ellipsoidal relaxations of the stable set problem: theory ... · mising over the ellipsoid is equal...
TRANSCRIPT
Ellipsoidal Relaxations of the Stable Set Problem:
Theory and Algorithms
Monia Giandomenico∗
Fabrizio Rossi∗Adam N. Letchford†
Stefano Smriglio∗
To appear in SIAM Journal on Optimization
Abstract
A new exact approach to the stable set problem is presented, whichattempts to avoid the pitfalls of existing approaches based on linearand semidefinite programming. The method begins by constructingan ellipsoid that contains the stable set polytope and has the prop-erty that the upper bound obtained by optimising over it is equalto the Lovasz theta number. This ellipsoid can then be used to con-struct useful convex relaxations of the stable set problem, which can beembedded within a branch-and-bound framework. Extensive compu-tational results are given, which indicate the potential of the approach.
Keywords: combinatorial optimisation, stable set problem, semidefi-nite programming.
1 Introduction
Given an undirected graph G = (V,E), a stable set is a set of pairwisenon-adjacent vertices. The stable set problem (SSP) calls for a stable set ofmaximum cardinality. The cardinality of this stable set is called the stabilitynumber of G and is denoted by α(G).
The SSP is NP-hard [22], hard to approximate [21], and hard to solvein practice (e.g., [10, 14, 29, 33, 34, 35]). Moreover, it is a remarkable factthat sophisticated mathematical programming algorithms for the SSP, suchas those in [14, 29, 33, 35], have not performed significantly better thanrelatively simple algorithms based on implicit enumeration, such as those in[10, 34].
∗Dipartimento di Informatica, Universita Di L’Aquila, Via Vetoio, 67010,Coppito (AQ), Italy. E-mail: [email protected], fabrizio.rossi,
[email protected]†Department of Management Science, Lancaster University, Lancaster LA1 4YX,
United Kingdom. E-mail: [email protected]
1
A possible explanation for the failure of mathematical programming ap-proaches is the following. Linear Programming (LP) relaxations can besolved reasonably quickly, but tend to yield weak upper bounds. Semidef-inite Programming (SDP) relaxations, on the other hand, typically yieldmuch stronger bounds, but take longer to solve. Therefore, branch-and-bound algorithms based on either LP or SDP relaxations are slow, due tothe large number of nodes in the search tree, or the long time taken toprocess each node, respectively.
In this paper we present a potential way out of this impasse. The keyconcept is that one can efficiently construct an ellipsoid that contains thestable set polytope, in such a way that the upper bound obtained by opti-mising over the ellipsoid is equal to the standard SDP bound, the so-calledLovasz theta number. This ellipsoid can then be used to construct usefulconvex programming relaxations of the stable set problem or, perhaps moreinterestingly, to derive cutting planes. These cutting planes turn out to bestrong and easy to generate.
We remark that our approach can be applied to the variant of the SSP inwhich vertices are weighted, and one seeks a stable set of maximum weight.
The paper is structured as follows. Some relevant literature is reviewedin Section 2. The ellipsoids are analysed theoretically in Section 3. Section4 discusses how to use the ellipsoids within exact algorithms for the SSP.Some computational results are presented in Section 5, and concluding re-marks are made in Section 6.
Remark: An extended abstract of this paper appeared in the 2011 IPCOproceedings [16]. In this full version, we explore three different algorithmicschemes (see Section 4), rather than only one as in [16].
2 Literature Review
We now review the relevant literature. From this point on, n = |V | andm = |E|.
2.1 Linear programming relaxations
The SSP has the following natural formulation as a 0-1 LP:
max∑
i∈V xi
s.t. xi + xj ≤ 1 ({i, j} ∈ E) (1)
x ∈ {0, 1}n, (2)
where the variable xi takes the value 1 if and only if vertex i is in the stableset.
2
The convex hull in Rn of feasible solutions to (1)–(2) is called the stableset polytope and denoted by STAB(G). This polytope has been studiedin great depth (see, e.g., [3, 18, 31]). The most well-known facet-defininginequalities for STAB(G) are the clique inequalities of Padberg [31]. A cliquein G is a set of pairwise adjacent vertices, and the associated inequalitiestake the form: ∑
i∈Cxi ≤ 1 (∀C ∈ C), (3)
where C denotes the set of maximal cliques in G. Note that the cliqueinequalities dominate the edge inequalities (1).
The polytope: {x ∈ Rn+ : (3) hold
}is denoted by QSTAB(G). The upper bound on α(G) obtained by optimisingover QSTAB(G) is called the fractional clique covering number and denotedby χf (G). By definition, we have STAB(G) ⊆ QSTAB(G) and α(G) ≤χf (G). Unfortunately, computing χf (G) is NP-hard, since the separationproblem for clique inequalities is NP-hard [30]. On the other hand, somefast and effective separation heuristics exist, not only for clique inequalities,but also for various other inequalities (e.g., [3, 14, 29, 33, 35]). Nevertheless,LP-based cutting-plane approaches can run into difficulties when n exceeds250 or so, mainly due to the weakness of the upper bounds.
2.2 Semidefinite programming relaxations
Lovasz [25] introduced another upper bound on α(G), called the theta num-ber and denoted by θ(G), which is based on an SDP relaxation. The boundcan be derived in several different ways, and we follow the derivation pre-sented in [18]. We start by formulating the SSP as the following non-convexquadratically-constrained program:
max∑
i∈V xi (4)
s.t. x2i − xi = 0 (i ∈ V ) (5)
xixj = 0 ({i, j} ∈ E). (6)
In order to linearise the constraints, we introduce an auxiliary matrix vari-able X = xxT , along with the augmented matrix
Y =
(1
x
)(1
x
)T=
(1 xT
x X
).
3
We then note that Y is real, symmetric and positive semidefinite (psd),which we write as Y � 0. This leads to the following SDP relaxation:
max∑
i∈V xi (7)
s.t. x = diag(X) (8)
Xij = 0 ({i, j} ∈ E) (9)
Y � 0. (10)
The corresponding upper bound is θ(G). One can compute θ(G) to arbitraryfixed precision in polynomial time [18].
Lovasz [25] proved that α(G) ≤ θ(G) ≤ χf (G). (This is a remarkableresult, given that it is NP-hard to compute χf (G).) In practice, θ(G) isoften a reasonably strong upper bound on α(G) (e.g., [4, 9, 19, 33]). Unfortu-nately, solving large-scale SDPs can be rather time-consuming, which makesSDP relaxations somewhat unattractive for use within a branch-and-boundframework.
The above SDP can be strengthened by adding various valid inequalities(e.g., [3, 9, 13, 19, 26, 36]). We omit details, for the sake of brevity.
2.3 The Lovasz theta body and ellipsoids
The following beautiful result can be found in Grotschel, Lovasz & Schrijver[18]. Let us define the following convex set:
TH(G) ={x ∈ Rn : ∃X ∈ Rn×n : (8)− (10) hold
}.
