rooted graphs as before, except that now we use linearized masters.Consequently also the analysis of the running time is almost the same. Almost! Lemma 2.3does not refer to τ -linearized decompositions with τ ≥ / Templates with singleton-regions.
Each algorithm in this paper either only use linearizeddecompositions or only template decomposition, so without adding extra nodes. Our templatedecompositions only come from separations where the grouping consists of singletons. Insuch a setting, we denote a region A just by its union A = A ; in line of that, we then denotetemplates as triples ( G, Z,
Ω) where Ω consists of subsets of V .If Ω is a collection of sets in V , then G (Ω) denotes the graph obtained by adding a record W record for each W ∈ Ω. If C is a class of triples ( G, Z,
Ω) where (
G, Z ) s a rooted graph and Ωa collection of subsets in V , then C record denotes all rooted graphs ( G (Ω) , Z ) with ( G, Z, Ω) ∈ C . Outline.
In Section 2.2, we consider node cutsets that induce a 3-node path. A 3-node pathhas 5 stable sets, so we get records on 5 nodes then. Five-node records are already quite big,but for the graphs considered in Section 2.2 they work out fine, see Section 2.2. Besides these“3-node path inducing” cutsets, all node cutsets we use are 1-linearizable. We analyze thosein Section 2.1 and our usage of node cutsets inducing a 3-node path in Section 2.2. / -linearized decompositions Lemma 2.3 implies that a decomposition along 0-linearizable cutsets U , will generate aquadratic of number of graphs. To understand the structure of these cutsets, observe that,for [ G U , γ, σ ] to linearize a non-increasing function d on (cid:98) U forces the values σ = ¯ d ( (cid:98) ∅ ) and γ u = ¯ d ( (cid:100) { u } ) − ¯ d ( (cid:98) ∅ ) for all u ∈ U . So [ G U , γ, σ ] linearizes d when U is a clique, but nototherwise: if a and b are nonadjacent nodes in U , then the function S (cid:55)→ ¯ d ( S ) that takesvalue 1 if S = (cid:98) ∅ and 0 otherwise, has − d ( (cid:92) { a, b } ) − σ = γ a + γ b = −
2; which is absurd. If10 is a clique and V and V are both nonempty, then U is a clique cutset . If U is a clique and V = ∅ and |U | < | U | , then U contains pair of adjacent twins u, v ( u, v are twins if they havethe same neighbors in V \{ u, v } ). Actually if u, v are adjacent twins, then ( ∅ , { u, v } , V \{ u, v } )is a 0-linearizable separation. All-in-all, clique cutsets and adjacent twins are 0-linearizablecutsets and, conversely, each 0-linearizable cutset is a clique cutset or has an adjacent twin.Since clique cutsets can be found in polynomial time (Whitesides [15]) we can get the follow-ing result of Whitesides [15] from Lemma 2.3. We skip the proof as it is a simpler version ofwhat is written in the proofs of Theorems 2.14 and 2.17. Corollary 2.10 (Whitesides [15]) . Let G be a class of graphs closed under clique cutsetdecomposition. If P ⊆ G contains all members of G without clique cutsets, then the stable setproblem on graphs in G is solvable in polynomial time if and only if the stable set problem ongraphs in P is solvable in polynomial time. So 0-linearized decompositions of rooted graphs are well-understood: the node cutsets arecliques, they can be found in polynomial time, and the 0-linearized decomposition-lists haveonly quadratically many members and use only proper induced subgraphs. The same is truefor 1-linearized decompositions, except that it is not the node cutset but only the servant-rootthat is guaranteed to be a clique.
Lemma 2.11.
Let H be a graph on U ∪ { r } with r (cid:54)∈ U and let U be a a partition of U in H . Let P be the union of the two element stable sets in U . Then H linearizes H U with U ifand only if P is fully adjacent to r and lies in a set A ∈ U ∪ {∅} .Proof. First assume that H linearizes H U with U . Recall, that [ H, γ, σ ] linearizing a nonneg-ative non-increasing function d on S [ (cid:98) U ] means that γ ( S ) = ¯ d ( (cid:98) S ) − σ if S ∈ S [ U ] is adjacent to r, (6)max { , γ r } + γ ( S ) = ¯ d ( (cid:98) S ) − σ if S ∈ S [ U ] is not adjacent to r. (7)Applying this to stable sets with at most one element, this forces the values: σ = ¯ d ( (cid:98) ∅ ) − max { , γ r } and γ u = (cid:40) ¯ d ( (cid:100) { u } ) − σ if u ∈ U is adjacent to r, ¯ d ( (cid:100) { u } ) − ¯ d ( (cid:98) ∅ ) if u ∈ U is not adjacent to r. (8)Substituting these back in (6) and (7) gives for any nonadjacent pair u, v in U the identity:¯ d ( (cid:92) { u, v } ) + ¯ d ( (cid:98) ∅ ) − ¯ d ( (cid:100) { u } ) − ¯ d ( (cid:100) { v } ) = (cid:40) max { , γ r } if u and v are adjacent to r, u or v is not adjacent to r. (9)Consider the nonnegative and non-increasing function ¯ d on S [ (cid:98) U ] given by ¯ d ( (cid:98) ∅ ) = 3 , ¯ d ( (cid:100) { s } ) = 2for all s ∈ P , and ¯ d ( (cid:98) S ) = 0 for all other stable sets in U . Consider any nonadjacent pair u, v in U . The definition of ¯ d implies that¯ d ( (cid:92) { u, v } ) ∈ { , } and ¯ d ( (cid:92) { u, v } ) + ¯ d ( (cid:98) ∅ ) − ¯ d ( (cid:100) { u } ) − ¯ d ( (cid:100) { v } ) = ¯ d ( (cid:92) { u, v } ) − . So, we see from (9) that u and v are both adjacent to r , that ¯ d ( (cid:92) { u, v } ) = 2, and that (cid:92) { u, v } = (cid:100) { s } for some s ∈ P . However, (cid:92) { u, v } = (cid:100) { s } is only possible when (cid:100) { u } = (cid:100) { s } = (cid:100) { v } . That means that11he expression ¯ d ( (cid:92) { u, v } ) + d (cid:98) ∅ − ¯ d ( (cid:100) { u } ) − ¯ d ( (cid:100) { v } )) is identical to ¯ d ( (cid:98) ∅ ) − ¯ d ( (cid:100) { u } )—for all d , not justfor ¯ d . Hence, for every nonadjacent u, v ∈ U , condition (9) reads:max { , γ r } = ¯ d ( (cid:98) ∅ ) − ¯ d ( (cid:100) { u } ) . (10)As this does not depend on v , this condition on γ r applies to each u ∈ P . So the function u (cid:55)→ ¯ d ( (cid:100) { u } ) is constant on P . This must hold for each nonnegative non-increasing function d on S [ (cid:98) U ]. This can only be the case if (cid:100) { u } is the same set for all u ∈ P . Denote that set by A ;it is as claimed in the lemma.Now assume P is fully adjacent to r and that there is a set A ∈ U ∪ {∅} containing P . Let d be a nonnegative and non-increasing function on (cid:98) U . Define σ and γ u with u ∈ U such that: σ = ¯ d ( (cid:98) A ) , γ u = ¯ d ( (cid:98) ∅ ) − ¯ d ( (cid:98) A ) if u = r, ¯ d ( (cid:100) { u } ) − ¯ d ( (cid:98) A ) if u ∈ U is adjacent to r , ¯ d ( (cid:100) { u } ) − ¯ d ( (cid:98) ∅ ) if u ∈ U is not adjacent to r . Note that (cid:100) { u } = (cid:98) A if u ∈ A . So γ u = 0 if u ∈ P . Since ¯ d ( (cid:98) ∅ ) ≥ ¯ d ( (cid:98) A ), it is straightforward tosee that [ H, γ, σ ] linearizes d . Lemma 2.12.
Let ( V , U, V ) be a node cutset separation with grouping U of rooted graph ( G, Z ) .If ( V , U, V ) is a -linearizable cutset separation with grouping U , then there exist threedisjoint (possibly empty) sets A , K, A in V with the following properties:(a) K is a clique, A = U \ K and A ⊆ V .(b) A is fully adjacent to K ∪ A and not adjacent to V \ A .Conversely, if A , K, A are disjoint sets in V satisfying (a) and (b), then the following hold. • For each r ∈ A , the pair ( G V ∪ U ∪{ r } , Z ) , ( G V ∪ (cid:98) U , (cid:98) U ) is a -linearized node cutset de-composition such that G V ∪ U ∪{ r } and G V ∪ (cid:98) U are proper induced subgraphs of G and theservant-root (cid:98) U is a clique. • If A = ∅ , then ( G V ∪ U , Z ) , ( G V ∪ K , K ) is a -linearized node cutset decomposition suchthat G V ∪ U and G V ∪ K are proper induced subgraphs of G and the servant-root K is aclique.Proof. If ( V , U, V ) is a 1-linearizable cutset separation with U , it follows from Lemma 2.11,that there exists A ∈ U ∪ {∅} , so that K = U \ A is a clique that is fully adjacent to A .Let A = N G ( A ) ∩ V . Since A is a group in G V ∪ U , the sets A , K, A satisfy (a) and (b).Now, suppose A , K, A satisfy (a) and (b). If r ∈ A , then by Lemma 2.11 H = G U ∪{ r } linearizes G U with U . If A = ∅ , then K is a clique cutset. The rest is straightforward.A graph on n nodes with a clique of size n − near-clique . We call a decompositionof a rooted graph ( G, Z ) proper if G is not a near-clique. A decomposition list is proper if it is obtained by proper decompositions only. For G = ( V, E ) and Z ⊆ V , we define load ( G, Z ) = | V \ Z | −
1. 12 emma 2.13.
