The quadratic minimum spanning tree problem: lower bounds via extended formulations
aa r X i v : . [ m a t h . O C ] F e b The quadratic minimum spanning tree problem: lower bounds viaextended formulations
Renata Sotirov ∗ Mathieu Verch´ere † Abstract
The quadratic minimum spanning tree problem (QMSTP) is the problem of finding a spanning treeof a graph such that the total interaction cost between pairs of edges in the tree is minimized. We firstshow that most of the bounding approaches for the QMSTP are closely related. Then, we exploit anextended formulation for the minimum spanning tree problem to derive a sequence of relaxations forthe QMSTP with increasing complexity. The resulting relaxations differ from the relaxations in theliterature. Namely, our relaxations have a polynomial number of constraints and can be efficiently solvedby a cutting plane algorithm. Moreover our bounds outperform most of the bounds from the literature.
Keywords: quadratic minimum spanning tree problem, linearization problem, extended formulation, Gilmore-Lawler bound
The quadratic minimum spanning tree problem (QMSTP) is the problem of finding a spanning tree of aconnected and undirected graph such that the sum of interaction costs over all pairs of edges in the tree isminimized. The QMSTP was introduced by Assad and Xu [1] who have proven that the problem is stronglyNP-hard. The QMSTP remains NP-hard even when the cost matrix is of rank one [27]. The adjacent-onlyquadratic minimum spanning tree problem (AQMSTP), that is the QMSTP where the interaction costs ofall non-adjacent edge pairs are assumed to be zero, is also introduced in [1]. That special version of theQMSTP is also strongly NP-hard.The QMSTP can be seen as a generalization of several well known optimization problems, including thequadratic assignment problem [1] and the satisfiability problem [6]. There are also numerous variants of theQMSTP problem such as the minimum spanning tree problem with conflict pairs, the quadratic bottleneckspanning tree problem, and the bottleneck spanning tree problem with conflict pairs. For description of thoseproblems, see e.g., ´Custi´c et al. [8]. In the same paper the authors investigate the complexity of the QMSTPand its variants, and prove intractability results for the mentioned variants on fan-stars, fans, wheels, ( k, n )–accordions, ladders and ( k, n )–ladders. The authors also prove that the AQMSTP on ( k, n )–ladders and theQMSTP where the cost matrix is a permuted doubly graded matrix are polynomially solvable.´Custi´c and Punnen [7] prove that the QMSTP on a complete graph is linearizable if and only if asymmetric cost matrix is a symmetric weak sum matrix. An instance of the QMSTP is called linearizableif there exists an instance of the minimum spanning tree problem (MSTP) such that the associated costsfor both problems are equal for every feasible spanning tree. The authors of [7] also present linearizabilityresults for the QMSTP on complete bipartite graphs, on the class of graphs in which every biconnectedcomponent is either a clique, a biclique or a cycle, as well as on biconnected graphs that contain a vertexwith degree two. Note that linearizable instances for the QMSTP can be solved in polynomial time. ∗ Department of Econometrics and OR, Tilburg University, The Netherlands, [email protected] † Doctorant ´a UMA, ENSTA Paris, Institut Polytechnique de Paris, 91120 Palaiseau, France, [email protected]
Most of the lower bounding approaches for the QMSTP are closely related as we show in the next section.Namely, those bounds are based on Lagrangian relaxations obtained from the RLT type of bounds andsolved by specialized iterative methods. Even semidefinite programming lower bounds for the QMSTPinvolve Lagrangian relaxations.In this paper we use a different approach for solving the QMSTP. We exploit an extended formulationfor the minimum spanning tree problem to compute lower bounds for the QMSTP. Our relaxations have apolynomial number of constraints and we solve them using a cutting plane algorithm.Our first relaxation provides the strongest linearization based bound for the QMSTP. The linearizationbased relaxation finds the best possible linearizable matrix for the given QMSTP instance, by solving a linearprogramming (LP) problem. In particular, it searches for the best under-estimator of the quadratic objectivefunction that is in the form of a weak sum matrix. Linearization based bounds are introduced by Hu andSotirov [14] and further exploited by de Meijer and Sotirov [9]. Further, we consider the linearized QMSTPformulation from [1], which contains an exponential number of subtour elimination constraints. In order toavoid the subtour elimination constraints, we exploit a valid separation linear program for those constraintsand the corresponding extended formulation from Martin [20]. We prove that the resulting relaxation forthe QMSTP is equivalent to our linearization based relaxation.In order to improve the mentioned relaxation we add facet defining inequalities of the Boolean QuadricPolytope (BQP), Padberg [22]. We show that the linearization based relaxation with a particular subsetof the BQP inequalities is not dominated by the incomplete first level RLT relaxation from the literature[24, 29], or vice versa. However, after adding all BQP cuts to the linearization based relaxation the resultingbounds outperform even the incomplete second level RLT bounds from the literature on some instances. Sinceadding all BQP inequalities at once is computationally expensive, we implement a cutting plane approach2hat considers the most violated constraints.This paper is organized as follows. In Section 2 we formally introduce the QMSTP and present theextended formulation for the MSTP from [19]. In Section 3 we provide an overview of lower bounds forthe QMSTP. In particular, Section 3.1 presents the Gilmore-Lawler type bounds including Assad-Xu and¨Oncan-Punnen bounds, and Section 3.2 the RLT type bounds. New relaxations are presented in Section 4.Section 4.1 introduces a linearization based relaxation, and Section 4.2 provides several new relaxations ofincreasing complexity. Numerical results are presented in Section 5 and concluding remarks in Section 6.
