ON THE STAR FOREST POLYTOPE FOR TREES AND CYCLES

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP -hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP (G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP (G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M -tree inequalities, for SFP (G). We show that for the case when G is a tree, the M -tree and the nonnegativity inequalities give a complete characterization of SFP (G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP (G) when G is a cycle. Mathematics Subject Classification. 90C27, 90C10, 05Cxx. Received June 1, 2015. Accepted September 4, 2018.


Introduction
Given an undirected graph G = (V, E) where n = |V | and m = |E|, a star in G is either a single node of G or a subgraph of G where every edge shares one common end-node. The latter is called the center of the star when the star is not reduced to a single node. If the star is a single edge, then any of its end-nodes can be designated as the center. A star forest is a collection of vertex-disjoint stars in G. An edge dominating set in G is an edge subset F ⊆ E such that for any edge e in G either e ∈ F or e shares at least one common end-node with some edge in F . A dominating set in G is a node subset S ⊆ V such that for any node u ∈ V either u ∈ S or u is neighbor with some node in S. We suppose that the edges in G have non-negative weights (note that the unweighted case can be seen as a special weighted case when weights are 0 or 1), then the weight of a star forest or an edge dominating set is the sum of the weights of its edges. The Maximum Weight spanning Star Forest Problem (MWSFP) is to find a star forest spanning the nodes of G of maximum weight. The Minimum Weight Edge Dominating Problem (MWEDP) is to find an edge dominating set in G of minimum weight. If the nodes are weighted, the weight of a dominating set is the sum of the weights of its nodes. The Minimum Weight Dominating Set Problem (MWDSP) is to find a minimum weight dominating set in G. The two last problems are well-known to be N P -hard. They have been the subject of many works in the literature [20], [13]. The MWSFP, however, is a recent problem which has been introduced by Nguyen et al. in [17]. It has applications in several areas, especially in computational biology [17] and automobile industry [1]. In [17], the authors show the N P -hardness of MWSFP by observing that in a maximal star forest F (a maximal star forest is a star forest to which no more edge can be added), the set of the centers of the stars in F is a dominating set of G. Conversely, for any dominating set S we can build a maximal star forest with centers as the nodes belonging to S. Thus, given a maximal star forest F , there exists a dominating set S such that |S| = |V | − |F | and vice versa. Hence the case of 0/1 weights of the MWSFP is N P -hard by a reduction from the of 0/1 weights case of the MWDP. In [17], the authors also give a linear time algorithm to solve the MWSFP when G is a tree and a 1 2 -approximation algorithm for the general case. Since then, the MWSFP has been intensively investigated, in particular for the unweighted version. Nguyen et al. [17] prove that the problem is APX-hard by presenting an explicit inapproximability bound of 259/260, and present a combinatorial 0.6-approximation algorithm for the unweighted MWSFP. Polynomial-time algorithms are presented for special classes of graphs such as planar graphs and trees in the same paper. Chen et al. [12] present a better approximation algorithm with ratio 0.71 for unweighted MWSFP. Later, Athanassopoulos et al. [2] improve this approximation ratio to 0.803 by using the fact that the problem is a special case of the complementary set cover problem. Interesting generalizations including node-weighted and edge-weighted versions of the MWSFP have also been considered. In [12,17] the authors present approximation algorithms and APX-hardness results for these problems as well. Stronger inapproximability results for these problems recently appeared in [11,14]. For the weighted version, Nguyen [18] has given a linear time algorithm for solving the MWSFP when G is a cactus.
Let SF P (G) (respectively EDP (G)) be the convex hull of the incidence vectors of the star forests (respectively the edge dominating sets) in G. Let R n be the real space indexed by the nodes in V . Let D be any dominating set in G, let χ(D) ∈ R n be the incidence vector of D, defined as Let DP (G) be the convex hull of the incidence vectors of the dominating sets in G. To the best of our knowledge, no polyhedral investigation has been done for SF P (G) and EDP (G) though some integer formulations have been used in approximation algorithms for the MWSFP and the MWEDP. There are, however, several works on DP (G), in particular, Saxena [19] has given a complete description for DP (G) when G is a tree, and Bouchakhour et al. [8] have given a complete description for DP (G) when G is a cycle. In [16], Mahjoub has given a complete description of DP (G) in threshold graphs. And in [9], Bouchakour and Mahjoub have studied compositions for the polytope DP (G) in graphs that decompose by one-node cutsets.
In this paper, we present a polyhedral investigation of the MWSFP for trees and cycles. More precisely, we study the facial structure of the star forest polytope. In the first part of the paper, we give a complete characterization of SF P (G) when G is a tree, which is obtained by projection of a simple extended formulation issued from the work of Baïou and Barahona [3] on the uncapacitated facility location polytope. Also, we show that the facet-defining inequalities for SF P (G) when G is a tree can be generalized to valid inequalities for SF P (G) when G is a arbitrary graph. These inequalities define facets for SF P (G) under certain conditions, and can be separated in polynomial time.
In the second part of the paper, we give a complete description for SF P (G) when G is a simple cycle C. More precisely, we establish the relation between spanning star forests and dominating sets when the graph is a simple cycle C and give a complete linear description for SF P (C) based on the one for DP (C) given by Bouchakhour et al. [8].
The paper is organized as follows. In the next section, we describe some properties of SF P (G) and give an integer programming formulation for the MWSFP. In section 3, we introduce a class of valid inequalities, called the M -tree inequalities, for SF P (G) and give a complete description for SF P (G) when G is a tree. In Section 4, we present a complete description for SF P (G) when G is a cycle.
In the rest of this section, we give some notations that will be used in the paper. For x ∈ R m , given any F ⊆ E, we let x(F ) denote e∈F x e . For x ∈ R n , given a set S ⊆ V , we let x(S) denote v∈S x v . Given a set of vertices S, we denote by E(S) the set of edges with both ends belonging to S. Let v ∈ V , the neighborhood of v, denoted by N (v), is the vertex set consisting of v and the nodes which are adjacent to v. Given any edge subset F ⊆ E, let V (F ) denote the set of the end-nodes of the edges in F . We call a 3-path a simple path having 3 edges in G and a 3-cycle a triangle in G. Let P 4 (respectively C 3 ) denote the collection of the 3-paths (resp. 3-cycles) in G.