A proper -linearized decomposition-list of a rooted graph with n nodes and aclique as root has most n members.Proof. Consider a proper 1-linearized decomposition-list G . Let G > consist of the membersof G with positive load. We analyze the impact of a single proper 1-linearized decompositionin G on the following parameters: • The total load ( G > ) = (cid:80) ( G (cid:48) ,Z (cid:48) ) ∈G > load ( G (cid:48) , Z (cid:48) ) of the positive loads in G . • The number |G > | of members of G with positive load. • The total root - size ( G > ) = (cid:80) ( G (cid:48) ,Z (cid:48) ) ∈G > | Z (cid:48) | of the members of G with positive load.Clearly, these numbers satisfy: |G > | ≤ load ( G > ) . (11)Consider a proper 1-linearized decomposition ( G , Z ), ( G , (cid:98) U ) of a member ( G, Z ) ∈ G comingfrom a separation ( V , U, V ) of ( G, Z ) with a grouping U . Then Z is a clique and G is not anear-clique. Recall from the definition of τ -linearized decompositions that G = G V ∪ U ∪{ r } for some node r ∈ V , if U is not a clique, and that G = G V ∪ U , otherwise. This gives thefollowing identity: load ( G , Z ) + load ( G , (cid:98) U ) = (cid:40) load ( G, Z ) − U is a clique, load ( G, Z ) if U is not a clique . (12)We distinguish between special decompositions, when load ( G , Z ) = −
1, and normal decom-positions, when load ( G , Z ) ≥
0. First we analyze special decompositions. For those G hasall nodes in Z , so U is a clique. Hence (12) gives: load ( G , (cid:98) U ) = load ( G, Z ), which is positive.Since | V ∪ U | > | (cid:98) U | and Z = V ∪ U , the root of ( G , (cid:98) U ) is smaller than the root of ( G, Z ).Hence a special decomposition in G gives a list H with: load ( H > ) = load ( G > ) , |H > | = |G > | , root - size ( H > ) < root - size ( G > ) . (13)Next we analyze normal decompositions in G , so when load ( G , Z ) ≥
0. Since Z is a cliqueand G is not a near-clique, we have that load ( G, Z ) ≥
2. So, by (12), at least one of ( G , Z )and ( G , (cid:98) U ) has positive load. So a normal decomposition in G gives a list H with: load ( H > ) ≤ load ( G > ) , |H > | ≥ |G > | , root - size ( H > ) ≤ root - size ( G > ) + n − . (14)Moreover: load ( H > ) < load ( G > ) or |H > | > |G > | . (15)Indeed, if load ( H > ) = load ( G > ), then load ( G , Z ) + load ( G , (cid:98) U ) = load ( G, Z ), so by (12), U is not a clique. That means that load ( G , (cid:98) U ) = | V | − ≥ U contains a nodethat is not in Z . Since the extra node in G is not in Z and not in U , we see that also load ( G , Z ) ≥
1. Hence, |H > | > |G > | . This proves (15).Now (11)-(15) tell that when decomposing a rooted graph on n nodes by proper 1-linearizeddecompositions we will make at most n normal decompositions, and that we will, over time,create no more than n ( n −
1) root nodes. So we do at most n ( n −
1) special decompositions.So |G| ≤ n + n ( n −
1) = n , as claimed. 13 heorem 2.14. Let R and P be classes of rooted graphs such that there exists a polynomial-time algorithm that for each input from R\P finds a proper -linearized decomposition into R and such that the stable set problem on rooted graphs in P is solvable in polynomial time.Then the stable set problem on rooted graphs in R is solvable in polynomial time.Proof. It follows from Lemma 2.13, that there is a polynomial time algorithm that finds forany input (
G, Z ) ∈ R on n nodes a proper 1-linearized decomposition list { ( G , Z ) , . . . , ( G m , Z m ) } in P with m ≤ n . With such list at hand, a stable set problem on ( G, Z ) reducesto solving a stable set problem on each of the rooted graphs { ( G , Z ) , . . . , ( G m , Z m ) } . Foreach i = 1 , . . . , m the node weights for the stable set problem on for ( G i , Z i ) can be determinedfrom the solutions of the problems on { ( G i +1 , Z i +1 ) , . . . , ( G m , Z m ) } . As each ( G i , Z i ) is in P , we can solve the full list of problems in polynomial-time, provided that we do that goingfrom right-to-left along the list, starting by ( G m , Z m ) and ending with ( G , Z ). Having afound a solution to all problems on the list, we can construct a solution of the original stableset problem on ( G, Z ) by scanning the list of solutions from left-to-right.
Amalgam decomposition
An array ( V , A , K, A , V ) of disjoint sets with union V is an amalgam separation for G =( V, E ), with amalgam ( A , K, A ), if it has the following properties: • A and A are nonempty fully adjacent sets. • K is a (possibly empty) clique that is fully adjacent to A ∪ A . • V is not adjacent to V ∪ A and V is not adjacent to V ∪ A . • | V ∪ A | ≥ | V ∪ A | ≥ K has neighbors in V ∪ V .Amalgams were introduced by Burlet and Fonlupt [2] to design a polynomial-time algorithmto recognize Meyniel graphs, a special class of perfect graphs. Cunningham and Cornu´ejols[10] designed a polynomial-time algorithm that finds an amalgam separation or decides thatnone exists. In [2, 10], a graph with an amalgam separation ( V , A , K, A , V ) is decomposedinto two amalgam blocks G and G , where, for both i = 1 and i = 2, the graph G i is obtainedfrom G V i ∪ A i ∪ K by adding a single new node that is fully adjacent to A i ∪ K . We call the pair G , G an amalgam decomposition of G corresponding to ( A , K, A ).Amalgams give rise to 1-linearizable cutsets and amalgam decompositions are 1-linearizeddecompositions; this is the next lemma. Lemma 2.15.
Let ( A , K, A ) be an amalgam of a graph G . Then A ∪ K is a -linearizablecutset of ( G, ∅ ) with grouping U = { A } ∪ {{ u } : u ∈ K } . Moreover, ( G, ∅ ) has a -linearized decomposition such that the master-graph and the server-graph form an amalgamdecomposition of G corresponding to amalgam ( A , K, A ) , the master-root is empty, and theservant-root is the clique (cid:98) U . roof. Let ( V , A , K, A , V ) be an amalgam separation. Then ( V , A ∪ K, V ∪ A ) is a nodecutset separation of ( G, ∅ ) with grouping U = { A } ∪ {{ u } : u ∈ K } . Observe that A , K, A satisfy (a) and (b) of Lemma 2.12 with respect to that node cutset separation. So A ∪ K is 1-linearizable. By the definition of amalgam, A (cid:54) = ∅ . Take r ∈ A . Now the secondpart of Lemma 2.12 tells that the pair ( G V ∪ A ∪ K ∪{ r } , ∅ ), ( G V ∪ A ∪ (cid:98) U , (cid:98) U ) is a 1-linearizedcutset decomposition. It is obvious that G V ∪ A ∪ K ∪{ r } and G V ∪ A ∪ (cid:98) U = G V ∪ A ∪ K ∪{ (cid:99) A } is an amalgam decomposition corresponding to amalgam decomposition ( V , A , K, A , V ).Clearly, (cid:98) U is a clique. Corollary 2.16. If G is an amalgam decomposition-list of a graph G = ( V, E ) , then |G| ≤| V | .Proof. By Lemma 2.15, there is a map H (cid:55)→ Z H from G to cliques, so that { ( H, Z H ) : H ∈ G} is a 1-linearized decomposition-list of ( G, ∅ ). Now Lemma 2.13 yields |G| ≤ | V | .Theorem 1.3 uses amalgams to describe the structure of cap-free graphs with no even holes.When we wanted to use that to design an algorithm for the stable set problem on thesegraphs, we ran into the multiple-problem aspect of the servant. Cornu´ejols and Cunningham[10] can find an amalgam separation with minimal servant, but as illustrated by Figure 1,that does not guarantee that the amalgam blocks will have no amalgams. This lead us to therooted graph approach of this paper. It is essential here and to our knowledge new; till todaywe know of no other way to use amalgam decompositions in polynomial-time algorithms forfinding maximum weight stable sets. Theorem 2.17.