Then we have:STAB(G) ⊆ TH(G) ⊆ QSTAB(G).
This provides an alternative proof that α(G) ≤ θ(G) ≤ χf (G).The set TH(G) is called the theta body. A characterisation of TH(G) in
terms of convex quadratic inequalities was given by Fujie & Tamura [12]. Fora given vector µ ∈ Rm, let M(µ) denote the symmetric matrix with µij/2in the ith row and jth column whenever {i, j} ∈ E, and zeroes elsewhere.Then, given vectors λ ∈ Rn and µ ∈ Rm such that Diag(λ) + M(µ) is psd,the set
E(λ, µ) ={x ∈ Rn : xT (Diag(λ) +M(µ))x ≤ λ · x
}(11)
is an ellipsoid that contains STAB(G). The result we need is the following:
Theorem 1 (Fujie & Tamura, 2002) For any graph G, we have:
TH(G) =⋂
λ,µ:Diag(λ)+M(µ)�0
E(λ, µ).
4
2.4 Relaxations of non-convex quadratic problems
For what follows, we will also need to briefly review Lagrangian and SDPrelaxations of general non-convex quadratic problems. Given a problem ofthe form:
inf xTQ0x+ c0 · xs.t. xTQjx+ cj · x = bj (j = 1, . . . , r) (12)
x ∈ Rn,
Shor [37] proposed to relax the constraints (12) in Lagrangian fashion, us-ing a vector φ ∈ Rr of Lagrangian multipliers. The Lagrangian objectivefunction is then:
f(x, φ) = xT
Q0 +r∑j=1
φjQj
x+
c0 +r∑j=1
φjcj
· x− r∑j=1
φjbj ,
and the Lagrangian dual is:
supφ
infxf(x, φ).
Shor also introduced the following SDP, which we call the dual SDP:
sup t
s.t.
(−t− φ · b
(c0 +
∑rj=1 φjc
j)T/ 2(
c0 +∑r
j=1 φjcj)/ 2 Q0 +
∑rj=1 φjQ
j
)� 0
φ ∈ Rr, t ∈ R.
He showed:
Theorem 2 (Shor [37]) The dual SDP yields the same lower bound asthe Lagrangian dual. Moreover, a vector φ∗ is an optimal solution to theLagrangian dual if and only if there exists an optimal solution (φ∗, t∗) to thedual SDP.
Note that, if such a vector φ∗ exists, then the function f(x, φ∗) is convex,since the matrix Q0 +
∑rj=1 φ
∗jQ
j is psd.As noted in [11, 32], Shor’s SDP is the dual of the following more natural
SDP relaxation of the original problem, which we call the primal SDP:
inf Q0 •X + c0 · xs.t. Qj •X + cj · x = bj (j = 1, . . . , r)
Y � 0,
5
where Qj • X denotes∑n
i=1
∑nk=1Q
jikXik. Thus, if either or both of the
primal and dual SDP satisfy Slater’s condition, then the primal SDP yieldsthe same lower bound as the Lagrangian dual.
In our recent paper [17], these results are applied to the SSP, yielding asubgradient algorithm that approximates θ(G) from above. Luz & Schrijver[27] showed that the optimal value of the Lagrangian dual remains equal toθ(G) even if one forces all of the Lagrangian multipliers for the constraints(5) to equal 1.
3 On Ellipsoids
From now on, when we say “ellipsoid”, we mean one of the ellipsoids inthe family defined by Fujie & Tamura [12] (see Subsection 2.3). In thissection, we investigate the ellipsoids in more detail. In Subsection 3.1, weshow how to construct ellipsoids that yield strong upper bounds on α(G).In Subsection 3.2, we consider two other desirable properties for ellipsoidsto have. In Subsection 3.3, we show that, perhaps surprisingly, the bestellipsoid according to the first criterion can be bad according to the othertwo criteria. This analysis is based on a special class of graphs, includingthe perfect graphs.
We note in passing that every point x ∈ Rn satisfying (5), (6) alsosatisfies at equality the constraint in (11). This implies that every extremepoint of STAB(G) lies on the boundary of every ellipsoid.
3.1 Optimal ellipsoids
Recall that Theorem 1 expresses TH(G) as the intersection of the entireinfinite family of ellipsoids. We will show that there exists at least oneellipsoid with a very desirable property. First, however, we need the dual ofthe SDP (7)–(10), which can be written in the form:
min t (13)
s.t.
(t −(e+ λ)T /2
−(e+ λ)/2 Diag(λ) +M(µ)
)� 0 (14)
λ ∈ Rn, µ ∈ Rm, t ∈ R, (15)
where e is the vector of all ones. Here, λ and µ are the vectors of dualvariables for the constraints (8) and (9), respectively. (As before, t is thedual variable for the constraint Y00 = 1.)
We are now ready to present our main result:
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Theorem 3 There exists at least one optimal solution to the dual SDP(13)-(15). Moreover, for any such solution (λ∗, µ∗, t∗), we have:
θ(G) = t∗ = max
{∑i∈V
xi : x ∈ E(λ∗, µ∗)
}. (16)
Proof. The SDP (7)–(10) has the form specified in Theorem 3.1 of Tuncel[38], and therefore satisfies the Slater condition. As a result, an optimaldual solution (λ∗, µ∗, t∗) exists, strong duality holds, and t∗ = θ(G).
Now consider a Lagrangian relaxation of the non-convex quadratic prob-lem (4)–(6), in which all constraints are relaxed. This relaxation can bewritten as:
max
{∑i∈V
xi − xT (Diag(λ) +M(µ))x+ λ · x : x ∈ Rn},
where λ ∈ Rn and µ ∈ Rm are the vectors of Lagrangian multipliers for theconstraints (5) and (6), respectively. Applying Theorem 2 in Subsection 2.4,but switching signs to take into account the fact that the objective (7) is ofmaximisation type, an optimal solution to the Lagrangian dual is obtainedby setting λ to λ∗ and µ to µ∗, and the optimal value of the Lagrangiandual is equal to θ(G). In other words, θ(G) is equal to:
max
{∑i∈V
xi − xT (Diag(λ∗) +M(µ∗))x+ λ∗ · x : x ∈ Rn}. (17)
Moreover, the matrix Diag(λ∗) +M(µ∗) is psd from (14), and therefore theobjective function in (17) is concave.
Now, let us write the relaxation on the right-hand side of (16) as:
max
{∑i∈V
xi : xT (Diag(λ∗) +M(µ∗))x ≤ λ∗ · x, x ∈ Rn}. (18)
We relax this further by moving the (one and only) constraint to the objec-tive function, with non-negative Lagrangian multiplier φ. The result is:
max
{∑i∈V
xi + φ(−xT (Diag(λ∗) +M(µ∗))x+ λ∗ · x
): x ∈ Rn
}.
