The multi-terminal vertex separator problem: Polytope ... - LAMSADE

The multi-terminal vertex separator problem: Polytope characterization and TDI-ness Youcef Magnouche and Sébastien Martin [email protected] Lamsade, Université Paris Dauphine [email protected] LCOMS, Université de Lorraine, Metz http://www.lamsade.dauphine.fr/ http://lcoms.univ-lorraine.fr/

Abstract. In this paper we discuss a variant of the well-known k-separator problem. Consider the simple graph G = (V ∪ T, E) with V ∪ T the set of vertices, where T is a set of distinguished vertices called terminals, inducing a stable set and E a set of edges. Given a weight function w : V → N, the multi-terminal vertex separator problem consists in finding a subset S ⊆ V of minimum weight intersecting every path between two terminals. We characterize the convex hull of the solutions of this problem in two classes of graph which we call, star trees and clique stars. We also give TDI systems for the problem in these graphs. Keywords: Vertex separator problem, Total dual integrality, Combinatorial optimization, Polytope characterization.

1

Introduction

Let G = (V ∪ T, E) be a simple graph with V ∪ T the set of vertices, where T is a set of distinguished vertices called terminals, inducing a stable set and E a set of edges. Given a weight function w : V → N, the multi-terminal vertex separator problem (MTVSP) consists in finding a subset S ⊆ V of minimum weight such that each path between two terminals intersects S. The problem can be solved in polynomial time when |T | = 2, [3] but when |T | ≥ 3, the MTVSP is NPhard ([6],[9]). In this paper we deal with the MTVSP in two specific classes of graph, star trees and clique stars, showing that this problem can be solved in polynomial time for any size of T in these two classes. We show also that the associated polytope is integer and we give a min-max relation for each class. The MTVS problem has applications in different areas like VLSI design, linear algebra, connectivity problems and parallel algorithms. It has also applications in network security, for instance, consider a graph G = (V ∪ T, E) representing a telecommunication network, with V the set of routers , T the set of customers and an edge between two vertices represents the possibility of transferring data between each other. We search to set up a monitoring system of minimum cost

2

The Multi-terminal vertex separator problem

on some routers, in order to monitor all data exchanged between customers. The set of these routers represents a minimum multi-terminal vertex separator. The MTVSP is a variant of the k-serparator problem that consists in partitioning the set of vertices of a graph G, into k + 1 subsets {S, V1 , . . . , Vk }, such that S has a minimum weight and no vertex in Vi is adjacent to a vertex in Vj . Many other variants of the k-serparator problem have been considered in the literature ([8], [2]). In [1], the authors discuss the following problem. Given a simple graph G = (V, E) and an integer β(n) with n = |V |, partition V into three subsets A, B and C such that |C| is minimum, no vertex in A is adjacent to a vertex in B and max{|A|, |B|} ≤ β(n). In [3] authors consider another variant of the problem. Given a simple graph G = (V, E) with a, b ∈ V two terminals, the problem here is to partition V into three subsets A, B and C such that a ∈ A, b ∈ B, no edge connecting A and B and the size of the cut induced by C is minimum. They show that this problem can be reduced to a minimum cut problem in an auxiliary graph and then, it can be solved in polynomial time.

Fig. 1. Star tree and clique star

The MTVS problem was considered in [6], in which the authors present several valid inequalities and develop a branch-and-cut algorithm to solve the problem. They also present two classes of graph, called star trees, see Figure 1.(a) and clique stars, see Figure 1.(b), on which our work is based. In [10], authors give a linear system for the MTVSP and characterize the class of graph for which it is total dual integral for any size of T , i.e., the dual problem has an integer optimal solution for any integer vertex weight vector. The main motivation of this paper is to derive TDI descriptions for other classes of graph. This is a preliminary work on two specific classes, star trees and clique stars. In this paper we first characterize the polytope of the multi-terminal vertex separators for these two classes of graph and then we give TDI linear systems. The paper is organized as follows, in Section 2, we introduce some notations and definitions that we use in the remainder of the paper, in Section 3, we characterize the polytope of the multi-terminal vertex separators for two classes of graph, in Section 4, we give a total dual integral system for each of these classes of graph.