Let G be a class of graphs closed under amalgam decomposition. If P ⊆ G contains all near-cliques and all members of G without amalgams, then the stable set problemon graphs in G is solvable in polynomial time if and only if the stable set problem on graphsin P is solvable in polynomial time.Proof. Let R be the class of rooted graphs ( G, Z ) such that G ∈ G and Z is a clique. Thestable set problem on a graph G ∈ G is a rooted stable set problem on ( G, ∅ ), which is in R . So it suffices to prove that the stable set problem on rooted graphs in R is solvable inpolynomial time. Let Q the class of rooted graphs ( G, Z ) with G ∈ P and such that Z is aclique. As a clique Z has only | Z | + 1 stable sets, the stable set problem on rooted graphs in Q is solvable in polynomial time. So by Theorem 2.14 it suffices to design a polynomial-timealgorithm that finds a proper 1-linearized decomposition for any rooted graph ( G, Z ) in R\ Q .Here is this algorithm: If G is a near-clique, ( G, Z ) is in Q . Otherwise, use the algorithmof Cornu´ejols and Cunningham [10] to search for amalgams in G . If none is found: G ∈ P ,so in ( G, Z ) ∈ Q . If an amalgam separation ( V , A , K, A , V ) is found, proceed as follows tofind a 1-linearized decomposition for ( G, Z ), which will be proper as G is not a near-clique.If the clique Z meets V ∪ V , the root Z is contained in V ∪ A ∪ K or in V ∪ A ∪ K , so thenone of the node cutsets A ∪ K and A ∪ K yields a 1-linearized decomposition of ( G, Z ). If Z does not meet V ∪ V , it lies in A ∪ K ∪ A and thus K ∪ Z is a clique. Since G is not anear-clique, it has at least 3 nodes outside K ∪ Z . Assume two of those lie in A ∪ V . Thenthe node cutset A ∪ K ∪ Z yields a 1-linearized decomposition of ( G, Z ).15 ba u Figure 1: The amalgam of the graph on the left is unique (in this case the clique K is empty).Its blocks have amalgams, for instance ( { a , a } , { u } , { b } ). To prove Theorem 1.1 we use Theorem 1.3, which decomposes cap-free graphs into triangle-free graphs, and Theorem 2.18, which further decomposes triangle-free “odd-signable” graphsinto “the cube” and “fan-templates”.A graph is odd-signable if it contains a set F of edges such that | F ∩ C | is odd for eachchordless cycle C . Triangle-free graphs with no even holes are clearly odd-signable. The cube is the unique 3-regular bipartite graph on 8 vertices, so that is the 1-skeleton ofthe three dimensional cube. The cube is odd-signable.A fan with base ( u, c, v ) consists of an uv -path P together with a node c adjacent to asubset of nodes of P including u and v . If Z is a subset of the base of a fan G = ( V, E )and Ω is a collection of triples in V , then we call ( G, Z,
Ω) a fan-template . If the triples in Ωeach induce a subpath of one of the holes of G , we call the fan-template good . The followingresults say that good fans do come up in decomposing cap-free odd-signable graphs and thatthey are well tractable. Theorem 2.18 ([7, Theorems 2.4 and 6.4]) . If G = ( V, E ) is a triangle-free odd-signablegraph that is not isomorphic to the cube and has no clique cutset, then the template ( G, ∅ , ∅ ) can in polynomial time be decomposed into list of at most | V | good fan-templates.Proof. By [7, Theorem 2.4], G has no induced subgraph isomorphic to the cube. Now [7,Theorem 6.4] says that G can be obtained from the hole by a sequence of “good ear additions”(defined in [7, Definition 6.1]). An ear addition is the reverse of a node cutset decompositionwhere the servant graph is a fan and the node cutset is the base of the fan. So reversing thesequence of good ear additions amounts to a decomposition of G into fan-templates. Theregions of these templates are the locations where the ears are added and the goodness of theseear additions means that the fan-templates are good. Since adding an ear increases the sizeof the graph, we obtain at most at most | V | good fan-templates. As the node cutsets neededfor this decomposition are triples, we can find them in polynomial time, by enumeration.16 emma 2.19. If ( G, Z, Ω) is a good fan-template with base ( u, c, v ) and n nodes, then G (Ω) − c can be decomposed along 2-node clique cutsets into a list of at most | Ω | + n graphs, each withat most 8 nodes.Proof. The graph G (Ω) − c consists of the path G − c together with all the records. As eachregion of a fan-template is a 3-node subpath of one of the holes of G , each record has 5 nodesand is attached in G (Ω) − c to a 2- or 3-node subpath of G (Ω) − c . Note that each edge ofthe path G (Ω) − c forms a 2-node clique cutset of G (Ω) − c (except maybe the first or thelast edge of G (Ω) − c ). If we decompose G (Ω) − c along all these 2-node clique cutsets, weobtain collection of graphs, each consisting of a 2- or 3-node subpath of G (Ω) − c togetherwith at most one of the records. Such graphs have at most 8 nodes. Lemma 2.20.