Notation
Given a subset S ⊆ V of vertices in a graph G = ( V, E ), we denote the set of edges with both endpoints in S by E ( S ) := { e ∈ E | e = { i, j } , i, j ∈ S } . Further, we use δ ( S ) ⊆ E to denote the set of edges with exactlyone endpoint in S .We introduce the operator Diag : R n → R n × n that maps a vector to a diagonal matrix whose diagonalelements correspond to the elements of the input vector. For two matrices X = ( x ij ) , Y = ( y ij ) ∈ R n × m , X ≥ Y means x ij ≥ y ij for all i, j . We denote by the all-zero vector of appropriate size. Let us formally introduce the quadratic minimum spanning tree problem. We are given a connected, undi-rected graph G = ( V, E ) with n = | V | vertices and m = | E | edges, and a matrix Q = ( q ef ) ∈ R m × m ofinteraction costs between edges of G . We assume without loss of generality that Q = Q T . The QMSTP canbe formulated as follows: min X e ∈ E X f ∈ E q ef x e x f | x ∈ T , (1)where T denotes the set of all spanning trees, and each spanning tree is represented by an incidence vector x of length m , i.e., T := x ∈ { , } m | X e ∈ E x e = n − , X e ∈ E ( S ) x e ≤ | S | − , S ⊂ V, | S | ≥ . (2)We denote by T R the convex hull of the elements from T .If the matrix Q is a diagonal matrix i.e., Q = Diag( p ) for some p ∈ R m , then the QMSTP reduces to theminimum spanning tree problem: min (X e ∈ E p e x e | x ∈ T ) . (3)It is a well known result by Edmonds [10] that the linear programming relaxation of (3) has an integerpolyhedron. However, the spanning tree polytope has an exponential number of constraints. Nevertheless,the MSTP is solvable in polynomial time by e.g., algorithms developed by Prim [26] and Kruskal [16].There exists a polynomial size extended formulation for the minimum spanning tree problem, due toMartin [20]. He derived the polynomial size formulation for the MSTP by exploiting a known result (seee.g., [19]) that one can test if a vector violates subtour elimination constraints by solving a max flow problem.Martin [20] uses the following valid separation linear program for subtour elimination constraints that containvertex k : ( SP k ( x )) max P e ∈ E x e α ke − n P i =1 ,i = k θ ki s.t. α ke − θ ki ∀ e ∈ δ ( i ) , i ∈ Vθ ki > ∀ i. (4)3he above max flow problem formulation is from [28]. Thus, for ˜ x ∈ R m , ˜ x ≥
0, the objective value of SP k (˜ x )equals zero if and only if P e ∈ E ( S ) ˜ x e ≤ | S | − k ∈ S , see [20]. By exploiting the dual problem of (4)for all vertices k ∈ V, Martin [20] derived the following extended formulation for the minimum spanning treeproblem: min X e ∈ E p e x e (5a)s . t . X e ∈ E x e = n − z kij + z kji = x e k = 1 , . . . , n, e ∈ E ( { i, j } ) (5c) X s>i z kis + X hk z kks + X h We consider here the classical Gilmore-Lawler type bound for the QMSTP. The GL procedure is a well-knownapproach to construct lower bounds for binary quadratic optimization problems. The bounding procedurewas introduced by Gilmore [12] and Lawler [17] to compute a lower bound for the quadratic assignmentproblem. Nowadays, the GL bounding procedure is extended to many other optimization problems includingthe quadratic minimum spanning tree problem [29], the quadratic shortest path problem [30], and thequadratic cycle cover problem [9].The GL procedure for the QMSTP is as follows. For each edge e ∈ E , solve the following optimizationproblem: z e := min X f ∈ E q ef x f | x e = 1 , x ∈ T . (9)The value z e is the best quadratic contribution to the QMSTP objective with e being in the solution. Then,solve the following minimization problem: GL ( Q ) := min (X e ∈ E ( z e + q e,e ) x e | x ∈ T ) . (10)The optimal solution of the above optimization problem is the GL type lower bound for the QMSTP. Notethat in (9) and (10) one can equivalently solve the corresponding continuous relaxations, thus optimize over T R or T E , or use one of the efficient algorithms for solving the MSTP.Rostami and Malucelli [29] prove that one can also compute the GL type bound by solving the followingLP problem: GL ( Q ) := min X e,f ∈ E q ef y ef (11a)s . t . X f ∈ E y ef = ( n − x e ∀ e ∈ E (11b) X f ∈ E ( S ) y ef ≤ ( | S | − x e ∀ e ∈ E, ∀ S ⊂ V, S = ∅ (11c) y ee = x e ∀ e ∈ E (11d) y ef ≥ ∀ e, f ∈ E (11e) x ∈ T R . (11f)Note that from (11c) it follows that y ef ≤ x e for all e, f ∈ E . Clearly, it is more efficient to compute theGL bound by solving (9)–(10), than solving (11).The Gilmore-Lawler type bound can be further improved within an iterative algorithm that exploitsequivalent reformulations of the problem. The so obtained iterative bounding scheme is known as thegeneralized Gilmore-Lawler bounding scheme. The generalized GL bounding scheme was implemented forseveral optimization problems, see e.g., [3, 9, 14, 30]. Hu and Sotirov [14] show that bounds obtained bythe GL bounding scheme are dominated by the first level RLT bound. The bounding procedure in the nextsection can be seen as a special case of the generalized Gilmore-Lawler bounding scheme. Assad and Xu [1] propose a method that generates a monotonic sequence of lower bounds for the QMSTP,which results in an improved GL type lower bound. The approach is based on equivalent reformulations ofthe QMSTP. 