Note that this reduces to (17) when φ is 1. Therefore, the upper boundfrom (18) must be at least as strong as the one from (17), which as we sawis equal to θ(G).
To complete the proof, it suffices to show that the optimal value of φ,which we denote by φ∗, is 1. Suppose that it was not equal to 1. Then wecould obtain an upper bound that is better than θ(G) by taking (17) andmultiplying λ∗ and µ∗ by φ∗. But this contradicts the fact that (λ∗, µ∗) isan optimal solution to the Lagrangian dual for (4)–(6). �
7
In other words, an optimal dual solution to the SDP can be used toconstruct a relaxation of the SSP that has a linear objective function anda single convex quadratic constraint, whose corresponding upper bound isequal to θ(G). (We remark that a similar proof technique was used in [2]to reformulate 0-1 quadratic programs. The main difference between theirmethod and ours, apart from the fact that they apply to different problems,is that they perturb the objective function, whereas we generate a validconvex quadratic constraint.)
We say that an ellipsoid E(λ∗, µ∗) is optimal if there exists a t∗ such that(λ∗, µ∗, t∗) is an optimal solution to the dual of the SDP (or, equivalently,if (λ∗, µ∗) is an optimal solution to the Lagrangian dual). We illustrate thisconcept on three simple examples:
Example 1: Let G be a stable set on n nodes, for which α(G) = θ(G) = n.The unique optimal dual solution has λ∗i = 1 for all i ∈ V . The correspond-ing convex quadratic constraint is:∑
i∈Vx2i ≤
∑i∈V
xi.
This can be written in the alternative form∑i∈V
(xi − 1/2)2 ≤ n/4.
One then sees that it defines a sphere of radius√n/2 centred at (1/2, . . . , 1/2)T .
The maximum value that∑
i∈V xi can take over this sphere is n, attainedwhen xi = 1 for all i ∈ V . �
Example 2: Let G be a clique on n nodes, for which α(G) = θ(G) = 1.The unique optimal dual solution has λ∗i = 1 for all i ∈ V and µ∗e = 2 forall e ∈ E. The corresponding convex quadratic constraint is:(∑
i∈Vxi
)2
≤∑i∈V
xi.
This is equivalent to the linear inequalities 0 ≤∑
i∈V xi ≤ 1. �
Note that the “ellipsoid” in Example 2 is unbounded. We say more aboutthis in the next subsection. We close this subsection with a less trivialexample:
Example 3: Let G be the 5-hole, i.e., let V = {1, 2, 3, 4, 5} and E ={{1, 2}, {2, 3}, {3, 4}, {4, 5}, {1, 5}}. In this case, α(G) = 2 [31] and θ(G) =√
5 [18]. The unique optimal dual solution has λ∗i = 1 for all i ∈ V andµ∗e =
√5 − 1 for all e ∈ E. The corresponding convex quadratic constraint
8
is: ∑i∈V
x2i + (√
5− 1)∑{i,j}∈E
xixj ≤∑i∈V
xi.
The maximum value that∑
i∈V xi can take over the corresponding ellipsoid
is√
5, attained when xi = 1/√
5 for all i ∈ V . �
3.2 Degenerate and weak ellipsoids
At this point, it is helpful to define two special kinds of ellipsoids, as follows:
Definition 1 An ellipsoid E(λ, µ) will be called “weak” if there exists aninteger k > 1 and distinct vector pairs (λ1, µ1), . . . , (λk, µk) such that
∩ki=1E(λi, µi) ⊆ E(λ, µ).
Otherwise, it will be called “strong”.
Definition 2 An ellipsoid E(λ, µ) will be called “degenerate” if it has infi-nite volume, or, equivalently, if the matrix Diag(λ) + M(µ) is psd but notpositive definite (pd).
In principle, instead of using a weak ellipsoid to construct a relaxationof STAB(G), we could take the intersection of two or more strong ellip-soids to obtain a tighter relaxation. Unfortunately, we are not aware of anypolynomial-time algorithm to test if a given ellipsoid E(λ, µ) is strong.
As for degenerate ellipsoids, we will see in Section 4 that they couldcause numerical problems for our algorithms. Fortunately, we found that,for graphs of practical interest, optimal ellipsoids are very rarely degenerate.(One can easily check if E(λ, µ) is degenerate, since one can easily checkwhether a rational matrix is psd or pd [18].)
To make these concepts more concrete, we apply them to the same threeexamples as in the previous subsection:
Example 1 (cont.): We saw that, when G is a stable set on n nodes, theunique optimal ellipsoid is a sphere of radius
√n/2 centered at (1/2, . . . , 1/2)T .
This sphere is a non-degenerate ellipsoid. On the other hand, observe that,for i = 1, . . . , n, the following convex set is an ellipsoid that is both degen-erate and strong:{
x ∈ Rn : x2i ≤ xi}
= {x ∈ Rn : xi ∈ [0, 1]} .
The intersection of these n degenerate and strong ellipsoids is the unit hy-percube, which is strictly contained in the sphere mentioned. (In fact it isequal to STAB(G).) Therefore the optimal ellipsoid is weak in this case. �
Example 2 (cont.): We saw that, whenG is a clique on n nodes, the unique
9
optimal ellipsoid is defined by the two linear inequalities 0 ≤∑
i∈V xi ≤ 1.So, the ellipsoid is degenerate. On the other hand, the inequality
∑i∈V xi ≤
1 is a clique inequality, which defines a facet of STAB(G). Since no otherellipsoid has this facet on its boundary, the optimal ellipsoid is strong. �
Example 3 (cont.): We saw that, when G is the 5-hole, the unique optimaldual solution has λ∗i = 1 for all i ∈ V and µ∗e =
√5 − 1 for all e ∈ E. The
minimum eigenvalue of the associated matrix Diag(λ)+M(µ) is zero, whichmeans that the optimal ellipsoid is degenerate. We do not know whether itis strong or weak, but conjecture that it is strong. �
The following additional example shows that the unique optimal ellipsoidcan be both degenerate and weak.
Example 4: Let G be the union of an isolated node and an edge. That is,let n = 3 and let E contain the edge {2, 3}. For this graph, α(G) = θ(G) = 2.The unique optimal dual solution has λ∗i = 1 for all i ∈ V and µ∗23 = 2. Thecorresponding convex quadratic constraint is:
x21 + (x2 + x3)2 ≤ x1 + x2 + x3.
Now observe that any point x∗ ∈ R3 with x∗1 ∈ {0, 1} and x∗3 = −x∗2 lies onthe boundary of this ellipsoid. Therefore the ellipsoid is degenerate. On theother hand, using the same arguments as in Example 1 and Example 2, thefollowing four ellipsoids are both degenerate and strong:{
x ∈ R3 : xi ∈ [0, 1]}
(i = 1, 2, 3){x ∈ R3 : 0 ≤ x2 + x3 ≤ 1
}.