The stable set problem on rooted record graphs of good fan-templates is solvablein polynomial time.Proof.
Let ( u, c, v ) be the base of a good fan-template (
G, Z,
Ω). By Corollary 2.10 andLemma 2.19, for each of the (at most 5) stable sets T in Z , we can find in polynomial time, astable set S T in G (Ω) that has maximum weight among those that intersect Z in T . Amongthese stable sets S T , we choose the best one. Theorem 2.21.
The stable set problem on cap-free odd-signable graphs is solvable in poly-nomial time.Proof.
If node u in graph G is adjacent to all other nodes, then we can set it aside to compareit with a maximum weight stable set in G − u , once we found that. The stable set problemon the cube can be found by enumeration. So the result follows from Theorems 1.3, 2.17,2.6, 2.18, Corollary 2.10, and Lemma 2.20. We examine extended formulations for the stable set polytope of a graph that admits certaindecompositions into smaller graphs and combine formulations for these smaller parts to onefor the whole graph. We apply this to cap-free odd-signable graphs and thus prove Theorem 2.
Notation.
We denote the convex hull of characteristic vectors of stable sets in a graph G by P [ G ]. If L is a collection of cliques in G , we denote the collection of stable sets in G that intersect each member of L by S [ G, L ] and the convex hull of the characteristic vectorsof these stable sets by P [ G, L ]. So P [ G ] = P [ G, ∅ ] and P [ G, L ] is the face of P [ G ] obtainedby setting at equality all clique constraints associated to the cliques in L . If x ∈ R V and H = G U , we denote the restriction of x to U by x H , of by x U . Extra variables used in extended formulations here mostly come from records.Consider a set of nodes U in a graph G . Recall from Section 2, that G ( U ) denotes thegraph obtained by adding to G a clique U record (the record) consisting of new nodes r S , onefor each S ∈ S [ G U ], and connect each such r S to all nodes in U \ S . We denote by L ( U ) the17ollection consisting of the clique U record together with all the cliques { v }∪{ r T : v (cid:54)∈ T ∈ S [ G U ] } with v ∈ U .The following result says that any (extended) formulation for P [ G ( U ) , L ( U )] is an ex-tended formulation for P [ G ]. Lemma 3.1.
Let U be a set of nodes in a graph G . Then each stable set S in G has a uniqueextension to a member of S [ G ( U ) , L ( U )] , namely S ∪ { r S ∩ U } , and P [ G ] = { x G : x ∈ P [ G ( U ) , L ( U )] } . If, moreover, L is a collection of cliques in G , then P [ G, L ] = { x G : x ∈ P [ G ( U ) , L ∪ L ( U )] } . Proof.
Proving that S ∪ { r S ∩ U } is the only extension of S in S [ G ( U ) , L ( U )] is straightfor-ward. Moreover, if S meets L , then so does S ∪ { r S ∩ T } . The rest now follows as each of P [ G ] , P [ G ( U ) , L ( U )], P [ G, L ], and P [ G ( U ) , L ∪ L ( U )] are convex hulls of stable sets. Lemma 3.1 expresses the stable set polytope of a graph G as a particular face of the stableset polytope of the record graph of U and G . Our next result is that those particular facesadmit a simple composition rule when U is a node cutset. Theorem 3.2.
Let ( V , U, V ) be a node cutset separation of graph G = ( V, E ) . Moreover,let L be a collection of cliques in G = G V ∪ U and let L be a collection of cliques in G = G V ∪ U . Then, each x ∈ R V ∪ U record satisfies: x ∈ P [ G ( U ) , L ∪ L ∪ L ( U )] if and only if x G ( U ) ∈ P [ G ( U ) , L ∪ L ( U )] and x G ( U ) ∈ P [ G ( U ) , L ∪ L ( U )] . Hence P [ G ] = { x ∈ R V : ∃ y ∈ R U record [ ( x G , y ) ∈ P [ G ( U ) , L ( U )] and ( x G , y ) ∈ P [ G ( U ) , L ( U )] ] } . Proof.