5ssad and Xu [1] introduce first the following transformation of costs: q ef ( γ ) = q ef + γ f , ∀ e, f ∈ E, e = fq e,e ( γ ) = q e,e − ( n − γ e , ∀ e ∈ E, (12)where γ e ( e ∈ E ) is a given parameter, and then apply the GL procedure for a given γ . In particular, Assadand Xu [1] solve the following problem: f e ( γ ) := min X f ∈ E q ef ( γ ) x f | x e = 1 , x ∈ T , (13)for each edge e ∈ E . The value f e ( γ ) is the best quadratic contribution to the reformulated QMSTP objectivewith e being in the solution. Then f e ( γ ) is used in the following optimization problem: AX ( γ ) := min (X e ∈ E ( f e ( γ ) + q e,e ( γ )) x e | x ∈ T ) . (14)The optimal solution of the above optimization problem is a GL type lower bound for the QMSTP thatdepends on a parameter γ . In the case that γ is all zero-vector, the corresponding lower bound AX ( γ ) isequal to the GL type lower bound described in the previous section.The function AX ( γ ) is a piecewise linear, concave function. In order to find γ that provides the bestpossible bound of type (14), that is to solve max γ AX ( γ ), the authors propose Algorithm 1. Algorithm 1 Assad-Xu leveling procedure ǫ > γ = , i ← while (max e f e ( γ i ) − min e f e ( γ i ) > ǫ , e ∈ E ) do Compute transformed costs using (12). Compute f e ( γ i ) and AX ( γ i ) using (13) and (14), resp. Update γ i +1 e ←− γ ie + n − ( f e ( γ i )+ q e,e ( γ i )) for e ∈ E . i ← i + 1 return AX ( Q ) ← AX ( γ i − ). Algorithm 1 resembles the Generalized Gilmore-Lawler bounding scheme. Since the algorithm starts with γ = , the first computed bound equals the classical GL type bound. Note that Algorithm 1, in Step 3 addsto the off diagonal elements of the cost matrix Q a weak sum matrix and then subtracts its linearizationvector on the diagonal. For a relation between linearizable weak sum matrices and corresponding linearizationvectors see Section 4.1. Thus Algorithm 1 generates a sequence of equivalent representations of the QMSTPwhile converging to the optimal γ ∗ . The following exact, linear formulation for the QMSTP is presented in [1]:min X e,f ∈ E q ef y ef (15a)s . t . X f ∈ E y ef = ( n − x e ∀ e ∈ E (15b) X e ∈ E y ef = ( n − x f ∀ f ∈ E (15c) y ee = x e ∀ e ∈ E (15d)0 ≤ y ef ≤ ∀ e, f ∈ E (15e) x ∈ T . (15f)6Oncan and Punnen [21] propose a lower bound for the QMSTP that is derived from (15) and also includesthe following valid inequalities: X e ∈ δ ( i ) y ef ≥ x f ∀ i ∈ V, f ∈ E (16) X f ∈ δ ( i ) y ef ≥ x e , ∀ i ∈ V, e ∈ E, (17)where δ ( i ) denotes the set of all incident edges to vertex i . In particular, the authors from [21] propose tosolve the Lagrangian relaxation of (15a)–(15f), (16)–(17) obtained by dualizing constraints (16). For a fixedLagrange mulitiplier λ , the resulting Lagrangian relaxation is solved in a similar fashion as the GL typebound, where costs in objectives of problems (9) and (10) are: q ef ( λ ) := q ef − X i ∈ V, f ∈ δ ( i ) λ i,f ∀ e, f ∈ E, e = f and q ee ( λ ) := q ee + X i ∈ V, e ∈ δ ( i ) λ i,e ∀ e ∈ E, respectively. After the solution of the Lagrangian relaxation with a given λ is obtained, a subgradientalgorithm is implemented to update the multiplier, and the process iterates. Numerical results show that¨Oncan-Punnen bounds provide, in general, stronger bounds than Assad-Xu bounds, see also [29]. In [21],the authors also consider replacing the condition x ∈ T by the multicommodity flow constraints from [19],but only for graphs up to 20 vertices. Pereira et al. [24] present a relaxation for the QMSTP that is derived by applying the first level reformulationlinearization technique on (1). Nevertheless, their numerical results are obtained by solving the following incomplete first level RLT relaxation: ^ RT L ( Q ) := min P e,f ∈ E q ef y ef s . t . y ef = y fe ∀ e, f ∈ E (11b) − (11f) . (18)We denote by ^ RLT the relaxation (18). To complete the above model to the first level RLT formulation ofthe QMSTP, one needs to add the following constraints: X f ∈ E ( S ) ( x f − y ef ) ≤ ( | S | − − x e ) ∀ e ∈ E, ∀ S ⊂ V, | S | ≥ . (19)It is written in [24] that preliminary computational experiments show that bounds obtained from the (com-plete) first level RLT relaxation ( RLT ) do not significantly differ from bounds obtained from the incompletefirst level RLT relaxation. However, computational effort to solve the full RLT relaxation significantly in-creases due to the constraints (19). Therefore, Pereira et al. [24] compute only the incomplete first level RLTbound.The (complete) first level RLT formulation for the QMSTP needs not to include constraints: y ef ≤ x e , y fe ≤ x e , ∀ e, f ∈ E, (20) y ef ≥ x e + x f − , ∀ e, f ∈ E, (21)since these are implied by the rest of the constraints. Namely, constraints (20)–(21) readily follow byconsidering S with two elements in (11c) and (19). However, constraints (21) are not implied by the7able 1: RLT type bounds n d U B GL ^ RLT RLT ^ RLT 10 33 350 299 350 350 344.110 67 255 149 202.2 204.1 226.115 33 745 445 578.2 603.3 637.815 67 659 283 385.4 385.7 488.920 33 1379 690 888 891.7 1056.7constraints of the incomplete RLT relaxation (18). Nevertheless, Pereira et al. [24] do not impose thoseconstraints to the incomplete model for the same reason that constraints (19) are not added; that is thatconstraints (21) do not significantly improve the value of the bound but are expensive to include.To approximately solve (18), Pereira et al. [24] dualize constraints y ef = y fe ( e, f ∈ E ) for a givenLagrange multiplier and then apply the GL procedure. Then, they use a subgradient algorithm to derive asequence of improved multipliers and compute the corresponding bounds. The so obtained bounds convergeto the optimal value of the incomplete first level RLT lower bound. It is clear from the above discussion thatthe bound from [24] is related to the Gilmore-Lawler type bounds.Pereira et al. [24] prove P ^ RLT ⊆ P OP ⊆ P AX , where P AX denotes the convex hull of the QMSTP formulation (15), P OP denotes the convex hull of (15b)–(15f) and (16)–(17), and P ^ RLT the convex hull of the incomplete first level RLT relaxation (18). We