2

3

Preliminaries

In this paper, we denote by n the cardinality of V and k the number of terminals in T . A path is a set of p distinct vertices v1 , v2 , . . . , vp such that for all i ∈ {1, . . . , p − 1}, vi vi+1 is an edge. The vertices v2 , . . . , vp−1 are called internal vertices of the path. A terminal path Ptt0 is the set of internal vertices of a path P between two terminals t, t0 ∈ T , such that P ∩ T = {t, t0 }. A terminal path Ptt0 is minimal if there does not exist another terminal path Pti tj in the graph, such that Pti tj ⊂ Ptt0 . The support graph of an inequality is the graph induced by the vertices of variables having positive coefficient in the inequality. Given a vertex v ∈ V ∪ T , we denote by N (v) ⊆ V ∪ T the set of vertices adjacent to v. Given a graph H, we denote by V (H) its set of verticesPand E(H) its set of edges. Given x ∈ RV and W ⊆ V ∪ T , we let x(W ) = x(v). Consider a v∈W ∩V

graph G = (V ∪ T, E) and two subgraphs G1 = (V1 ∪ T1 , E1 ), G2 = (V2 ∪ T2 , E2 ) of G. Graph G1 is said to be completely included in G2 , if V1 ∪ T1 ⊆ V2 ∪ T2 . A star tree Hk = (VHk ∪ THk , EHk ), Figure 1.(a), where THk = {t1 , . . . , tk }, is a tree that is the union of k paths Pt1 , . . . , Ptk , such that one end of each Pti is a common node vr ∈ VHk , called the root, and the other end is a terminal ti . A clique star Qk = (VQk ∪ TQk , EQk ), Figure 1.(b), is a graph defined by a clique Kk of k vertices and k disjoint paths Pt1 , . . . , Ptk between all terminals of TQk = {t1 , t2 , . . . , tk } and vertices of Kk . In the star trees and clique stars the path Pt is refereed as a branch. For any star tree H (resp. clique star Q), we denote by t(H) (resp. t(Q)) the number of branches of H (resp. Q).

Fig. 2. Star trees and clique stars

Figure 2 gives some star trees and clique stars where the terminals are given by triangles. If k = 1, the star tree and the clique star are reduced to a single branch, see Figure 2.(a). If k = 2, the star tree and the clique star are reduced to a path between two terminals, see Figure 2.(b).

4


If k ≥ 3, the star tree (resp. clique star) with k terminals contains kk0 star trees (resp. clique stars) as subgraphs with k 0 ∈ {1, . . . , k} terminals. Let Π (resp. Θ) be the set of all star trees subgraphs of Hk (resp. clique stars subgraphs of Qk ). Note that Π (resp. Θ) contains Hk (resp. Qk ). Let Π v (resp. Θv ) be the star trees (resp. clique stars) of Π (resp. Θ) containing vertex v ∈ VHk (resp. v ∈ VQk ) . Let x ∈ {0, 1}V be a vector of variables such that for a vertex v ∈ V , x(v) = 1 if v belongs to the separator and x(v) = 0 otherwise. The vector xS is called an incidence vector of the separator S. Consider the vertex weight vector w ∈ NV , the MTVSP is equivalent to the following integer linear program P 0 X min w(v)x(v) (1) v∈V

x(Ptt0 ) ≥ 1

∀Ptt0 ∈ Γ

(2)

x(v) ≤ 1

∀v ∈ V

(3)

x(v) ≥ 0

∀v ∈ V

(4)

x(v) integer

(5)

where Γ is the set of all terminal paths in G. Let P (G, T ) = conv(x ∈ [0, 1]V | x satisfies (2)) be the polytope given by inequalities (2) − (4). In this paper we consider a star tree Hk and a clique star Qk satisfying the following hypothesis 1. the number of terminals is at least three, otherwise the linear system (2), (4) is TDI, [10]. 2. each branch of the star tree Hk contains at least one internal vertex, otherwise the linear system (2), (4) is TDI, [10]. In star trees and clique stars, under the above hypothesis, the polytope P (G, T ) is full-dimensional [6].