By Lemma 3.1, the second assertion follows from the first one. The “only if” directionof the first assertion is obvious. For the “if” direction, it suffices to consider x ∈ Q V ∪ U record .Assume x G ( U ) ∈ P [ G ( U ) , L ∪ L ( U )] and x G ( U ) ∈ P [ G ( U ) , L ∪ L ( U )]. Then, for i = 1 ,
2, there exists a positive integer n i , so that n i x G i ( U ) is the sum of the characteristicvectors of a collection of (not necessarily distinct) stable sets S i , . . . , S in i in G i ( U ). Byreplicating members in these two collections of stable sets (if necessary), we may assumethat n = n ; let n = n = n .Since x G i ( U ) ∈ P [ G i ( U ) , L ( U )], each S i , . . . , S in meets each clique in L ( U ) exactly once.So, for each stable set S in U , the number of sets among S i , . . . , S in that intersect U in S isequal to n ( x G i ( U ) ) r S = nx r S . As this applies to both i = 1 ,
2, we can renumber S , . . . , S n so that S j ∩ U = S j ∩ U for j = 1 , . . . , n . Doing so, x is a convex combination of thecharacteristic vectors of the stable sets S ∪ S , . . . , S n ∪ S n . Since S j ∈ P [ G ( U ) , L ∪ L ( U )]and S j ∈ P [ G ( U ) , L ∪ L ( U )] for all j , each S j ∪ S j is in S [ G ( U ) , L ∪ L ∪ L ( U )]. Hence, x ∈ P [ G ( U ) , L ∪ L ∪ L ( U )], as claimed. 18onsider Theorem 3.2 in case U is a clique cutset. Then U record = { r ∅ } ∪ { r { v } : v ∈ U } .Moreover, for ( x, y ) ∈ R V × R U record , we have that ( x, y ) ∈ P [ G ( U ) , L ∪ L ( U )] if and only if: x G ∈ P [ G, L ] , y r ∅ = 1 − (cid:88) v ∈ U x v , and y r { v } = x v ( v ∈ U ) . (16)Applying this to each of the three graphs G, G , G in Theorem 3.2, we obtain the followingresult of Chv´atal. Corollary 3.3 (Chv´atal [5]) . Let ( V , U, V ) be a clique cutset separation of a graph G =( V, E ) and let G = G V ∪ U and G = G V ∪ U . Then: P [ G ] = { x ∈ R V : x G ∈ P [ G ] and x G ∈ P [ G ] } . If, moreover, L is a collection of cliques in G and L is a collection of cliques in G , theneach x ∈ R V satisfies: x ∈ P [ G, L ∪ L ] if and only if x G ∈ P [ G , L ] and x G ∈ P [ G , L ] . In Corollary 3.3, we can not drop the condition that U is a clique. Indeed, let u and v betwo nonadjacent nodes in U and suppose G has a chordless even uv -path Q and G has achordless odd uv -path Q . Consider the vector x ∈ R V with x v = 1 / v lies on Q ∪ Q and x v = 0 otherwise. Then x (cid:54)∈ P [ G ], but x G ∈ P [ G ] and x G ∈ P [ G ].Balas [1] has shown how to obtain an extended formulation for the convex hull of polytopes P , . . . , P k , whose size is approximately the sum of the sizes of the descriptions for thesepolytopes. If A i x + B i y ≤ d i is an extended formulation for P i ( i = 1 , . . . , k ), then Balas’sformulation for the convex hull reads: x = x + · · · + x k , λ + · · · + λ k = 1; A i x i + B i y i − λ i d i ≤ , λ i ≥ i = 1 , . . . , k ) . (17)This formula can be used to construct a formulation for P [ G ] from such formulations forparts of a node cutset decomposition of G . Let H be one of these parts and let U denotethe node cutset. For every stable set S in U , a description of the face of P [ H ( U )] given by x r S = 1 can be inferred from any linear description of the face { x ∈ P [ H ] : x v = 1 ( v ∈ S ) } of P [ H ]. Since P [ H ( U ) , L ( U )] is the convex hull of these faces, Balas’s formula (17) givesan extended formulation for P [ H ( U ) , L ( U )] whose size is in the order of | U record | = | S [ H U ] | times the size of the linear description of P [ H ]. If we apply this to each part H of thedecomposition and combine the resulting formulations into one list of linear inequalities, weobtain, by Theorem 3.2, an extended formulation for P [ G ]. This leads to the following result. Theorem 3.4.
Let G be a graph and { ( G , Z , Ω ) , . . . , ( G k , Z k , Ω k ) } be a decomposition-listof ( G, ∅ , ∅ ) . Assume we are given for each i = 1 , . . . , k an extended formulation with size m i for P [ G i ( { Z i } ∪ Ω i )] . Then there exists an extended formulation for P [ G ] with size at most O ( k ) + m + · · · + m k .Proof. Recursively apply the following immediate corollary of Theorem 3.2: if ( V , U, V ) is acutset separation of template ( G, Z,
Ω) with master template ( G , Z, Ω ) and servant template( G , U, Ω ), then a vector x lies in P [ G ( { Z, U } ∪ Ω) , L ( U ))] if and only if x G ( { Z,U }∪ Ω ) ∈ P [ G ( { Z, U } ∪ Ω ) , L ( U )] and x G ( { U }∪ Ω ) ∈ P [ G ( { U } ∪ Ω ) , L ( U )].19n alternative for adding a record to a graph G is lifting a node set U to a clique . Thisamounts to deleting U from G and replacing it by a clique with node set U record \{ r ∅ } , andconnecting each r S ∈ U record \{ r ∅ } with each node in N G ( S ) \ U . We call the new graph the clique lift of U from G . An advantage of clique lifts over records is that clique lifts yieldextended formulations for stable sets that do not involve “tight clique constraints”: x ( K ) =1 ( K ∈ L ( U )). Lemma 3.5.
Let G + be the clique lift of U ⊆ V from a graph G = ( V, E ) . Then the stableset polytope P [ G ] is the image of P [ G + ] under the projection p : R ( V \ U ) ∪ ( U record \{ r ∅ } ) → R V defined by p v ( x ) = (cid:40)(cid:80) S ∈ S [ G U ] ,S (cid:51) v x r S if v ∈ Ux v otherwise . Lifting a node cutset to a clique turns it into a clique cutset. So we get the followingconsequence of Corollary 3.3.
Corollary 3.6.