remarkthat in [24] the authors add to P AX and P OP constraints (11c) that are not in the original description ofthose polyhedrons, see [21]. Nevertheless, their result as well as the statement above are correct. Remark . It is interesting to note that the continuous relaxation (11) and the incomplete first level RLTrelaxation differ only in the symmetry constraints y ef = y fe ( ∀ e, f ) that are included in (18). In other words,the dual of the relaxation ^ RLT contains also dual variables say δ ef and δ fe ( ∀ e, f ), that correspond to thesymmetry constraints. Thus, for each e, f there is a constraint in the dual of the incomplete first level RLTrelaxation that has a term q ef + ( δ ef − δ fe ). Since δ ef − δ fe = − ( δ fe − δ ef ), this term corresponds to addinga skew symmetric matrix to the cost matrix Q . This implies that the optimal solution of the dual providesthe best skew symmetric matrix that is added to the quadratic cost to improve the GL type bound. Thusone can obtain the optimal solution for the incomplete first level RLT relaxation for the QMSTP in one stepof the GL bounding scheme.Further, in [24] it is proposed a generalization of the RLT relaxation that is based on decomposingspanning trees into forests of a fixed size. The larger the size of the forest, the stronger the formulation.The resulting relaxations are solved by using a Lagrangian relaxation scheme. In [29] it is implemented adual-ascent procedure to approximate the value of the (incomplete) level two RLT relaxation ( ^ RLT ).Since we couldn’t find results on the first level RLT relaxation in the literature, we computed them forseveral instances. In Table 1 we compare GL , ^ RLT , RLT and ^ RLT bounds for small CP1 instances. Forthe description of instances see Section 5. U B stands for upper bounds. Bounds for the (incomplete) firstand second level RLT relaxation are from [29]. Table 1 shows that the GL type bounds are significantlyweaker than the other bounds. The results also show that difference between ^ RLT and RLT bounds isnot always small. ^ RLT bounds dominate all presented bounds in all but one case. This discrepancy shouldbe due to the way that those bounds are computed, i.e., by implementing a dual ascent algorithm on theincomplete second level RLT relaxation. For more such examples see [29].8 New lower bounds for the QMSTP ´Custi´c and Punnen [7] prove that the QMSTP on a complete graph is linearizable if and only if its cost matrixis a symmetric weak sum matrix. In this section we exploit weak sum matrices to derive lower bounds forthe QMSTP. Hu and Sotirov [14] introduce the concept of linearization based bounds, and show that manyof the known bounding approaches including the GL type bounds are also linearization based bounds. DeMeijer and Sotirov [9] derive strong and efficient linearization based bounds for the quadratic cycle coverproblem.A matrix Q ∈ R m × m is called a sum matrix if there exist a, b ∈ R m such that q ef = a e + b f for all e, f ∈ { , . . . , m } . A weak sum matrix is a matrix for which this property holds except for the entries on thediagonal, i.e., q ef = a e + b f for all e = f . However for a symmetric weak sum matrix we have that vectors a and b are equal, i.e., q ef = a e + a f for all e = f .It is proven in [7, Theorem 5] that a symmetric cost matrix Q of the QMSTP on a complete graph islinearizable if and only if it is a symmetric weak sum matrix. In particular, the authors show that for asymmetric weak sum matrix of the form q ef = a e + a f and a tree T in a complete graph one has: P e ∈ T P f ∈ T q ef = P e ∈ T P f ∈ Tf = e ( a e + a f ) + P e ∈ T q ee = 2( n − P e ∈ T a e + P e ∈ T q ee = P e ∈ T p e , where p ∈ R m with p e = 2( n − a e + q ee (22)is a linearization vector. Thus, solving the QMSTP in which the cost matrix is a symmetric weak sum matrixcorresponds to solving a minimum spanning tree problem.In [9, 14] it is proven that one can obtain a lower bound for the optimal value OP T ( Q ) of a minimizationquadratic optimization problem with the cost matrix Q from a linearization matrix ˆ Q , which satisfies Q ≥ ˆ Q .In particular, we have: OP T ( Q ) = min x ∈ X { x T Qx } ≥ min x ∈ X { x T ˆ Qx } = min x ∈ X { x T ˆ p } = OP T (ˆ p ) , where ˆ p is a linearization vector of ˆ Q , X is the feasible set of the optimization problem, and OP T (ˆ p ) is theoptimal solution of the corresponding linear problem.Thus, for a weak sum matrix ˆ Q such that Q ≥ ˆ Q and the corresponding linearization vector ˆ p , the optimalsolution for the MSTP (3) with the cost vector ˆ p is a lower bound for the QMSTP (1). By maximizing theright hand side in the above inequality over all linearization vectors of the form (22), one obtains the strongestlinearization based bound for the QMSTP. Therefore, the strongest linearization based lower bound for the9MST is the optimal solution of the following problem: LBB ( Q ) := max − ( n − ǫ − n X i =1 n X j =1 j = i µ ij (23a)s . t . a e + a f ≤ q ef ∀ e, f ∈ E, e = f (23b) n X k =1 θ ke − ǫ ≤ n − a e + q ee ∀ e ∈ E (23c) µ ki + X e ∈ E ( { i,j } ) θ ke ≥ k, i, j = 1 , . . . , n (23d) µ ki ≥ k, i = 1 , . . . , n (23e) ǫ ∈ R , θ ke ∈ R , a e ∈ R k = 1 , . . . , n, e ∈ E. (23f)We denote the linear programming relaxation (23) by LBB . The LBB is derived from the dual of theextended formulation of the MSTP, see (8). Note that in computations one can exploit the fact that Q = Q T to reduce the number of constraints in (23b). In the next section we relate relaxation (23) with a relaxationobtained from the linear formulation of the QMSTP. The QMSTP formulation (15) is introduced in [1]. We exploit that QMSTP formulation