3

Polytope characterization

In this section we will characterize the multi-terminal vertex separator’s polytope in the star trees and clique stars. 3.1

Star trees

Proposition 3.11 In star trees, the polytope P (G, T ) is not integral. Proof. Consider a star tree Hk = (VHk ∪ THk , EHk ) with at least one vertex in each branch. Let x ∈ [0, 1]Vk be a solution of P (Hk , THk ) defined as follows – x(v) = 0.5

∀v ∈ N (vr )


– x(v) = 0

5

∀v ∈ VHk \ N (vr )

The vector x represents a fractional extreme point of P (Hk , THk ), since there is k fractional variables and k terminal path inequalities that are linearly independent and tight by x. Consider the following valid inequalities presented in [6] x(VHk0 \ {vr }) + (k 0 − 1)x(vr ) ≥ k 0 − 1

Hk0 ∈ Π

(6)

We recall that terminal paths are star trees of two terminals and inequalities (2) are included in (6). We notice that all inequalities (6) associated with the star trees of Π with one terminal are dominated by trivial inequalities. Theorem 3.11 For any star tree, the polytope given by inequalities (6) and trivial inequalities is integer. Proof. Let us assume the contrary and let x∗ be a fractional extreme point of the polytope P (Hk , Tk ) associated with the star tree Hk , where |VHk | is minimum (i.e., for all star trees of n0 vertices whith n0 < |VHk |, the associated polytope is integer). Thus x∗ satisfies a unique system of linear independent equalities A x(VHk0 \ {vr }) + (k 0 − 1)x(vr ) = k 0 − 1

∀Hk0 ∈ Π1

(7)

x (v) = 1

∀v ∈ V1

(8)

x∗ (v) = 0

∀v ∈ V2

(9)

∗

such that |Π1 | + |V1 | + |V2 | = |VHk |, Π1 ⊆ Π, V1 ⊆ VHk and V2 ⊆ VHk . Moreover we have the following claims Claim 3.11 For all v ∈ VHk \ {vr }, x∗ (v) > 0. Proof of claim 3.11. Otherwise, |VHk | cannot be minimum. Claim 3.12 For each branch Pt , x∗ (Pt ) ≤ 1. Proof of claim 3.12. Otherwise, the variables of internal vertices of Pt must belong to (8) and to no other equality. Thus, |VHk | cannot be minimum. Claim 3.13 For the root vertex vr , x∗ (vr ) < 1. Proof of claim 3.13. Otherwise, x∗ cannot be fractional. Claim 3.14 Each branch Pt contains at most one internal vertex. Proof of claim 3.14. Otherwise, from claims 3.11 and 3.12 the variables associated with the internal vertices of Pt are fractional and cannot appear separately in (7) (i.e., if a variable of a vertex vi ∈ Pt appears in an equality (7), all the variables associated with the vertices of Pt appear in the same equality). It is easy to construct another feasible solution for the system of equalities A, which contradicts the extremity of x∗ .

6


Claim 3.15 If there exists a branch Pt such that x∗ (Pt ) < 1, then all support graphs of equalities (7), contain the branch Pt . Proof of claim 3.15. Otherwise, let ax∗ = b be an equality (7) not containing the variables associated with the vertices of Pt \ {vr }. Thus, the star tree inequality x∗ (Pt ) + ax∗ ≥ b + 1 is violated. Claim 3.16 For the root vertex vr , x∗ (vr ) > 0. Proof of claim 3.16. Otherwise, there must exists two branches Pt and Pt0 such that x∗ (Pt ) < 1 and x∗ (Pt0 ) < 1. It follows that these two branches must belong to all support graphs of equalities (7). It is easy to construct another solution for the system of equalities A, which contradicts the extremity of x∗ . From the claims 3.11, 3.12, 3.13 and 3.16 we deduce that for all v ∈ VHk , 0 < x∗ (v) < 1. We distinguish two cases a. There exists a branch Pt such that x∗ (Pt ) < 1. Let y ∗ ∈ RV defined as follows - y ∗ (v) = x∗ (v) for v ∈ Pt \ {vr , t} - y ∗ (vr ) = x∗ (vr ) + - y ∗ (v) = x∗ (v) − for all v ∈ VHk \ (Pt ∪ {vr }) From claim 3.15, the vector y ∗ satisfies all equalities of A. Contradiction with x∗ a fractional extreme point. b. For each branch Pt , x∗ (Pt ) = 1 It follows that for each pair of vertices vi , vj ∈ VHk \{vr }, x∗ (vi ) = x∗ (vj ) = 1 − x∗ (vr ). Since all the variables are fractional, the variable associated with each vertex v ∈ VHk \ {vr } must belong to at least one equality (7) x(VHk0 \ {vr }) + (k 0 − 1)x(vr ) = k 0 − 1