Let ( V , U, V ) be a node cutset separation of graph G = ( V, E ) . Moreover,let G + , G +1 , and G +2 be the clique lifts of U from G, G V ∪ U , respectively G V ∪ U . Then each x ∈ R ( V \ U ) ∪ ( U record \{ r ∅ } ) satisfies: x ∈ P [ G + ] if and only if x G +1 ∈ P [ G +1 ] and x G +2 ∈ P [ G +2 ] . We give a decomposition rule for stable set polytopes of graphs that admit a generalizedamalgam separation ; for a graph with node set V , this is a pair ( U, W ) with U ⊆ V such that W is a partition of V \ U into nonempty sets W that each have the property that each nodein W with a neighbor outside W ∪ U is fully adjacent to U .Generalized amalgam separation unifies a great variety of known separations. Cliquecutset separation and amalgam separation are obvious special cases. A notable other ex-ample is the “strip-structure for trigraphs” introduced by Chudnovsky and Seymour [4].Faenza, Oriolo, and Stauffer [11] used strip-structures to obtain extended formulations andpolynomial-time algorithms for stable sets problems in “claw-free” graphs. The “2-clique-bonds” that Galluccio, Gentile, and Ventura [12] use to compose linear formulations of stableset problems are generalized amalgam separations as well.Before actually decomposing a graph along a generalized amalgam separation ( U, W ) we firstlift U to a clique, K (say). Then ( K, W ) is a generalized amalgam separation of the clique liftand all structure of ( U, W ) and the original graph fully carries over to ( K, W ) and the cliquelift, except for the internal structure of U resp. K . Since Lemma 3.5 explains the effect ofclique lifts to the stable set polytope, it is enough to investigate ( K, W ) in the clique lift; wecall the clique lift G .So ( K, W ) is a generalized amalgam separation of a graph G and K is a clique.For W ∈ W , we denote by A W the collection of equivalence classes in B G − K ( W ) of therelation “having the same neighbors outside W ”. Related to A W we will consider a clique A power W consisting of new nodes r X , one for each subcollection X of A W .20he generalized amalgam decomposition of G along ( K, W ) consists of a collection ofgraphs G ( K, W ), one for each W ∈ W , together with a “connecting” graph G connect ( K, W ).Each graph G ( K, W ) is obtained from the disjoint union of G K ∪ W and A power W by connect-ing each r X ∈ A power W to all nodes in B G − K ( W ) \ (cid:83) X and to all nodes in K . The graph G connect ( K, W ) is obtained from the disjoint union of the clique K and all cliques A power W with W ∈ W , by adding edges from each node in K to all nodes in all cliques A power W and byadding all edges r X r X (cid:48) such that ∪X and ∪X (cid:48) are adjacent in G and X ⊆ A W , X (cid:48) ⊆ A W (cid:48) , W, W (cid:48) ∈ W , W (cid:54) = W (cid:48) . We also define L ( K, W ) = { K ∪ A power W : W ∈ W} . Theorem 3.7.
Let ( K, W ) be a generalized amalgam separation of a graph G = ( V, E ) suchthat K is a clique. Moreover, let L = L ( K, W ) . Then P [ G ] consists of the restrictions x G ofthose x ∈ R V ∪ ( (cid:83) {A power W : W ∈W} ) with x G connect ( K, W ) ∈ P [ G connect ( K, W ) , L ] and x G ( K,W ) ∈ P [ G ( K, W )] for all W ∈ W . (18) Proof.
Let H be the graph defined as follows: the node set of H is V ∪ ( (cid:83) {A power W : W ∈ W} )and the edge set of H is the union of the edge set of G connect ( K, W ) with the edge sets of allgraphs G ( K, W ) with W ∈ W .Since each member of L is a clique cutset of H , it follows from Corollary 3.3 that x ∈ P [ H, L ] if and only if x satisfies (18). Hence, it suffices to prove that P [ G ] = { x G : x ∈ P [ H, L ] } . For that it suffices to prove that the function S (cid:55)→ S G maps S [ H, L ] onto S [ G ].First consider S ∈ S [ H, L ]. We prove that S G ∈ S [ G ]. If S ∩ K (cid:54) = ∅ , then S ⊆ V \ (cid:83) { B G ( W ) : W ∈ W} , so S ∈ S [ G ]. Hence we may assume that S ∩ K = ∅ . Thenthere exists, for each W ∈ W , a collection X W ⊆ A W with S ∩ A power W = { r X W } . Since S is astable set in H , we have that S ∩ B G − K ( W ) ⊆ B G − K ( W ) \ N H ( r X W ) = (cid:83) X W . Now consider W, W (cid:48) ∈ W with W (cid:48) (cid:54) = W . Then in H , node r X W is not adjacent to node r X W (cid:48) . Hence (cid:83) X W and (cid:83) X W (cid:48) are not adjacent in G . From this it follows that S G is a stable set in G , asclaimed.Next consider S (cid:48) ∈ S [ G ]. We prove that there exists an S ∈ S [ H, L ] with S (cid:48) = S G . If S (cid:48) ∩ K (cid:54) = ∅ , we just take S = S (cid:48) . Indeed, in that case, S (cid:48) ⊆ V \ (cid:83) { B G ( W ) : W ∈ W} , so S (cid:48) ∈ S [ H, L ]. Hence, we may assume S (cid:48) ∩ K = ∅ . For each W ∈ W , let X W be the membersof A W that contain an element of S (cid:48) . Define S = S (cid:48) ∪ { r X W : W ∈ W} . Then S ∈ S [ H, L ]and S G = S (cid:48) , as required. Amalgams