to derive a sequenceof linear relaxations for the QMSTP. First, by replacing T by T E , and adding the symmetry constraints y ef = y fe ( ∀ e, f ), we obtain the following relaxation for the QMSTP: V S ( Q ) := min X e,f ∈ E q ef y ef (24a)s . t . X f ∈ E y ef = ( n − x e ∀ e ∈ E (24b) y ee = x e ∀ e ∈ E (24c) y ef = y fe ∀ e, f ∈ E (24d)( x, z ) ∈ T E (24e)0 ≤ y ef ≤ ∀ e, f ∈ E. (24f)We denote by V S the above relaxation. In the above optimization problem, we omit constraints: X e ∈ E y ef = ( n − x f ∀ f ∈ E, − ( n − ǫ − n X i =1 n X j =1 j = i µ ij (25a)s . t . α e + γ e ≥ ∀ e ∈ E (25b) − α e − δ ef + δ fe ≤ q ef ∀ e, f ∈ E, e = f (25c) n X k =1 θ ke + ( n − α e + γ e − ǫ ≤ q ee ∀ e ∈ E (25d) µ ki + X e ∈ S ( { i,j } ) θ ke ≥ k, i, j = 1 , . . . , n (25e) µ ki ≥ k, i = 1 , . . . , n (25f) ǫ ∈ R , θ ke , α e , γ e , δ e ∈ R k = 1 , . . . , n, e ∈ E. (25g)To derive the above dual problem we remove upper bounds on x e and y ef in the corresponding primalproblem, since it is never optimal to set the values of these variables larger than one. The following resultshows that the strongest linearization based bound equals the bound obtained by solving (24), or equivalently(25). Theorem 4.1. Let G be a complete graph. Then, optimization problems (23) and (25) are equivalent.Proof. We show that for every feasible solution for (25), we can find a feasible solution for (23) with thesame objective value, and conversely for every feasible solution for (23) we can find a feasible solution for(25) with the same objective value.Let ( α , γ , δ , ǫ , µ , θ ) be a feasible solution for the optimization problem (25). Let us find a feasible solution(ˆ a ,ˆ ǫ ,ˆ µ ,ˆ θ ) for (23). We define ˆ ǫ := ǫ , ˆ µ := µ and ˆ θ := θ . Thus, the objective values (25a) and (23a) are equal,and constraints (23d) (resp., (23e)) correspond to constraints (25e) (resp., (25f)).Let us define ˆ a e := − n − (( n − α e + γ e ) for e ∈ E and note thatˆ a e = − ( n − α e n − − γ e n − ≤ − ( n − α e n − 2) + α e n − 2) = − α e , where α e ≥ − γ e follows from (25b). Further from (25c) we have that − α e − ( δ ef − δ fe ) ≤ q ef − α f − ( δ fe − δ ef ) ≤ q fe , for every pair of edges ( e, f ) such that e = f . By adding these two inequalities, we obtain − ( α e + α f ) ≤ q ef + q fe . However, the inequality ˆ a e ≤ − α e implies ˆ a e + ˆ a f ≤ − ( α e + α f ) ≤ ( q ef + q fe ) = q ef for everypair ( e, f ) such that e = f . Note that we assumed that Q = Q T . Thus constraints (23b) are also satisfied.To show that constraints (23c) are satisfied, we use (25d) from where it follows: n X k =1 θ ke − ǫ ≤ q ee − ( n − α e − γ e = q ee + 2( n − a e ∀ e ∈ E. Conversely, let (ˆ a ,ˆ ǫ ,ˆ µ ,ˆ θ ) be a feasible for (23). Below, we find a feasible solution ( α , γ , δ , ǫ , µ , θ ) for (25).We define ǫ := ˆ ǫ , µ := ˆ µ by θ := ˆ θ . Thus, we have that the objective values (25a) and (23a) are equal, andconstraints (23d) (resp., (23e)) correspond to constraints (25e) (resp., (25f)). We define α e := − a e and γ e := 2ˆ a e , for all e ∈ E , from where it follows α e + γ e = 0. Thus, constraint (25b) is satisfied.11rom the definitions of α and γ we have − ( n − α e − γ e = 2( n − a e , and from (23c) it follows n X k =1 θ ke − ǫ ≤ q ee + 2( n − a e = q ee − ( n − α e − γ e e ∈ E. Thus, for each e ∈ E constraint (25d) is satisfied.It remains to verify constraints (25c). For that purpose we consider constraints (23b) for a pair of edges( e, f ) such that e = f . After multiplying the constraint with two, and using q ef = q fe and α e = − a e weobtain − α e − q ef ≤ q fe + α f . Thus, there exist δ ef and δ fe such that − α e − q ef ≤ ( δ ef − δ fe )( δ ef − δ fe ) ≤ q fe + α f . This finishes the proof.Theorem 4.1 shows that the linearization based relaxation for the QMSTP is equivalent to the relaxationof an exact linear formulation for the QMSTP. Our preliminary numerical results show that the bound V S ( Q ) is not dominated by GL ( Q ), or vice versa.To strengthen the relaxation V S , see (24), one can add the following facet defining inequalities of theBoolean Quadric Polytope, see e.g., [22]: 0 ≤ y ef ≤ x e (26a) x e + x f ≤ y ef (26b) y eg + y fg ≤ x g + y ef (26c) x e + x f + x g ≤ y ef + y eg + y fg + 1 , (26d)where e, f, g ∈ E , e = f = g . Table 2 introduces relaxations with increasing complexity that are obtainedby adding subsets of the BQP inequalities to the linear program (24).Table 2: VS relaxationsname constraints complexity V S (24b) – (24f) O ( n + m ) V S (24b) – (24e), (26a) – (26b) O ( n + m ) V S (24b) – (24e), (26a) – (26d) O ( n + m )To solve the relaxations V S and V S we use a cutting plane scheme that iteratively adds the mostviolated inequalities, see section on numerical results for more details.The following result relates ^ RLT and V S relaxations. Proposition 4.2. The relaxation V S is not dominated by the incomplete first level RLT relaxation (18) ,or vice versa.Proof. We consider a feasible solution ( x, Y ) of (18) for an instance on complete graph with 6 vertices.Let x := (0 . , . , , , , , , , . , , , . , , , T and define the 15 × 15 symmetric matrix Y whosediagonal elements correspond to the elements of vector x . Further, elements on the positions y ef = y fe where e f . 