for k 0 ∈ {2, . . . , k}

By the variable changing presented before k 0 − k 0 x(vr ) + (k 0 − 1)x(vr ) = k 0 − 1 Hence, x(vr ) = 1 which contradicts the extremity of x∗ . 3.2

Clique stars

Proposition 3.21 For clique stars, P (G, T ) is not integral. Consider the following valid inequalities presented in [6] x(Qk0 ) ≥ k 0 − 1

∀Qk0 ∈ Θ

(10)

We recall that Θ contains all the terminal paths in Qk since they are clique stars of two terminals and inequalities (2) are included in (10). Theorem 3.21 For clique stars, the polytope given by inequalities (10) and trivial inequalities is integer.


4

7

TDI-ness

In this section we give a TDI descriptions for the multi-terminal vertex separator problem in star trees and clique stars. 4.1

Star trees

We first introduce some notations. Consider two star trees Hli and Hsj subgraphs of Hk , such that s ≥ 2, l ≥ 2, i ∈ {1, . . . , kl } and j ∈ {1, . . . , ks }. We denote i∩j by Hl,s the star tree subgraph of Hk , whose branches are all those in common i∪j i with Hl and Hsj . We denote by Hl,s a star tree subgraph of Hk with min{s + i∩j l − t(Hl,s ), s + l − 1} terminals, whose branches belong either to Hli or to Hsj . i∩j i∪j If t(Hl,s ) = 0, Hl,s is any star tree of s + l − 1 branches.

Fig. 3. Star trees, subgraphs of H4 in Figure 1.(a)

To illustrates these notations, if Hli is the graph in Figure 3.(c) and Hsj the i∪j graph in Figure 3.(e), then Hl,s is the graph in Figure 3.(a) or the graph in i∩j Figure 3.(b) and Hl,s does not exist. If Hli is the graph in Figure 3.(a) and Hsj i∪j i∩j the graph in Figure 3.(b), then Hl,s is the graph in Figure 1.(a) and Hl,s is the graph in Figure 3.(c). Let P ∗ be the linear program defined by the variable vector x, the objective function (1), the trivial inequalities (4) and inequalities (6). Let y ∈ RΠ + be the dual variable vector associated with inequalities (6). Consider the dual D∗ of P ∗ X max (k 0 − 1)yHk0 Hk0 ∈Π

X

yH ≤ w(v)

∀v ∈ V \ {vr }

(11)

H∈Π v

X

(k 0 − 1)yH ≤ w(vr )

(12)

H∈Π

yH ≥ 0

∀H ∈ Π

(13)

8


We notice that D∗ consists in packing star trees of Π in Hk satisfying the ca∗ pacity w of each vertex. Let y ∗ ∈ RΠ + be an optimal solution of D . The solution y ∗ is called maximal optimal if for each other optimal solution y ∈ RΠ + there exists s ∈ {1, . . . , k} satisfying the following conditions 1. 2.

fl P i=1 f P i=1

y Hli = y Hsi
0, l

i j j ∗ yH j > 0 either Hl is completely included in Hs or Hs is completely included in s

Hli . Corollary 4.11 For s ∈ {1, . . . , k}, there exists at most one star tree Hsj with ∗ a value yH j > 0 over all star trees with s terminals. s