If graph G = ( V, E ) has an amalgam separation ( V , A , K, A , V ), then ( K, { V ∪ A , V ∪ A } )is a generalized amalgam separation and K is a (possibly empty) clique. By Theorem 3.7, P [ G ] consists of the restrictions x G of all vectors x ∈ R V ∪{ r { A } ,r ∅ ,r { A } ,r ∅ }} + with x V ∪ A ∪ K ∪{ r { A } ,r ∅ } ∈ P [ G ( K, V ∪ A )] , (19) x V ∪ A ∪ K ∪{ r { A } ,r ∅ } ∈ P [ G ( K, V ∪ A )] , (20) x K ∪{ r { A } ,r ∅ ,r { A } ,r ∅ } ∈ P [ G connect ( K, { V ∪ A , V ∪ A } ) , L ] , (21)where L consists of the two cliques K ∪ { r { A } , r ∅ } and K ∪ { r { A } , r ∅ } .21or x ∈ R K ∪{ r { A } ,r ∅ ,r { A } ,r ∅ } + , condition (21) is equivalent to x ( K ) + x r { A } + x r ∅ = 1 , x ( K ) + x r ∅ + x r { A } = 1 , x ( K ) + x r { A } + x r { A } ≤ , so, with x r { A } = 1 − x ( K ) − x r ∅ , x r { A } = 1 − x ( K ) − x r ∅ , x ( K ) + x r ∅ + x r ∅ ≥ . (22)We now eliminate x r { A } and x r { A } . In (19), this amounts to deleting r { A } from G ( K, V ∪ A ), In (20), this amounts to deleting r { A } from G ( K, V ∪ A ). Since G ( K, V ∪ A ) − r { A } and G ( K, V ∪ A ) − r { A } are the blocks of the amalgam decomposition of G , we get thefollowing result. Theorem 3.8. If G and G are the blocks of an amalgam decomposition of G = ( V, E ) usingthe amalgam separation ( V , A , K, A , V ) , then the stable set polytope P [ G ] of G satisfies: P [ G ] = { x G x ∈ R V ∪{ r ∅ ,r ∅ } , x G ∈ P [ G ] , x G ∈ P [ G ] , x ( K ) + x r ∅ + x r ∅ ≥ } . If we have original space descriptions for P [ G ] and P [ G ], Theorem 3.8 yields an extendedformulation for P [ G ] with x r ∅ and x r ∅ as the only extra variables. With Fourier-Motzkinelimination it easy to remove x r ∅ and x r ∅ from that extended formulation. This leads to anew proof of the following result of Burlet and Fonlupt (see [14] for an extension). Corollary 3.9 (Burlet and Fonlupt[3]) . Let the stable set polytopes of the blocks of an amal-gam decomposition of G be described by the following systems: x ≥ , D x ≤ δ , x r ∅ ≥ , and x r ∅ + c ,i x ≤ γ ,i ( i = 1 , . . . , n ) , (23) and x ≥ , D x ≤ δ , x r ∅ ≥ , and x r ∅ + c ,i x ≤ γ ,i ( i = 1 , . . . , n ) (24) where r ∅ and r ∅ are the nodes that are not in G . Then P [ G ] is given by the followingsystem: x ≥ , D x ≤ δ , D ≤ δ , (25) (cid:2) c ,i + c ,j (cid:3) x − x ( K ) ≤ γ ,i + γ ,j − i = 1 , . . . , n , j = 1 , . . . , n ) . (26) where K is the clique in the amalgam separation.Proof. Let G and G be the blocks of the amalgam decomposition, where (23) describes P [ G ] and (24) describes P [ G ]. By Theorem 3.8, P [ G ] consists of all x for which there exists x r ∅ and x r ∅ such that ( x, x r ∅ , x r ∅ ) satisfies (23), (24), and x ( K ) + x r ∅ + x r ∅ ≥ . (27)Since (23) describes P [ G ], we get that (23) implies “ x ( K ) + x r ∅ ≤ x r ∅ ≥ x r ∅ ≥
0” isredundant in the system of linear inequalities given by (23), (24) and (27). By symmetry,22he same applies to “ x r ∅ ≥ x r ∅ ≤ γ ,i − c ,i x ( i = 1 , . . . , n ) (28) x r ∅ ≤ γ ,j − c ,j x ( j = 1 , . . . , n ) (29)1 − x ( K ) − x r ∅ ≤ x r ∅ (30)Eliminating x r ∅ , replaces (28)-(30) by x r ∅ ≤ γ ,i − c ,i x ( i = 1 , . . . , n ) (31)1 − x ( K ) + c ,j x − γ ,j ≤ x r ∅ ( j = 1 , . . . , n ) (32)Eliminating x r ∅ , replaces (31) and (32) by (26). Lemma 3.10.
Stable set polytopes of record graphs of fan-templates have compact extendedformulations that can be constructed in polynomial time.Proof.
Let H be the record graph of a fan-template with base ( u, c, v ). Then P [ H ] is theconvex hull of P [ H − c ] and of a face of the convex hull of the characteristic vector of { c } and P [ H − N H ( c ) − c ]. Since, by Lemma 2.19, the graphs H − c and H − N H ( c ) − c aredecomposable by 2-node clique sets into a list of at most | V | graphs, each with at most 8nodes, the lemma follows from Corollary 3.3 and Balas’s formula (17). Theorem 3.11.
The stable set polytopes of cap-free odd-signable graphs have a compactextended formulation that can be constructed in polynomial time.Proof.
When graph G has as a node u adjacent to all other nodes, P [ G ] is the convex hull ofthe characteristic vector of { u } and P [ G − u ]. Hence in that case it follows from (17), that P [ G ] has an extended formulation with only three more variables than any such formulationfor P [ G − u ]. Recall from Section 2.1, that clique cutset separations and amalgam separationsare 1-linearizable and that a 1-linearized decomposition-list of a graph G = ( V, E ) can haveat most | V | members. Hence, by Lemma 3.10, the result follows from the decompositionresults Theorem 1.3, 2.18 and the polyhedral composition results Corollary 3.3 and Theorems3.4 and 3.8. References [1] E. Balas. Disjunctive programming: Properties of the convex hull of feasible points.
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