The intersection of these four ellipsoids is equal to STAB(G) in this case.Therefore the optimal ellipsoid is weak. �
3.3 Ellipsoids, clique covers and perfect graphs
Given a graph G, the minimum number of cliques needed to cover thenodes of G is called the clique covering number and denoted by χ(G).By definition, χf (G) ≤ χ(G), and we already saw in Subsection 2.2 thatα(G) ≤ θ(G) ≤ χf (G). A graph G is called perfect if α(G′) = χ(G′) holdsfor every node-induced subgraph G′ ⊆ G. The following proposition statesthat only degenerate ellipsoids are needed to describe the stable set polytopein the perfect graph case:
Proposition 1 If G = (V,E) is a perfect graph, then STAB(G) is the in-tersection of a finite number of ellipsoids that are strong, but degenerate.
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Proof. Using the same argument as in Example 2 in the previous subsec-tions, given any clique C, the convex quadratic inequality(∑
i∈Cxi
)2
≤∑i∈C
xi
defines the following strong and degenerate ellipsoid:{x ∈ Rn : 0 ≤
∑i∈C
xi ≤ 1
}.
Similarly, using the same argument as in Example 1 in the last subsection,for each i ∈ V , the convex quadratic inequality x2i ≤ xi defines the followingstrong and degenerate ellipsoid:
{x ∈ Rn : xi ∈ [0, 1]} .
The intersection of these ellipsoids satisfies all clique and non-negativityinequalities. When G is perfect, STAB(G) = QSTAB(G), and is thereforecompletely defined by clique and non-negativity inequalities. �
Next, consider the class of graphs for which α(G) = χ(G). This includesall perfect graphs of course, but it includes many other graphs besides. (Forexample, let G be the graph obtained from the 5-hole by adding a new node6 and a new edge {1, 6}. Then α(G) = χ(G) = 3, yet G is not perfect.)The following proposition shows that, for the graphs in this class, there existoptimal ellipsoids of a particularly simple form.
Proposition 2 Suppose that α(G) = χ(G). Let C1, . . . , Cα(G) ⊂ V be anyfamily of cliques in G that form a clique cover. If necessary, delete verticesfrom the cliques in the family until each vertex is included in exactly oneclique in the family. Let C ′1, . . . , C
′α(G) be the resulting cliques. Then the
convex quadratic inequality
α(G)∑k=1
∑i∈C′k
xi
2
≤α(G)∑k=1
∑i∈C′k
xi (19)
defines an optimal ellipsoid.
Proof. First, observe that the inequality (19) can be obtained from (11)by setting λi to 1 for all i ∈ V and setting µij to 2 if {i, j} ⊆ C ′k for some k,and to 0 otherwise. Moreover, the left-hand side of (19) is a sum of squares,and therefore the inequality is convex, as stated.
Now, let f(x) denote the left-hand side of (19) minus the right-hand side,so that the specified ellipsoid is defined by the constraint f(x) ≤ 0. Also let
11
x∗ ∈ Rn be an arbitrary point such that∑
i∈C′kx∗i = 1 for k = 1, . . . , α(G).
Note that f(x∗) = 0, and therefore x∗ is on the boundary of the ellipsoid.A first-order Taylor series expansion of f(x) at x∗ shows that the tangenthyperplane to the ellipsoid at x∗ is given by the equation∇f(x∗)·(x−x∗) = 0.Now, for k = 1, . . . , α(G) and any i ∈ C ′k, we have ∂f/∂xi = 2
∑j∈C′k
xj−1,
which takes the value 1 at x∗. So, the tangent hyperplane is defined by theequation
∑i∈V xi −
∑i∈V x
∗i = 0. But
∑i∈V x
∗i = α(G), and therefore the
equation reduces to∑
i∈V xi = α(G). So, all points in the ellipsoid satisfy∑i∈V xi ≤ α(G). This implies that the given pair (λ, µ) is an optimal
solution to the Lagrangian dual, and therefore that the ellipsoid is optimal.�
Counter-intuitively, the ellipsoids described in Proposition 2 are almostalways both degenerate and weak:
Proposition 3 The optimal ellipsoids obtained via Proposition 2 are:
• degenerate unless G is a stable set;
• weak unless G is a clique.
Proof. Note that the matrix of quadratic terms on the left-hand side of(19) is block-diagonal, with α(G) blocks. This implies that its rank is α(G).Now, if G is a stable set, the matrix is of full rank (in fact, it is the identitymatrix). It is therefore pd, and the ellipsoid is non-degenerate. If, on theother hand, G is not a stable set, then α(G) < n, and the matrix is singular.It therefore cannot be pd, and the ellipsoid is degenerate.
For the second point, recall that Proposition 1 gave a complete char-acterisation of the convex quadratic inequalities needed to define strongellipsoids for perfect graphs. The inequality (19) is one of them if and onlyif α(G) = 1, i.e., if G is a clique. �
4 Using Ellipsoids Computationally
Our goal is to design branch-and-cut algorithms that exploit the existenceof (near) optimal ellipsoids. In this section, we discuss ways in which thiscan be done.
4.1 Computation of a near-optimal ellipsoid
In practice, the dual SDP can be solved only to limited precision. Therefore,we must be content with a “near-optimal” dual solution. It is however crucialfor the dual solution to be feasible, i.e., for the matrix Diag(λ) + M(µ) tobe psd, since otherwise the resulting set E(λ, µ) would not be an ellipsoid.In the conference version of this paper [16], the dual solution was obtained
12
using the SDP solver described in [28]. Afterwards, however, we showedthat a tailored implementation of the subgradient method may be muchfaster, if a slight impairment of the upper bound is accepted [17]. This isindeed crucial to the application of the ellipsoidal method to large graphs.Therefore, for this paper we used our subgradient approach to compute thedual solution, which we denote by (λ, µ).
4.2 An initial convex relaxation of the SSP
We start by constructing a collection C of cliques covering all the edges of G,using a standard greedy algorithm (see, e.g., [14]). Now, let xTAx ≤ b ·x bethe convex quadratic constraint associated with the ellipsoid E(λ, µ). Ourinitial convex relaxation of the SSP is:
max∑
i∈V xi
s.t. xTAx ≤ b · x (20)∑i∈C xi ≤ 1 (C ∈ C)
x ∈ [0, 1]n.
Note that this is a (convex) quadratically constrained program (QCP). Inprinciple, this relaxation could be used directly within a plain branch-and-bound algorithm. We however use a more sophisticated approach, as weexplain in the following subsections.
4.3 Alternative representations of the quadratic constraint
In practice, it is helpful to reformulate the quadratic constraint in (20). Wefound that three options worked well.