Fig. 4. A maximal optimal solution structure

Figure 4 illustrates the structure of the maximal optimal solution y ∗ (each star tree is included in another, except Hk , and no more than one star tree with the same number of terminals). Theorem 4.11 For star trees, the linear system of P ∗ is TDI. Proof. We should prove that D∗ has an integer optimal solution. For this, we need to show the claims below. Claim 4.11 If for a star tree Hsj ∈ Π,

k k (P l) P

l=s i=1

∗ yH i < wv for each vertex v ∈ l

∗ i V (Hsj ) \ {vr } then yH i = 0 for each star tree Hp ∈ Π with p ≤ s − 1 terminals. p


9

Proof of claim 4.11. Let us assume the contrary, then there exists Hsj ∈ Π such that k k (P l) P ∗ j yH i < wv for all vertices v ∈ V (Hs ) \ {vr } and there exists p ∈ {2, . . . , s − l l=s i=1 i ∗ 1} and i ∈ {1, . . . , kp } such that yH i > 0 (from lemma 4.11, Hp is a subgraph p

of Hsj ). We suppose that p is maximum. To prove the claim, we will show that y ∗ cannot be maximal optimal by constructing another solution y ∈ RΠ + from ∗ y ∗ . Indeed, proving that y, obtained by adding α > 0 to yH and subtracting j s ∗ ∗ β > 0 from yH i , is feasible and optimal, will contradicts the maximality of y . p To guarantee the optimality of y we should have α × (s − 1) = β × (p − 1). Then k ∗ k (P l) yH i (p−1) P α(s−1) p ∗ if α = min{ (s−1) , min {c(v) − yH i }} then α > 0 and β = p−1 . v∈V

j \{vr } Hs

l=s i=1

l

Since p is maximum, thus y must be feasible optimal solution for D∗ . Thus our claim holds. Claim 4.12 If y ∗ is fractional then there exists exactly one star tree Hsj ∈ Π ∗ such that yH j is fractional. s

Proof of claim 4.12. We suppose that there exists two different star trees Hli ∗ ∗ and Hsj , such that yH j and yH i are fractional. We suppose that s is maximum s l ∗ (i.e., for all p ∈ {s + 1, . . . , k}, yH is integer). From corollary 4.11, s > l. We p distinguish two cases a. There exists a vertex v ∈ VHsj \ {vr } such that

k p) k (P P

p=s q=1

∗ yH q = w(v). Since p

∗ s is maximum, we know that yH is integer for any star tree Hp with p ∈ p k p) k (P P ∗ ∗ {s + 1, . . . , k}, yH q = w(v) and w(v) is integer. Thus, y j is integer. p H s

p=s q=1

∗ Contradiction with yH j fractional. s

b. For all vertex v ∈ VHsj \ {vr }, we have

k p) k (P P

p=s q=1

∗ yH q < w(v). From the claim p

∗ 4.11, yH = 0 for any star tree Hp ∈ Π with p ≤ s−1 terminals. Contradiction p ∗ with yH i fractional. l

∗ Thus there exists at most one star tree Hsj ∈ Π such that yH j is fractional. s

∗

Claim 4.13 If y is fractional then there exists another optimal solution y that is integer. ∗ Proof of claim 4.13. Let Hsj ∈ Π be the star tree such that yH j is fractional. We s distinguish three cases ∗ ∗ - If s = 1 then let y ∈ RΠ + be the solution obtained from y by setting yHsj = 0. The vector y represents an integer feasible optimal solution.

10


∗ ∗ ∗ - If s ≥ 2 then it is clear that (s−1)yH j is integer. We denote by = y j −by j c. Hs Hs s ∗ It follows that (s − 1)byH j c + (s − 1) is integer. Thus (s − 1) is integer. Let s

∗ ∗ y ∈ RΠ + be another solution obtained from y by subtracting from yHsj and ∗ by adding 1 to yH×(s−1)+1 for an arbitrary star tree H×(s−1)+1 .

Thus y is an integer optimal solution for D∗ . Then the proof is ended and the linear system of P ∗ is TDI. As consequence, we obtain the following min-max relation: In star trees, the minimum number of vertices covering all terminal paths is equal to the maximum packing of star trees.