Basic properties of SF P (G) and integer programming formulation
for the MWSP 2.1. Basic properties of SF P (G).
The following remark is about zero vector 0 ∈ R m which is the incidence vector associated with the single node star forests.
Remark 2.1. The zero vector 0 ∈ R m is an extreme point of SF P (G).
Proof. We can see that 0 is the incidence vector associated with the single node star forests and as for any x ∈ SF P (G) and any e ∈ E, x e ≥ 0, 0 is an extreme point of SF P (G).
Hence, SF P (G) is a polytope pointed at 0. Moreover, the following theorem shows that SF P (G) is full dimensional.
Proof. Suppose that the incidence vectors of all the star forests in G satisfy some equality α t x = β. As α t 0 = β, β = 0. As any edge e ∈ E is a star forest in G, we have α e = β = 0 for all e ∈ E. Theorem 2.3. All the facet-defining inequalities of SF P (G), that are different from x e ≥ 0 for some e ∈ E, are of the form a t x ≤ b with a ∈ R m + and b ≥ 0 scalar.
Proof. Let a t x ≤ b be any facet-defining inequality for SF P (G) which is not x e ≥ 0 for some e ∈ E. As a t 0 ≤ b, we have b ≥ 0. Suppose that for some edge e, a e < 0. As a t x ≤ b defines a facet different from x e ≥ 0, there exists a star forests F in G containing e such that a t χ F = b. As F = F \ {e} is also a star forest, we have This contradicts the fact that a t x ≤ b is valid for SF P (G).
Given a t x ≤ b any facet-defining inequality of SF P (G), the support graph of a t x ≤ b is the subgraph G a = (V a , E a ) of G induced by the edges e ∈ E such that a e > 0. A tight star forest F with respect to a t x ≤ b is a star forest with which the associated incidence vector satisfies a t x ≤ b at equality. A star forest F is maximal with respect to an edge subset E ⊆ E if for any edge e ∈ E \ F , F ∪ {e} is not anymore a star forest. Theorem 2.3 implies the following corollary.
its support graph, all the tight star forests with respect to a t x ≤ b are maximal with respect to E a .
Lemma 2.5. LetḠ = (V ,Ē) be any induced subgraph of G, then any facet-defining inequality for SF P (Ḡ) also defines a facet for SF P (G).
Proof. First notice that the theorem trivially holds for the trivial inequalities x(e) ≤ 1 for all e ∈Ē, since these inequalities define facets for SF P (G) for any graph G. Let I be any facet-defining inequality for SF P (Ḡ), different from x(e) ≤ 1 for all e ∈ E. Then I defines a facet for SF P (G) if we are able to show that for any edge ij ∈ E \Ē, there always exists a tight star forest F inḠ with respect to I such that F ∪ {ij} is also star forest in G. Suppose that for some edge ij ∈ E \Ē, no such F exists. This implies that for every tight star forest F inḠ w.r.t I, F ∪ {ij} is not a star forest in G. In this case, exactly one node i or j, say i, belongs toV . Moreover, asḠ is an induced subgraph of G, every tight forest F inḠ w.r.t I should contain an edge ik such that i is of degree 1 in F and k is of degree at least 2 in F . Otherwise, F ∪ {ij} would be a star forest of G.
Then every tight star forest F inḠ w.r.t I also satisfies x(δḠ(i)) = 1. This, together with Theorem 2.2, implies that I should be x(δḠ(i)) ≤ 1. Since I is different from x(e) ≤ 1 for all e ∈ E, i should be of degree at least 2 inḠ. But then x(δḠ(i)) ≤ 1 is not valid for SF P (Ḡ), a contradiction.