The first option is to convert the quadratic constraint into a second-orderconic (SOC) constraint. This can be profitable since such constraints can behandled quite efficiently via interior-point algorithms (see, e.g., [1, 24]). Ifwe let A be factorised as BTB, this representation amounts to adding a newvector of variables y ∈ Rn and two more variables z1 and z2, and replacingthe original constraint with:
Bx = y
z1 = 1− b · xz2 = 1 + b · xz2 ≥
∥∥∥(z1y )∥∥∥ .Notice that all these new constraints are linear, apart from the last, whichis a SOC constraint.
The second option is to approximate the SOC constraint in Rn by thewell-known linear description of Ben-Tal and Nemirovski [5]. They define
13
an extended formulation in the space Rn+d with d+ 1 inequalities, where dis a parameter. Reformulation accuracy is proportional to 2d+1 (see [5] fordetails). Our motivation for doing this is that one then needs to solve onlyLP relaxations within a branch-and-bound algorithm.
The third option is to work in the space of the original x variables,and approximate the original quadratic inequality directly by means of lin-ear inequalities. We use the cutting-plane method of Kelley [23]. Given aconvex constraint of the form f(x) ≤ 0, it constructs a polyhedral outer-approximation via a set of linear constraints of the form ∇f(xi)·(x−xi) ≤ 0,for a collection of suitably chosen points xi. In our implementation, thesepoints are chosen to lie on the boundary of the ellipsoid, so that the linearconstraints define tangent hyperplanes on the boundary of the ellipsoid.
Here are the details. Since the ellipsoid is invariably non-degenerate inpractice, it has a centre, which we denote by x. One can easily show thatx = 1
2A−1b. Then, to generate a collection of Kelley inequalities, one simply
runs Kelley’s cutting-plane algorithm, using the following separation routineto generate cuts in each iteration:
1. Let x∗ ∈ [0, 1]n be a point lying outside the ellipsoid.
2. Find the value 0 < ε < 1 such that the point
x = εx∗ + (1− ε)x
lies on the boundary of the ellipsoid. (This can be done by solving aquadratic equation in the single real variable ε.)
3. Generate the Kelley cutting plane corresponding to x, which takes theform (2Ax− b) · x ≤ (2Ax− b) · x.
We will call the resulting cutting-plane algorithm “Algorithm 0”. This ap-proach has the advantage that the branch-and-cut algorithm needs to solveonly LPs that involve the original x variables.
The outer approximation can be further strengthened by adding well-known valid inequalities for STAB(G). In particular, one can imagine im-proving the outer approximation by adding violated clique inequalities ofthe form
∑i∈C xi ≤ 1 (for some maximal clique C in G) along with each
Kelley cut generated. Note that by adding clique inequalities alone, one can-not expect a bound improvement, since TH(G) already satisfies all cliqueinequalities. Nevertheless, we have found that the addition of clique inequal-ities can improve the convergence of the cutting-plane algorithm, probablybecause they can be generated more quickly than Kelley cuts and are morewell-behaved numerically. Algorithm 1 describes the Kelley cutting-planealgorithm with additional generation of clique inequalities.
14
Algorithm 1 Kelley cutting plane algorithm with clique inequalities
Input: A linear formulation P, an ellipsoid E(λ, µ)Output: An updated formulation P
P ← Pfor i := 1 to maxiter do
optimize over P, get x∗
evaluate x, ∇f(x)if ∇f(x) · (x∗ − x) > 0 then
scale ∇f(x) · (x− x) ≤ 0 by a constant factor Γ→ a · x ≤ bround down bac · x ≤ bbcP ← P ∪ {bac · x ≤ bbc}optimize over P, get x∗
execute clique cutting plane routine thatreturns a collection of clique inequalities C;P ← P ∪ C
elseSTOP
end ifend forreturn P
Remark: In the extended abstract of this paper [16], we pointed out thatthe Kelley cuts can be strengthened using a simple “sequential lifting” pro-cedure. In our experiments, however, we found this to be of little benefit.For this reason, and also for the sake of brevity, we do not consider thisoption further in this paper.
4.4 Branch-and-cut
Finally, we discuss branch-and-cut algorithms. In principle, one can embedthe relaxation (20) directly within a branch-and-bound algorithm, and thenadd known valid inequalities for the SSP (such as clique inequalities) at eachnode of the enumeration tree. Unfortunately, reoptimising after branchingis slow when using interior-point methods. For the same reason, a branch-and-cut algorithm based on SOCP relaxations is not viable either. Not onlythat, but we have found that the Ben-Tal and Nemirovski [5] relaxation doesnot work well in a branch-and-cut context either, due partly to the presenceof additional variables and constraints, and partly to numerical difficulties.
Our preferred choice, therefore, is to use a branch-and-cut algorithmwith LP relaxations in the space of the original x variables, in which Kelleycuts and/or clique inequalities can be generated at each node.
15
5 Computational Results
We now perform an experimental comparison of the different algorithmicoptions. Two major issues are addressed: (i) showing that the θ-bound iscomputationally accessible by linear descriptions and (ii) checking whethersuch descriptions can be of use within a branch-and-cut framework.
5.1 Test bed and implementation details
The test bed consists of the graphs from the DIMACS Second Challenge[20] with up to 700 vertices, available at the web site [7]. Among them, wedo not consider the “easy” ones, that is, those for which the initial set ofclique inequalities included in the relaxation (20) provides an LP relaxationyielding the integer optimum, or showing a negligible integrality gap. Theinstance features are reported in Table 1.
The cutting plane/branch-and-cut algorithms are implemented by theGurobi Optimizer 5.6.2 framework in a linux x86 64 architecture (Ubuntu12.04), compiler g++ 4.8. Gurobi is also used as QCP solver. The com-putations are run on a machine with two Intel Xeon 5150 processors (for atotal of 4 cores) clocked at 2.6 GHz and 8GB RAM.
5.2 Strength of the linear descriptions
Table 2 compares θ(G) with four upper bounds:
1. UBE achieved by the first formulation mentioned in Subsection 4.3, inwhich the quadratic constraint is converted into a SOC constraint andthe relaxation is solved by the Gurobi barrier algorithm;
2. UBBTN corresponding to the optimal value of the Ben-Tal and Ne-mirovski reformulation; the parameter d has been fixed to d = 8 forall graphs, as a result of a preliminary investigation.
3. UBK returned by Algorithm 0 with iteration limit set to 300 (thealgorithm terminates early if the target ellipsoid bound is achieved);
4. UBC returned by Algorithm 1 with iteration limit set to 50 (the algo-rithm terminates early if the target ellipsoid bound is achieved). Theseparation heuristic for clique inequalities is executed “aggressively”(i.e., repeatedly invoked until no further clique violations are found)if the graph density ≤ 25%, but “moderately” (i.e., just one call foreach Kelley cut) if the graph density is > 25%.