75, while y , = 1, and all other elements are zero. By direct verification it follows that ( x, Y ) isfeasible for (18). On the other hand, we have that 1 + y − y − y < 0. Thus, (26b) are violated and( x, Y ) is not feasible for ( V S ). The (incomplete) RLT bound for this particular instance is 386.5, while V S ( Q ) = 372 . V S are strictly greater thanthe optimal values of ] RLT .The results of the previous proposition are not surprising since constraints (26b) are not included in therelaxation ^ RLT . It is not difficult to show the following results. Corollary 4.3. The relaxation V S is dominated by the first level RLT relaxation for the QMSTP. Corollary 4.4. The relaxation V S is not dominated by the first level RLT relaxation for the QMSTP, orvice versa. Note that Corollary 4.3 follows from the discussion in Section 3.2, and Corollary 4.4 from the fact thatconstraints (26d) are not present in the RLT relaxation. In this section we compare several lower bounds from the literature with the bounds introduced in Section4. Numerical experiments are performed on an Intel(R) Core(TM) i7-9700 CPU, 3.00 GHz with 32 GBmemory. We implement our bounds in the Julia Programming Language [2] and use CPLEX 12.7.1.To solve V S relaxation we implement a cutting plane algorithm that starts from V S relaxation anditeratively adds the most violated n · m cuts. The algorithm stops if no more violated cuts or after two hours.To compute upper bounds for the here introduced benchmark instances V S we implement the tabusearch algorithm and variable neighbourhood search algorithm from [6]. In the mentioned paper the authorssuggest restarting the tabu search algorithm for better performance of the algorithm. We notice that a smallnumber of restarts (i.e., at most 5) and then running the neighbourhood search algorithm can be beneficialfor large instances. The total number of iterations of the tabu search algorithm is 5000. We test our bounds on the following benchmark sets.The benchmark set CP is introduced by Cordone and Passeri [6] and consists of 108 instances. Theseinstances consist of graphs with n ∈ { , , . . . , } vertices and densities d ∈ { , , } . Thereare four types of random instances denoted by CP1, CP2, CP3, CP4. In CP1 instances, the linear andthe quadratic costs are uniformly distributed at random in { , . . . , } . In CP2 instances, the linear (resp.,quadratic) costs are uniformly distributed at random in { , . . . , } (resp., { , . . . , } ). In CP3 instances,the linear (resp., quadratic) costs are uniformly distributed at random in { , . . . , } (resp., { , . . . , } ).In CP4 instances, the linear and the quadratic costs are uniformly distributed at random in { , . . . , } .The benchmark set OP is introduced by ¨Oncan and Punnen [21] and consists of 480 instances. Theseinstances consist of complete graphs with n ∈ { , , . . . , , , , , , } vertices and are divided intothree types. In particular, in OPsym the linear (resp., quadratic) costs are uniformly distributed at randomin { , . . . , } (resp., { , . . . , } ). In OPvsym the linear costs are uniformly distributed at random in { , . . . , } , while the quadratic costs are obtained associating to the vertices i ∈ V random values w ( i )uniformly distributed in { , . . . , } and setting q { i,j } , { k,l } = w ( i ) w ( j ) w ( k ) w ( l ). In OPesym the verticesare spread uniformly at random in a square of length side 100, the linear costs are the Euclidean distancesbetween the end vertices of each edge, while the quadratic costs are the Euclidean distances between themidpoints of the edges. 13e introduce the benchmark set VS. Our benchmark consists of 24 instances. For given a size n ∈{ , , , , , , , } , density d ∈ { , , } , a maximum cost for the diagonal entries, and amaximum cost for the off-diagonal entries, we generate an instance in the following way. 10% of the rows arerandomly chosen. These rows will have high costs with each other (between 90% and 100% of the maximumoff-diagonal cost), and low costs with rest (between 20% and 40% of the maximum off-diagonal cost). Therows that are not selected have an interaction cost of between 50% and 70% of the maximum off-diagonalcost. Finally, the impact of diagonal entries is greatly minimized, between 0 and 20% of the maximumdiagonal cost. The cost matrices obtained this way are not symmetric, but they can be made so at the user’sconvenience. We first present results for our benchmark set VS. Table 3 reads as follows. In the first two columns we listthe number of vertices and density of a graph, respectively. In the third column we provide upper boundscomputed as mentioned earlier. In the following three columns we list the incomplete first level RLT bound ^ RLT see (18), V S bound that is (24b) – (24e), (26a) – (26b), and V S bound that is (24b) – (24e), (26a) –(26d), see also Table 2. In columns 7–9 we present gaps by using the formula 100( U B − LB ) /U B where LB stands for the value of the lower bound. Note that the gap we present in our numerical results differs fromthe gap used in other QMSTP papers, where the authors use 100( U B − LB ) /LB . In last two columns ofTable 3 we presents time in seconds required to solve our relaxations. We do not report time for computing ^ RLT as we implement (18) directly, while the other authors that compute RTL type bounds use moreefficient way to compute those bounds.The results in Table 3 show that V S bounds are stronger than ^ RLT bounds for all instances except forthe VS instance with n = 25 and d = 100, and the VS instance with n = 30 and d = 100. The differencebetween gaps for ^ RLT and V S bounds for both instances is less than 1%. Moreover V S relaxationprovides a better bound than ^ RLT for the VS instance with n = 25 and d = 100 within 2 minutes of thecutting plane algorithm. Similarly, V S relaxation provides a better bound than ^ RLT relaxation for theVS instance n = 30 and d = 100 after a few iterations of the cutting plane algorithm. Recall that the V S relaxation is not dominated by the incomplete first level RLT