Integer programming formulation for the MWSFP
In this subsection, we give an integer programming formulation for MWSFP. First we state the following lemma.
Lemma 2.6. A graph is a star forest iff it does not contains 3-paths and 3-cycles.
Proof. It can be immediately verified from the definition of a star forest.
Let us consider the following integer program.
x integer Inequalities (2.1), called the 3-path inequalities, state the fact that a star forest can only take at most 2 edges in a 3-path. Similarly, inequalities (2.2), called the 3-cycle inequalities, state the fact that a star forest can only take at most 2 edges in a 3-cycle. Inequalities (2.3) are the trivial inequalities.

M -tree inequalities and complete description of SFP(G) in trees
In this section, we shall give a complete description of SF P (G) in trees. For this, let us first describe a class of inequalities which are valid for any graph, not only for trees.

M -tree inequalities
In this subsection, G is an arbitrary graph (not necessarily a tree).  We can remark that the M -tree inequalities generalize inequalities x e ≤ 1 for e ∈ E and the 3-path inequalities. These are M -tree inequalities with |M | = 1 and |M | = 2, respectively. Let us first prove the validity. Let F be any star forest of G. We will prove the validity by showing that |F ∩ τ | ≤ |M |. If F only contains the edges in M , then |F ∩ τ | ≤ |M |. If F only contains edges in τ \ M then |F ∩ τ | ≤ |τ \ M | and, from the definition of a M -tree, |τ \ M | = |M | − 1 < |M |. Now suppose that F contains edges in both M and τ \ M . If F ∩ τ is a matching then |F ∩ τ | ≤ |M | since M covers all the nodes of τ . So suppose that F ∩ τ is not a matching. Thus, F ∩ τ should contain a (sub)star S with, say u 2 , as center, which contains at least two edges in F : one edge, say u 1 u 2 , which belongs to M and one other, say u 2 v 2 , which belongs to τ \ M . As v 2 should be also covered by M , there exists an edge v 1 v 2 ∈ M . As u 1 u 2 and u 2 v 2 are in F , v 1 v 2 / ∈ F . Moreover, by definition of M -tree, v 1 should be of degree 1 in τ . It follows that each edge in F which belongs to τ \ M (e.g. u 2 v 2 ) correspond exactly to another edge (e.g. v Let us prove now that (3.1) defines a facet for SF P (G). Suppose that there exists a facet-defining inequality α t x ≤ β for SF P (T ) such that all the star forests satisfying (3.1) at equality, satisfy also α t x ≤ β at equality.
Since by Theorem 2.2, SF P (T ) is full dimensional, it suffices to show that α t x ≤ β is a positive multiple of (3.1).
Let us remark that M is a star forest satisfying (3.1) at equality and hence also statisfies α t x ≤ β at equality, i.e. α(M ) = β. For any edge e ∈ E \ τ , we can see that M ∪ {e} is also star forest satisfying (3.1) at equality. Hence, α(M ∪ {e}) = β. This implies that α e = 0 for all e ∈ E \ τ .
Let u 2 v 2 be any edge in τ \ M and u 1 u 2 and v 1 v 2 be the edges in M incident to u 2 and v 2 respectively. We can see that M , (M \ {u 1 u 2 }) ∪ {u 2 v 2 } and (M \ {v 1 v 2 }) ∪ {u 2 v 2 } are all star forests satisfying (3.1) at equality. This implies that α u1u2 = α u2v2 = α v1v2 . If we extend this to all the edges in τ \ M , we obtain that α e = α e for all e, e ∈ τ . Hence, α t x ≤ β is a positive multiple of (3.1), which ends the proof of the theorem.