The chosen iteration limits avoid a significant tailing-off effect for all in-stances and yield, on average, a profitable trade-off between quality of theupper bound and efficiency. The columns headed “%gap(θ(G))” report the
16
Name |V | Density α(G) θ(G)
brock200 1 200 0.25 21 27.46brock200 2 200 0.5 12 14.23brock200 3 200 0.39 15 18.82brock200 4 200 0.34 17 21.29brock400 1 400 0.25 27 39.7brock400 2 400 0.25 29 39.56brock400 3 400 0.25 31 39.48brock400 4 400 0.25 33 39.7C.125.9 125 0.1 34 37.81C.250.9 250 0.1 44 56.24DSJC125.1 125 0.09 34 38.4DSJC125.5 125 0.5 10 11.47DSJC125.9 125 0.9 4 4gen200 p0.9 44 200 0.1 44 44keller4 171 0.35 11 14.01p hat300-1 300 0.76 8 10.07p hat300-2 300 0.51 25 26.96p hat300-3 300 0.26 36 41.17p hat500-1 500 0.75 9 13.07p hat500-2 500 0.5 36 38.98p hat500-3 500 0.25 50 58.57p hat700-1 700 0.75 11 15.12p hat700-2 700 0.5 44 49.02p hat700-3 700 0.25 62 72.74san400 0.5-1 400 0.5 13 13san400 0.7-1 400 0.3 40 40san400 0.7-2 400 0.3 30 30san400 0.7-3 400 0.3 22 22san400 0.9-1 400 0.1 100 100sanr200 0.7 200 0.3 18 23.84sanr200 0.9 200 0.1 42 49.27
Table 1: Graph characteristics
percentage gap UB−θ(G)θ(G) % w.r.t. the Lovasz θ bound. Notice that the ap-
proximation error is small for most of the instances. Notable exceptions arethe graphs p hat300-1, p hat500-1 and p hat700-1, representing the dens-est members of the p hat class. To explain this fact, recall that the initialellipsoid E(λ, µ) is computed by the subgradient algorithm presented in [17].
17
When the graph is dense, the subgradient algorithm has to handle a largenumber of Lagrangian multipliers, which leads to very slow convergence. So,for those instances, the ellipsoid generated is quite far from being optimal.Happily, we will show in Section 4 that our branch-and-cut algorithm iseffective on those very instances, even if the ellipsoids are sub-optimal.
Table 2 also shows that the quality of the upper bound UBE is preservedby all the linearization techniques. Indeed, the approximations achieved byUBBTN and even by UBK, UBC are satisfactory. This was somehow ex-pected from the BTN reformulation as it provides a guaranteed approxima-tion. Interestingly, also Algorithm 0 achieves a strong bound, but Algorithm1 exhibits significantly better performance with respect to CPU time. Thisis clear from the details reported in Table 3, in which tK and tC denotesthe total CPU time (sum of separation and LP reoptimization times) ofAlgorithm 0 and Algorithm 1 respectively. Table 3 contains also the fol-lowing data for each graph: time required by the subgradient algorithm tocompute the initial ellipsoid; time required by the barrier algorithm to solvethe SOCP reformulation; number of variables and constraints of the BTNreformulation and time elapsed by the Gurobi LP solver to solve it; numberof Kelley cuts generated by Algorithm 0, upper bound and number of Kelleycuts by Algorithm 0 at tC; number of Kelley cuts and clique cuts generatedby Algorithm 1. All times are expressed in seconds.
Looking at the bounds computed by Algorithm 0 at tC, one can observethat Algorithm 1 shows a faster convergence. Indeed it computes betterbounds in 21 out of 31 cases, typically corresponding to hard instances.In the remaining 10 cases, even if the bound from Algorithm 0 is slightlystronger, it is achieved by a much larger number of Kelley cuts.
The Kelley cuts provide two additional benefits. The first benefit con-cerns dense graphs, in which the standard formulation size may becomeintractable. In fact, as the density goes over 40− 50%, the number of cliqueinequalities grows rapidly and overloads the simplex reoptimization. How-ever, by applying the Kelley cuts, many of the clique inequalities can bediscarded from the current formulation, while keeping safe the quality ofthe upper bound, yielding an important speed-up.
A second benefit is that using a small number of Kelley cuts protects themethod from a potential difficulty caused by the cut density. Kelley cuts arestructurally quite dense, a feature that typically degrades the computationalbehaviour of cutting planes (see the CPU times of the pure Kelley cutting-plane algorithm). On the contrary, a selected bunch of Kelley cuts is cost-effective even when used in branch-and-cut frameworks, as illustrated in thenext section.
The overall picture drawn by this experiment is that linearizing the ellip-soidal constraint yields strong and computationally tractable LP relaxationsof stable set problems with size |V | in the range [300, 700].
18
Ellip
soid
BT
Nre
form
ula
tion
Alg
ori
thm
0A
lgori
thm
1G
rap
hUB
E%
gap
(θ(G
))UB
BTN
%ga
p(θ
(G))
UB
K%
gap
(θ(G
))UB
C%
gap
(θ(G
))
bro
ck20
01
27.4
60.0
027
.46
0.00
28.4
53.
6128
.53
3.90
bro
ck20
02
14.5
32.1
114
.93
4.92
15.2
57.
1715
.16
6.54
bro
ck20
03
19.0
91.4
319
.44
3.29
19.6
04.
1419
.60
4.14
bro
ck20
04
21.5
31.1
321
.81
2.44
22.2
64.
5622
.35
4.98
bro
ck40
01
40.3
51.6
440
.72
2.57
42.1
16.
0741
.83
5.37
bro
ck40
02
41.7
15.4
342
.38
7.13
43.2
29.
2542
.82
8.24
bro
ck40
03
40.2
51.9
540
.34
2.18
42.1
16.
6641
.05
3.98
bro
ck40
04
40.8
62.9
240
.91
3.05
41.4
54.
4141
.26
3.93
C.1
25.9
37.9
90.4
938
.16
0.94
38.2
91.
2838
.48
1.79
C.2
50.9
57.0
21.3
957
.43
2.12
58.2
73.
6158
.09
3.29
DS
JC
125.1
38.5
40.
3738
.75
0.92
38.8
41.
1539
.16
1.99
DS
JC
125.5
11.8
23.
0511
.87
3.49
11.9
13.
8412
.24
6.71
DS
JC
125.9
4.1
23.
004.
133.
254.
287.
004.
317.
75ge
n20
0p
0.9
44
44.3
80.
8644
.66
1.50
44.6
61.
5044
.38
0.86
kell
er4
14.
281.
9314
.43
3.00
14.2
81.
9314
.55
3.85
ph
at300-
110.9
08.
2410
.95
8.74
12.1
120
.26
11.1
310
.53
ph
at300-
227.2
51.
0827
.83
3.23
28.1
04.
2328
.50
5.71
ph
at300-
341.6
91.