relaxation, or vice versa in general, seeProposition 4.3. Therefore our benchmark set VS can be used to test quality of QMSTP bounds that arenot RTL type bounds. Table 3 shows that computational times for solving the relaxation V S are very smallfor all instances. The computational effort for solving the relaxation V S is small for instances with n ≤ d ∈ { , } . The results also show that it is computationally more challenging to solve V S fordense instances. Note also that we can stop our cutting plane algorithm at any time and obtain a valid lowerbound.Tables 4 – 9 read similarly to Table 3. We do not report results for OPvsym instances since gaps for V S bounds for those instances are less than or equal to 0 . n > 35 since the corresponding gaps are (too) large for all bounds in the literature, including ours.Clearly, it is a big challenge to obtain good bounds for the QMSTP when n ≥ 20 for most of the instancesand approaches.Running times required to solve V S relaxation for most of the test instances in Tables 4 – 9 are similarto times given in Table 3. For several difficult dense instances and n = 35 computational times for solving V S relaxation reaches 4 minutes. Computational times required to solve V S relaxation for dense instancesand n ≥ 25 can be larger than the times given in Table 3. Note that we allow our algorithm to run up to 2hours.Results in Table 4 – 7 present bounds for CP instances and show that the GL bounds are significantlyweaker than the other listed bounds. The results also show that ^ RLT bounds are stronger than V S boundsfor all instances, while V S bounds dominate ^ RLT bounds for all instances except the CP1 instance with n = 35 and d = 100. The results also show that V S bounds are not dominated by ^ RLT bounds or viceversa. 14able 3: VS instances: bounds and gapsinstance lower bounds Gap (%) time (s) n d (%) U B ^ RLT V S V S ^ RLT V S V S V S V S 10 33 4217 4217 4217 4217 0 0 0 < . < . < . 05 0.310 100 3930 3499.1 3604.3 3857 11 8.2 1.9 0.1 1.912 33 6141 5859.4 6058.1 6136 4.6 1.3 0.1 0.0 0.212 67 6050 5759.6 5910.4 6038 4.8 2.3 0.2 0.1 3.112 100 6051 5726 5868.1 5991.8 5.4 3.0 1.0 0.3 5.014 33 8736 8495.2 8619.1 8709.1 2.8 1.3 0.3 0.1 0.414 67 8606 8096.8 8306.0 8542.8 5.9 3.5 0.7 0.2 4.814 100 8513 7556 7731.7 8267.4 11.2 9.2 2.9 2.4 61.116 33 11735 11231.5 11465 11711.5 4.3 2.3 0.2 0.1 1.016 67 11610 10559.8 10816.2 11335.3 9.0 6.8 2.4 1.7 50.416 100 11516 10089.6 10302.7 11003.5 12.4 10.5 4.5 8.4 241.118 33 15125 13995.8 14382.9 14916.5 7.5 4.9 1.4 0.3 3.618 67 15020 12976.9 13210.6 14204.6 13.6 12.0 5.4 3.2 200.118 100 14943 13217.6 13501.8 14336.0 11.5 9.6 4.1 2.6 89.620 33 19057 17777.2 18178.9 18778.2 6.7 4.6 1.5 0.3 7.120 67 18830 16325.5 16433.5 17130.4 13.3 12.7 9.2 4.8 163.220 100 18812 16026 16299.6 17528.9 14.8 13.4 6.8 4.1 185.1525 33 30747 28436.4 29102.6 30084.4 7.5 5.3 2.2 4.8 179.125 67 30554 27061.5 27546.6 29186.1 11.4 9.8 4.5 2.8 377.925 100 30405 24455.6 24257.7 26251.7 19.7 20.3 13.7 10.4 1260.730 33 45184 40946.1 41995.1 43889.3 7.1 9.4 2.7 34.3 2268.630 67 44989 37676.7 38089.8 41162.8 16.25 15.3 8.5 13.5 1359.930 100 44856 35116.9 35037.5 37489.8 21.7 21.9 16.4 41.8 270015able 4: CP1 instances: bounds and gapsinstance lower bounds Gap (%) n d (%) U B GL ^ RLT ^ RLT V S V S GL ^ RLT ^ RLT V S V S 10 33 350 299 350 344,1 350 350 14.6 0 1.7 0 010 67 255 149 202.2 226.1 166.4 248.8 41.6 20.7 11.3 34.7 2.410 100 239 120 159.7 199.0 140.7 201.3 49.8 33.2 16.7 41.1 15.815 33 745 445 578.2 637.8 487 709.2 40.3 22.4 14.4 34.6 4.815 67 659 283 385.4 488.9 324.4 478.9 57.1 41.5 25.8 50.8 27.315 100 620 246 320.9 442.9 281.5 386.1 60.3 48.2 28.6 54.6 37.720 33 1379 690 888 1056.7 740.9 1165.4 50.0 35.6 23.4 46.3 15.520 67 1252 454 603.1 843.1 523.3 742.7 63.7 51.8 32.7 58.2 40.720 100 1174 398 506.9 737.9 436.7 587.7 66.1 56.8 37.1 62.8 49.925 33 2185 985 1285.1 1594.9 1045.3 1615.2 54.9 41.2 27.0 52.2 26.125 67 2023 660 834.9 1239.6 717 1003.7 67.4 58.7 38.7 64.6 50.425 100 1943 596 719.1 1091.0 645.6 823.4 69.3 63.0 43.9 66.8 57.630 33 3205 1260 1617.9 2149.6 1352.6 2062.12 60.7 49.5 32.9 57.8 35.730 67 2998 916 1118.2 1681.4 956.6 1309.6 69.4 62.7 43.9 68.1 56.330 100 2874 854 986.3 1495.3 906.9 1069.5 70.3 65.7 48.0 68.4 62.835 33 4474 1597 2014.4 2812.1 1690.5 2529.2 64.3 55.0 37.1 62.2 43.535 67 4147 1215 1437.3 2208.2 1262.7 1653.2 70.7 65.3 46.8 69.6 60.135 100 4000 1156 1303.8 1953.1 1222.1 1286.4 71.1 67.4 51.2 69.4 67.8Numerical results in Table 8 provide bounds for OPesym instances and show that V S and V S boundsare weaker than ^ RLT and ^ RLT bounds for all OPesym instances. This is due to the fact that there arenot many violated cuts of type (26a) – (26d) for those instances.Table 9 presents results for OPsym instances and shows that V S bounds are stronger than ^ RLT bounds for instances with less than 16 vertices. Moreover, V S bounds are stronger than ^ RLT bounds forall presented instances except for n = 18. Note that results in Table 8 and Table 9 present an average over10 instances of a given size.Pereira et al. [24] derive strong bounds for the QMSTP by using the idea of partitioning spanning treesinto forests of a given fixed size. The resulting model can be seen as a generalization of the RLT relaxation.To obtain bounds, the authors introduce a bounding procedure based on Lagrangian relaxation. In [24],Table 1 the authors provide bounds for CP instances and n = 25. Our lower bound V S is better than theirbound only for CP3 instance with n = 25 density 33%. Namely, Pereira et al. [24] report bound 2652.5and we compute 2730.3. Although bounds that result from the generalized RLT relaxation are