Complete description of SFP(G) in trees
From now on and throughout this section, G will be a tree denoted by T . Proof. Let S be any maximal star forest in τ . Let m = |M | then |V (τ )| = 2m and |τ | = 2m − 1. By the validity of M -tree inequalities, we have |S| ≤ m. We will prove that |S| ≥ m by showing that each edge in M correspond to an edge in S. Let v 1 be any leaf in τ and let v 2 be the non-pendant node such that the edge v 1 v 2 ∈ M . We distinguish two cases: • v 1 / ∈ S. In this case, v 2 should belong to V (S) since otherwise we can add v 1 v 2 to S and S remains a star forest. Moreover, v 2 should be of degree 1 in S since otherwise we can also add v 1 v 2 to S and S remains a star forest. Thus, the edge v 1 v 2 correspond to the edge incident to v 2 in S.
• v 1 ∈ S. Then the edge v 1 v 2 should belong to S and hence it corresponds to itself.
We will prove the following theorem. Proof. Suppose that a t x ≤ b is any facet-defining inequality for SF P (T ) which is not a M -tree inequality neither the nonnegativity inequality. Let G a be the support graph of a t x ≤ b. We can suppose without loss of generality that G a is a tree. Let T a denote this tree. Hence, T a is a subtree of T . We have two possible cases.
• T a is a M -tree. Let F be any tight star forest with respect to a t x ≤ b. By Corollary 2.4, F is maximal with respect to T a . Consequently, by Proposition 3.4, F is tight with respect to the M -tree inequalities. Contradiction to the fact that a t x ≤ b is a facet defining inequality different from a M -tree inequality. • T a is not a M -tree. Hence, in T a there must be one non-pendant node of one of the two following types: Type 1. A non-pendant node not connected to a leaf of T a . Type 2. A non-pendant node connected to at least two leaves of T a .
We distinguish two cases: Case 1. T a contains only non-pendant nodes of Type 2. In this case, one can obtain a M -tree τ from T a by keeping for each non-pendant node, only one leaf connecting to it. Let x(τ ) ≤ |M τ | be the M -tree inequality associated with τ . The following remark can be easily proved.
Remark 3.6. Given a non-pendant node v of Type 2, any star forest F satisfying a t x ≤ b at equality must be maximal in τ and contains either all the leaves connected to v or no of them.
Let F be any tight star forest with respect to a t x ≤ b. We have F ∩ τ is a maximal star forest with respect to τ . Since by Proposition 3.4, every maximal star forest in τ safisfies x(τ ) ≤ |M τ | at equality, it follows that every star forest satisfying a t x ≤ b at equality satisfies also the M -tree inequality associated with τ at equality. This contradicts the fact that a t x ≤ b is a facet-defining inequality.
Case 2. T a contains at least one non-pendant node of Type 1. We will show the following lemma. We have the following remark. Thus, the procedure should be ended by finding a non-pendant node of Type 1 satisfying the condition stated in the lemma after at most |V (T a )| iterations since the kernel grows after each iteration.
Let s be a non-pendant node of Type 1 satisfying the condition stated in Lemma 3.7. Let T 1 be the connected component which contains the other non-pendant nodes of Type 1 obtained by removing s from T a . We can observe that s is connected to T 1 by only one edge and the subgraph H of T a induced by the edges in T a \ T 1 is a tree. Moreover, H is either a M -tree or a tree which contains some non-pendant nodes of Type 2 but no one of Type 1. Hence, in all the cases as we have proved in Case 1, H contains a M -tree τ which contains all the non-pendant nodes of H. Let u be the neighbour of s in T 1 , i.e. the edge su ∈ H. Observe that su is a bridge in T a that links H to T 1 . We can see that any maximal star forest in T a , whatever it contains su or not, contains a maximal star forest in H (if a maximal star forest in T a does not contain su, it must contain some edge sv where v is of degree at least 2 in the star forest). The latter, by Remark 3.6, contains a maximal star forest in τ . By Corollary 2.4, any star forest F satisfying a t x ≤ b at equality is maximal in T a . Hence F ∩ τ is a maximal star forest in τ . By Proposition 3.4, F ∩ τ satisfies the M -tree inequality associated with τ at equality. This contradicts the fact that a t x ≤ b is facet-defining.