2641
.73
1.36
43.0
74.
6243
.02
4.49
ph
at500-
114.5
311
.17
14.9
214
.15
16.4
726
.01
15.9
221
.81
ph
at500-
239.5
31.
4140
.43
3.72
40.8
54.
8041
.02
5.23
ph
at500-
359.9
72.
3960
.40
3.12
61.8
65.
6261
.79
5.50
ph
at700-
118.2
720
.83
18.2
920
.97
19.9
131
.68
19.3
928
.24
ph
at700-
251.1
73.
3951
.54
5.14
52.0
56.
1852
.05
6.18
ph
at700-
374.0
91.
8675
.27
3.48
77.3
66.
3576
.34
4.95
san
400
0.5
-113.4
13.
1513
.42
3.23
13.2
82.
1513
.46
3.54
san
400
0.7
-140.0
20.
0540
.12
0.30
40.0
20.
0540
.03
0.08
san
400
0.7
-230.0
40.
1330
.09
0.30
30.0
10.
0330
.04
0.13
san
400
0.7
-322.6
63.
0022
.66
3.00
22.9
54.
3222
.93
4.23
san
400
0.9
-110
0.0
00.
0010
0.24
0.24
100.
000.
0010
0.00
0.00
san
r200
0.7
23.9
50.
4624
.44
2.52
24.7
13.
6524
.75
3.82
san
r200
0.9
49.6
40.
7549
.85
1.18
50.3
32.
1550
.60
2.70
Tab
le2:
Ell
ipso
idvs.
LP
up
per
bou
nd
s
19
Ellip
soid
BT
Nre
form
ula
tion
Alg
ori
thm
0A
lgori
thm
1S
ub
gra
die
nt
Bar
rier
Col
um
ns
Row
sL
Pt K
Kel
ley
Bou
nd
Kel
ley
t CK
elle
yC
liqu
eG
rap
hti
me
tim
eti
me
cuts
att C
cuts
att C
cuts
cuts
bro
ck20
01
8.3
1.2
3,80
16,
839
2.89
27.6
730
028
.52
178
9.88
401,
996
bro
ck20
02
6.23
0.91
3,80
16,
624
7.73
22.9
830
015
.26
585.
85
4,05
0b
rock
200
319
.47
1.46
3,8
016,
828
7.02
28.7
930
019
.62
149
14.3
730
3,43
7b
rock
200
48.
491.
543,
801
6,87
07.
620
.45
300
22.2
929
620
.12
305,
550
bro
ck40
01
74.4
140
.28
7,6
0116
,051
49.8
840
8.34
300
42.7
514
291
.91
4011
,432
bro
ck40
02
45.3
538
.81
7,6
0116
,095
46.9
247
4.53
300
43.3
415
512
0.07
405,
877
bro
ck40
03
34.4
540
.82
7,6
0116
,095
83.5
936
0.51
300
42.1
912
084
.11
405,
440
bro
ck40
04
80.7
140
.25
7,6
0116
,033
85.3
529
7.53
300
41.5
312
188
.68
405,
829
C.1
25.9
3.45
0.08
2,37
63,
733
0.26
5.91
300
38.4
353
0.6
5038
C.2
50.9
45.3
61.
074,7
518,
132
4.9
56.4
430
059
.19
615.
0250
325
DS
JC
125.1
6.4
0.09
2,3
763,
708
0.24
4.94
300
39.1
154
0.44
5019
DS
JC
125.5
4.2
40.
192,3
763,
895
0.47
4.93
300
12.0
116
92.
275
1,91
6D
SJC
125.9
3.1
70.
032,3
763,
368
0.21
1.17
113
N/A
N/A
2.16
51,
073
gen
200
p0.
944
20.4
80.3
63,8
016,
212
0.87
0.01
2N
/AN
/A0.
072
25ke
ller
43.9
90.2
3,2
504,
959
0.58
7.34
300
14.3
959
0.64
1083
9p
hat
300-
129
.92
0.8
75,7
018,
926
2.72
43.8
330
012
.19
394.
235
8,42
4p
hat
300-
239
.97
2.7
75,7
019,
819
21.4
481
.99
300
28.3
627
3.35
51,
837
ph
at300-
385
.42
65,
701
10,9
4926
.37
73.9
830
043
.417
825
.94
402,
537
ph
at500-
116
6.8
76.
849,5
0115
,709
54.7
412
8.01
300
16.3
615
940
.46
523
,585
ph
at500-
224
1.3
824
.59
9,5
0117
,805
193.
1438
9.02
300
41.0
115
093
.69
512
,185
ph
at500-
321
1.5
788
.85
9,5
0120
,808
305.
5240
4.79
300
62.0
122
816
8.52
506,
082
ph
at700-
137
2.6
928
.42
13,
301
22,7
6823
9.73
1,39
2.81
300
19.9
714
921
5.79
543
,855
ph
at700-
247
9.2
512
3.8
313
,301
26,2
7033
8.15
3,34
4.90
300
52.2
515
723
0.92
1018
,215
ph
at700-
31,
427.9
660
2.9
313
,301
32,4
8677
8.75
2,04
5.77
300
77.8
119
829
1.83
5025
,834
san
400
0.5
-112.7
52.
737,
601
12,1
557.
913
.93
11N
/AN
/A23
.14
16,
774
san
400
0.7
-116.8
521
.88
7,60
115
,890
31.6
312
.73
1040
.18
64.
762
757
san
400
0.7
-220.8
219
.44
7,60
114
,782
26.2
342
.52
300
30.8
533
17.8
410
2,72
3sa
n400
0.7
-3758.
079.
727,
601
13,6
1837
.01
11.5
930
023
.83
103.
041
996
san
400
0.9
-14.2
916.4
97,
601
15,4
4524
.27
0.27
1N
/AN
/A0.
521
53sa
nr2
00
0.7
12.2
31.3
23,8
016,
873
9.99
26.4
930
025
.37
996.
0230
1,64
0sa
nr2
00
0.9
28.3
50.5
3,8
016,
332
3.91
25.0
330
050
.96
511.
4230
125
Tab
le3:
CP
Uti
mes
and
oth
erd
etai
ls
20
We remark that lift-and-project methods have been previously inves-tigated [15, 14] with the purpose of achieving target bounds comparableto those from SDP formulations. Compared to those, the method basedon ellipsoids shows a more robust and density-independent behavior. Infact, dense graphs with remarkably larger size could be tackled than thosewhich were tractable by lift-and-project (which, up to now, has been suc-cessfully applied to graphs with more than 300 vertices only if they are quitesparse). Interestingly, (building and) solving the BTN and Kelley reformu-lations seems to be faster than solving strong SDP relaxations, such as thoseinvestigated in [6], [9] and [19].