4.2

Clique stars

For this section we introduce some notations. Consider two clique stars Qil and Qjs subgraphs of Qk such that s ≥ 2, l ≥ 2, i ∈ {1, . . . , kl } and j ∈ {1, . . . , ks }. i∩i Let Ql,s be the clique star subgraph of Qk whose branches are all those in common with Qil and Qjs . We denote by Qi∪j l,s the clique star subgraph of Qk whose branches are all those in common either with Qil or with Qjs . Let P Q be the linear program defined by the variable vector x, the objective function (1) and inequalities (4), (10). Let DQ be the dual of P Q . We notice that the DQ consists in packing clique stars of Θ in Qk satisfying the capacity of each vertex. Let y ∈ RΘ + be the dual variables associated with inequalities (10) and y ∗ the optimal solution of DQ . The solution y ∗ is called maximal optimal if for each other optimal solution y ∈ RΘ + there exists s ∈ {1, . . . , k} satisfying the following conditions 1. 2.

f P i=1 f P i=1

y Qil = y Qis
0 s

∗ j i i and yQ j > 0, either Qs is completely included in Ql or Ql is completely included s

in Qjs .


11

Proof of claim 4.21. We suppose that there exists two subgraphs Qil and Qjs of ∗ ∗ Qk , such that yQ i > 0 and y j > 0 and no one is included in the other. There Q s

l

∗ ∗ exists > 0 such that y ∈ RΘ + , obtained from y by subtracting from yQi and l ∗ ∗ ∗ from yQ j and by adding to y i∪j and to y i∩j , is feasible and optimal solution Q Q s

l,s

l,s

for D∗ . Thus contradiction with y ∗ maximal optimal. Corollary 4.21 For s ∈ {1, . . . , k}, there exists at most one clique star Qjs with ∗ a value yQ j > 0 over all clique stars with s terminals. s

Claim 4.22 For all Qjs subgraph of Qk , there exists a vertex v ∈ V (Qjs ) such k p) k (P P ∗ that yQ q = w(v). p p=s q=1

Proof of claim 4.22. We suppose there exists Qjs subgraph of Qk , such that k p) k (P P ∗ q for all v ∈ V (Qjs ), yQ q < w(v). There must exist Qp ∈ Θ subgraph of p p=s q=1

∗ Qk such that 2 ≤ p < s and yQ q > 0, Otherwise the solution is not optimal. p Θ ∗ We suppose that p is maximum. There exists 0 < ≤ yQ q such that y ∈ R+ , p ∗ ∗ ∗ obtained from y by subtracting from yQqp and by adding to yQj , is feasible s and optimal solution for D∗ . Thus contradiction with y ∗ maximal optimal.

Then we deduce an algorithm to solve D∗ . We start by packing the clique star of Θ having a maximum number of terminals until the capacity of some vertex is all used. The branches containing a saturated vertex, are we subtract the number of packed clique stars from the capacities of the other vertices. The same operations are repeated until Qk becomes a branch. Algorithm 1: An exact algorithm for solving the D∗

1 2 3 4

5

Data: The graph Qk = (VQk ∪ TQk , EQk ), a vector w ∈ NV Result: A maximal optimal solution y ∗ begin Let Qk+1 ← Qk ; for (i = k → 1) do Qi ← clique star obtained from Qi+1 by deleting each branch Pt containing a vertex v with w(v) = 0; ∗ yQ min i {w(v)}; i = ∀v∈V (Q )

6 7

for (v ∈ V (Qi )) do w(v) = w(v) − min i {w(v)}; ∀v∈V (Q )

Corollary 4.22 From claims 4.21 and 4.22, the algorithm 1 gives an optimal solution y ∗ for D∗ , and since the capacities are integer, it follows that y ∗ is integer.

12


As consequence, we obtain the following min-max relation: In clique stars, the minimum number of vertices covering all terminal paths is equal to the maximum packing of clique stars.

5

Conclusion

In this paper we characterized the polytope of the multi-terminal vertex separators in two classes of graph, the star trees and the clique stars and we showed that the associated linear system is total dual integral. Hence, the multi-terminal vertex separator problem is polynomial in these two classes of graph. It would be interesting to extend the results on other classes of graph, for instance, the terminal cycles [6], the graph composed of a cycle C of k vertices and k disjoint paths between each vertex of C and k terminals, the terminal tree [6], which is a tree with all leaves int T .

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