Polyhedral results on cycles
In this section, G will be a chordless cycle C = (V (C), E(C)) of n nodes with V (C) = {1, 2, . . . , n} numbered clockwise, i.e. the n edges in E(C) will be e i = (i, i + 1) for i = 1, . . . , n − 1, and the edge e n = (n, 1). We will sometimes use |C| instead of n, |V (C)| or |E(C)| which are all equal. The edges in C are weighted by a vector c ∈ R n where c i is the weight associated with edge e i for i = 1, . . . , n. We also consider L(C) = (V L , E L ) the line graph of C where the nodes correspond to the edges of C, and two nodes of L(C) are adjacent if the corresponding edges are adjacent in C. Note that L(C) is also a cycle node-weighted by vector c. For the sake of convenience, given a node 1 ≤ i ≤ n and an integer t > 0, let i + t designate the node i + t if i + t ≤ n and the node (i + t) mod n if (i + t) > n. For two nodes u and v with v = u + t for some integer t > 0 in C, let C(u, v) denote the path (u + 1, . . . , u + t − 1) of C between u + 1 and u + t − 1 (note that the path does not contain In what follows, we will establish the relations between star forests, dominating sets and edge dominating sets on cycles. Then, using theses relations with the polyhedral results in [9] and [8], we will derive a complete description of SF P (C). Note that in [18], a linear time algorithm for the MWSFP when G is a cactus have been presented, hence the MWSFP can be solved in linear time in cycles.

Star forests, dominating sets and edge dominating sets on cycles
In the context of a cycle, Lemma 2.6 can be restated as follows.  (⇐) Let ED ⊆ C be a edge dominating set in C. Let F = C \ ED and suppose that F is not a star forest. Then F contains a 3-path (v 1 , v 2 , v 3 , v 4 ). We can see that the edge (v 2 , v 3 ) is not dominated by ED implying that ED is not an edge dominating set, a contradiction.
By the one-to-one correspondence between the nodes of L(C) and the edges of C, we have the following result. The following lemma reformulates these relations in polyhedral terms for the polytopes SF P (C), EDP (C) and DP (L(C)). (i) α t y ≥ β with y ∈ R n defines a facet for DP (L(C)), (ii) α t x ≥ β with x ∈ R n defines a facet for EDP (C), (iii) α t x ≤ e∈E(C) α(e) − β with x ∈ R n defines a facet for SF P (C).
Hence, the polytopes SF P (C), DP (L(C)) and EDP (C) are equivalent in the sense that there is a one-to-one correspondence between their facets.
Proof. Note that these polytopes are all defined in R n and are full dimensional. The lemma follows from the relations described in Lemmas 4.2 and 4.3 as they are all preserved under affine transformations.
As in [8], a complete linear description for DP (L(C)) is given, by Lemma 4.4, we can also derive complete descriptions for SF P (C) and EDP (C). We will explicit these complete descriptions in the following section.

Complete description of SF P (C)
Let us consider the graph L(C) and the polytope DP (L(C)) in R n whose component indexed by the nodes in V L . In [9] and [8], Bouchakour et al. give the following integer formulation for MWDSP, x v integer for all v ∈ V L They have also characterized two classes of facet-defining inequalities for DP (L(C)).
Theorem 4.5. [9] The inequality defines a facet for DP (L(C)) if and only if either |C| = 3 or |C| ≥ 4 and |C| is not a multiple of 3.
[8] Let W = {v 1 , . . . , v p } be a subset of p ≥ 3 nodes in V L satisfying the following conditions: Then the constraint defines a facet for DP (L(C)).
Let us apply Lemma 4.4 to derive facet-definining inequalities for SF P (C). It is clear that applying Lemma 4.4 to inequalities (4.1) yields the trivial inequalities 0 ≤ x e ≤ 1 for all e ∈ E(C), for SF P (C).
By applying Lemma 4.4 to inequalities (4.2), we get the 3-path inequalities x(P ) ≤ 2 for all path of length 3 in C, for SF P (C) which have been described in Section 2. The following proposition can be obtained by applying   define facets for SF P (C).
Proof. Given a matching M of C satisfying conditions M1 and M2, suppose that v 1 , . . . , v p are the nodes in L(C) corresponding respectively to f 1 , . . . , f p . It is easy to see that v 1 , . . . , v p satisfy conditions C1 and C2 of Theorem 4.6, and hence, we have that defines a facet for DP (L(C)). The result thus follows from Lemma 4.4.
In  As a direct consequence, we have the following result.
Corollary 4.10. When G is a cycle, SF P (G) is completely described by the trivial inequalities, the 3-path inequalities, the cycle inequality (4.5) and the matching-cycle inequalities (4.6).

Conclusions
In this paper, we have presented an IP formulation for MWSFP. We have also given two complete linear descriptions for SF P (G), the star forests polytope for the cases where G is a tree and G is a cycle. An interesting direction for future works would be to exploit these results to derive a complete linear description for SF P (G) when G is a cactus. Our complete description for SF P (G) when G is a tree could be helpful to find an exact solution for the MWSFP in general graphs. A star forest is always a subgraph of a spanning tree in G and a complete linear description of the spanning tree polytope is known.