5.3 Branch-and-cut
We then tested a straightforward branch-and-cut implementation. Specifi-cally, we load into an initial pool all the Kelley cuts along with the cliqueinequalities generated by Algorithm 1 and then run the Gurobi branch-and-cut algorithm with checking for violated cuts in the pool. All other cuttingplanes are turned off, as these turn out to be not effective.
Table 4 compares the branch-and-cut algorithm described in the previousparagraph with two other algorithms. The first consists of solving, by theGurobi branch-and-cut routine (default settings), the 0-1 LP formulationbased on the initial collection C of clique inequalities (see Section 4.2). Thisis referred to as the clique formulation and reads as:
max∑
i∈V xi
s.t.∑
i∈C xi ≤ 1 (C ∈ C) (21)
x ∈ {0, 1}n.
The second competitor is again Gurobi, but using its mixed-integer quadraticconvex programming (MIQCP) solver to solve a formulation obtained fromthe relaxation (20) by declaring all variables to be binary.
It is worth mentioning that Gurobi implements sophisticated strategiesto handle convex quadratic inequalities: conversion of quadratic constraintsto SOC constraints, linearization of quadratic terms on binary variables,and outer-approximation. The Gurobi parameter MIQCPMethod controls themethod used to solve MIQCP models. We set it to 1, which corresponds toa linearized, outer-approximation approach (instead of solving continuousQCP relaxations at each node). In our experiments, this value achieved thebest results.
For each algorithm we report the number of evaluated subproblems andthe CPU time (this includes the time to compute the ellipsoid and to gen-erate the Kelley cuts). The speed-up factor of Kelley based branch-and-cutw.r.t. the best competitor is also quoted.
We also include Figures 1 and 2, with the performance profiles corre-sponding to the results of Table 4. The performance profiles, introduced
21
by Dolan and More [8], provide a succinct view of the relative behaviour ofthe different algorithms upon the given test set. Specifically, they plot theprobability that the number of evaluated subproblems or the CPU time foran algorithm on a given instance is within a factor of β of the best compet-ing algorithm for that instance. Therefore, methods whose correspondingprofile lines are the highest are the most effective.
Gurobi linear formulationBranch and cut
Gurobi MIQCP formulation
0
0.2
0.4
0.6
0.8
1
1 10 100 1000 10000 100000 1e+06 1e+07 1e+08
Figure 1: Performance profile of number of subproblems
Gurobi linear formulationBranch and cut
Gurobi MIQCP formulation
0
0.2
0.4
0.6
0.8
1
1 10 100
Figure 2: Performance profile CPU time
22
Figure 1 shows that strengthening the clique formulation, either by thequadratic constraint or by the Kelley cuts generated with Algorithm 1, con-sistently reduces the size of the enumeration tree. Figure 2 shows a differentpicture when it comes to CPU time, probably due to the fact that formula-tion strengthening leads to additional effort in evaluating subproblems. Itis still the case that leaving Gurobi to handle internally the“raw” quadraticconstraint is generally not cost-effective. In fact, this method fails to solvefour large instances within the time limit (C.250.9, brock400 and p hat500,p hat700). On the other hand, branch-and-cut tends to be faster when theKelley cuts are not included. A more detailed examination, however, showsthat Kelley cuts lead to a significant speed-up for some specific families ofhard instances, namely, the brock, p hat and sanr instances.
The overall indication of this experiment is that the proposed Kelley cutsprovide a linear reformulation of the quadratic constraint which outperformsthe advanced reformulation technique embedded in Gurobi. Notice that thishappens despite the fact that our implementation is quite basic, as cuttingplanes are generated using only the initial ellipsoid (i.e., the one relatedto the root relaxation), and leaves the cut management to Gurobi. Anadvanced implementation could be devised by generating ellipsoids at eachsubproblem and dealing with related cut-selection issues.
6 Concluding Remarks
The key idea in this paper is that one can use an approximate dual solution tothe standard SDP relaxation of the SSP to construct an ellipsoid that wrapsreasonably tightly around the stable set polytope, and that this ellipsoid canbe used to construct quite strong cutting planes in the original (linear) space.The computational results, though preliminary, indicate that this approachis promising.
There is, however, room for further improvement. In particular:
• Instead of using the “optimal” ellipsoid to generate cutting planes, onecould use some other ellipsoid, or indeed a whole family of ellipsoids.
• As mentioned in Subsection 2.2, the SDP relaxation (7)-(10) can bestrengthened by adding valid inequalities. (For example, Schrijver[36] suggested adding the inequalities Xij ≥ 0 for all {i, j} /∈ E.) Letθ(G) < θ(G) be an improved upper bound obtained by solving such astrengthened SDP. The proof of Theorem 3 can be modified to showthe existence of an ellipsoid, say E, such that
θ(G) = max
{∑i∈V
xi : x ∈ E
}.
23
Such an ellipsoid might produce stronger cutting planes. (The ex-treme points of STAB(G) would no longer be guaranteed to lie on theboundary of the ellipsoid, but this may not matter.)
• In a branch-and-cut context, one could perhaps re-optimise the dualSDP solution (and therefore the corresponding ellipsoid) after branch-ing. This could yield stronger cuts at the non-root nodes of the enu-meration tree. Of course, one would not wish to solve an SDP todo this. Instead, one could perhaps perform a few iterations of thesubgradient method that we described in [17].
Finally, one could explore the possibility of adapting the ellipsoidal ap-proach to other combinatorial optimisation problems. In our view, this islikely to work well only for problems that have a “natural” formulationas a continuous optimisation problem with a linear objective function andnon-convex quadratic constraints, like the formulation (4)–(6) of the SSP.
Acknowledgment:
The second author was partially supported by the Engineering and PhysicalSciences Research Council under grant EP/D072662/1.
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27
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944
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09
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45
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ph
at30
0-1
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817
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75
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1.3
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hat
300-
26,
123
286.
603,6
89
356.5
14,9
33
164.0
81.7
5p
hat
300-
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498,
311
7,65
0.19
225,3
45
4,4
91.9
6558,8
61
3,3
05.6
32.3
1p
hat
500-
167
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1,41
8.18
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59,0
89
699.9
42.0
3p
hat
500-
226
8,86
55,
266.
73***
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173,6
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71.1
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700-
116
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61***
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94
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61.5
26
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731
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70
39.6
4—
san
400
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10
11.6
90
21.4
20
24.4
8—
san
400
0.7-
250
155
.15
037.2
61
41.3
3—
san
400
0.7-
30
2.7
20
764.5
30
763.6
5—
san
400
0.9-
10
0.9
80
5.4
80
5.3
3—
san
r200
0.7
118,
261
242.
1144,9
89
241.6
373,6
87
136.5
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nr2
000.
91,
171,
075
921.
88243,3
31
1,4
54.5
9302,0
31
303.1
53.0
4
Tab
le4:
Bra
nch
-an
d-c
ut
resu
lts
28