strongerthan the bounds obtained from the incomplete first level RLT relaxation, the former were not used within aBranch and Bound algorithm in [24]. Instead, the authors use ^ RLT relaxation to solve several instances tooptimality. Note that our V S bounds are stronger than ^ RLT bounds for all CP instances except for theCP1 instance with n = 35 and d = 100, and all OPsym instances except for n = 18.16able 5: CP2 instances: bounds and gapsinstance lower bounds Gap (%) n d (%) U B GL ^ RLT ^ RLT V S V S GL ^ RLT ^ RLT V S V S 10 33 3122 2562 3122 3045.9 3114.6 3122 17.9 0 2.4 0.2 010 67 2042 809 1399.6 1710.2 1014.9 1946.2 60.4 31.5 16.2 50.3 4.710 100 1815 553 937 1384.4 733 1388.4 69.5 48.4 23.7 59.6 23.515 33 6539 3272 4684.1 5329.3 3690.9 6133.7 50.0 28.4 18.5 43.6 6.215 67 5573 1555 2589.7 3760.5 1908.1 3637 72.1 53.5 32.5 65.8 34.715 100 5184 1070 1829.9 3236.0 1406 2581.3 79.4 64.7 37.6 72.9 50.220 33 12425 4801 7035.7 8849.7 5390.8 10106.7 61.4 43.4 28.8 56.6 18.720 67 10893 2352 3816.7 6573.9 2907.5 5415.4 78.4 65.0 39.6 73.3 50.320 100 10215 1676 2748.2 5442.2 2060.2 3720.7 83.6 73.1 46.7 79.8 63.625 33 19976 6642 10068.5 13460.9 7361.8 13722.1 66.8 49.6 32.6 63.1 31.325 67 18251 3196 5054.3 9666.8 3795.7 7015.5 82.5 72.3 47.0 79.2 61.625 100 17411 2123 3469 7921.3 2623.2 4759.9 87.8 80.1 54.5 84.9 72.730 33 29731 7953 12046.6 17942.7 9120.5 17066.6 73.3 59.5 39.6 69.3 42.630 67 27581 3991 6382 12840.3 4684.5 8621.6 85.5 76.9 53.4 83.0 68.730 100 26146 2712 4332.4 10624.1 3271.8 5880 89.6 83.4 59.4 87.5 77.535 33 42305 9995 14684.1 23568.2 10771 20378.3 76.4 65.3 44.3 74.5 51.835 67 38490 4778 7409 16274.8 5517.8 10165.3 87.6 80.8 57.7 85.7 73.635 100 36723 3388 5241.7 13687.3 4016.8 6782.8 90.8 85.7 62.7 89.1 81.517able 6: CP3 instances: bounds and gapsinstance lower bounds Gap (%) n d (%) U B GL ^ RLT ^ RLT V S V S GL ^ RLT ^ RLT V S V S 10 33 646 589 646 646 646 646 8.8 0 0 0 010 67 488 320 488 476.1 474.9 488 34.4 0 2.4 2.7 010 100 426 234 386.2 401.1 360.4 426 45.1 9.3 5.8 15.4 015 33 1236 845 1180.5 1176.0 1113.5 1218 31.6 4.5 4.9 9.9 1.515 67 966 508 848.1 886.2 771.7 965.3 47.4 12.2 8.3 20.1 0.115 100 975 421 780 849.3 724.9 910 56.8 20.0 12.9 25.7 6.720 33 1972 1131 1672.6 1759.1 1511.9 1911.6 42.6 15.2 10.8 23.3 3.120 67 1792 743 1307.1 1470.1 1190 1541 58.5 27.1 18.0 33.6 14.020 100 1544 532 1056.1 1220.6 949.7 1257.5 65.5 31.6 20.9 38.5 18.625 33 2976 1448 2289.2 2488.3 2071.7 2730.3 51.3 23.1 16.4 30.4 8.325 67 2546 888 1630 1917.2 1468.8 1932.1 65.1 36.0 24.7 42.3 24.125 100 2471 761 1409.6 1740.1 1284.1 1657.5 69.2 43.0 29.6 48.0 32.930 33 4070 1834 2856.1 3235.3 2574.2 3383.1 54.9 29.8 20.5 36.8 16.930 67 3649 1152 2053.5 2516.6 1854.9 2425.4 68.4 43.7 31.0 49.2 33.530 100 3483 986 1776.1 2257.3 1627.5 2048.2 71.7 49.0 35.2 53.3 41.235 33 5423 2060 3360 3946.9 3007.1 4016 62.0 38.0 27.2 44.5 25.935 67 4981 1430 2515.7 3195.0 2324.1 2938.9 71.3 49.5 35.9 53.3 41.035 100 4770 1288 2190.1 2866.6 1995.7 2504.1 73.0 54.1 39.9 58.2 47.518able 7: CP4 instances: bounds and gapsinstance lower bounds Gap (%) n d (%) U B GL ^ RLT ^ RLT V S V S GL ^ RLT ^ RLT V S V S 10 33 3486 2891 3486 3424.4 3486 3486 17.1 0 1.8 0 010 67 2404 1158 1794 2076.0 1408.8 2330.3 51.8 25.4 13.6 41.4 3.110 100 2197 823 1321.1 1743.7 1120.2 1794.7 62.5 39.9 20.6 49.0 18.315 33 7245 3859 5354.8 6017.4 4371.2 6819.9 46.7 26.1 16.9 39.7 5.915 67 6188 2003 3214.5 4339.4 2542.9 4276.8 67.6 48.1 29.9 58.9 30.915 100 5879 1567 2490 3865.2 2056.7 3236.1 73.3 57.6 34.3 65.0 45.020 33 13288 5646 7914.2 9741.9 6256.3 10966.6 57.5 40.4 26.7 52.9 17.520 67 11893 2949 4693.4 7442.4 3791.7 6303.3 75.2 60.5 37.4 68.1 47.020 100 11101 2103 3574.1 6219.0 2887 4556.5 81.1 67.8 44.0 74.0 59.025 33 21176 7631 11186.5 14584.0 8507.1 14859.1 64.0 47.2 31.1 59.8 29.825 67 19207 3821 6095.5 10652.8 4828.6 8061.4 80.1 68.3 44.5 74.9 58.025 100 18370 2534 4490.3 8874.4 3649.3 5808.4 86.2 75.6 51.7 80.1 68.430 33 31077 9255 13401 19278.5 10451 18403.3 70.2 56.9 38.0 66.4 40.830 67 28777 4830 7566.9 14017.0 5935.5 9885.3 83.2 73.7 51.3 79.4 65.630 100 27314 3198 5555 11803.8 4495.3 7120.8 88.3 79.7 56.8 83.5 73.935 33 43629 11107 16170.9 24988.0 12271.2 21875 74.5 62.9 42.7 71.9 49.935 67 39660 5631 8884.4 17970.1 6998.4 11635.8 85.8 77.6 54.7 82.4 70.735 100 38049 3917 6707 15039.1 5477.4 8271 89.7 82.4 60.5 85.6 78.3Table 8: OPesym instances: bounds and gapsinstance lower bounds Gap (%) n d (%) U B GL ^ RLT ^ RLT V S V S GL ^ RLT ^ RLT V S V S n d (%) U B GL ^ RLT ^ RLT V S V S GL ^ RLT ^ RLT V S V S This paper introduces a hierarchy of lower bounds for the quadratic minimum spanning tree problem. Ourbounds exploit an extended formulation for the MSTP from [20] and the linear, exact formulation forthe QMSTP from [1]. We prove that our simplest relaxation V S is equivalent to the linearization basedrelaxation derived in Section 4.1, see Theorem 4.1. To improve the relaxation V S we add facet defininginequalities of the Boolean Quadric Polytope. The resulting relaxations V S and V S are presented inTable 2. Our relaxations have a polynomial number of constraints and can be solved by a cutting planealgorithm.On the other hand, all relaxations in the literature have an exponential number of constraints and most ofthem belong to the RLT type of bounds, see Section 3. The fact that our relaxations differ in both mentionedaspects from the other relaxations, enables us to efficiently compute bounds that are not dominated by theRLT type of bounds. Acknowledgements. We would like to thank Frank de Meijer for an insightful discussions on liearizationbased bounds. 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