Complete description for the spanning tree problem with one linearised quadratic term

Operations Research Letters 41 (2013) 701–705

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Operations Research Letters journal homepage: www.elsevier.com/locate/orl

Complete description for the spanning tree problem with one linearised quadratic term Anja Fischer a,∗ , Frank Fischer b a

Technische Universität Dortmund, Fakultät für Mathematik, D-44227 Dortmund, Germany

b

Technische Universität Chemnitz, Fakultät für Mathematik, D-09107 Chemnitz, Germany

article

info

Article history: Received 25 June 2013 Received in revised form 26 September 2013 Accepted 26 September 2013 Available online 4 October 2013 Keywords: Polyhedral combinatorics Complete description Spanning tree problem Quadratic zero–one problem

abstract Given an edge-weighted graph the minimum spanning tree problem (MSTP) asks for a spanning tree of minimal weight. The complete description of the associated polytope is well-known. Recently, Buchheim and Klein suggested studying the MSTP with one quadratic term in the objective function resp. the polytope arising after linearisation of that term, in order to better understand the MSTP with a general quadratic objective function. We prove a conjecture by Buchheim and Klein (2013) concerning the complete description of the associated polytope. © 2013 Elsevier B.V. All rights reserved.

1. Introduction and models

a linear optimisation problem formulation (LP) for the MSTP. Its corresponding dual linear program (DP) reads

Let G = (V , E ) be an undirected, simple, complete, edgeweighted graph with node set V , |V | = n, set of edges E and weight function c: E → R. Then the minimum spanning tree problem (MSTP) asks for a spanning tree in G with minimal total weight,

maximise



(1 − |S |)zS

∅̸=S ⊆V

subject to

minimise c (T )



−

zS ≤ c (e),

e ∈ E,

(4)

S:e⊆S ⊆V

zS ≥ 0, ∅ ̸= S ( V .

subject to T ⊆ G is a spanning tree,

zV free,



where c (X ) := e∈E (X ) c (e), X ⊆ G, with E (X ) := {e = {u, v}: u, v ∈ X , u ̸= v}. It is well-known that using a binary variable for each edge e ∈ E indicating whether the edge is contained in the spanning tree or not a linear integer formulation reads

Although the linear spanning tree problem and its associated polytope are well understood, not much is known if the objective function depends on products of edge-variables, i.e., if we want to optimise

c (e) · x(e)



minimise



e∈E

e∈E

− x(E ) = 1 − |V |, −x(E (S )) ≥ 1 − |S |, ∅ ̸= S ( V ,

(1)

x(e) ∈ {0, 1},

(3)

subject to

e ∈ E.

(2)

Edmonds [5] proved that replacing x(e) ∈ {0, 1}, e ∈ E, by x(e) ≥ 0 yields a complete description of the associated polytope. So we get

∗

Corresponding author. E-mail addresses: [email protected] (A. Fischer), [email protected] (F. Fischer). 0167-6377/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.orl.2013.09.011

c (e) · x(e) +



cq (e, f ) · x(e) · x(f )

e,f ∈E ,e̸=f

with additional weight function cq : E × E → R. The so called Quadratic Minimum Spanning Tree Problem (QMSTP) is known to be NP-hard [1]. This is analogous to the Assignment Problem, which can be solved efficiently and whose polyhedral structure is wellknown, and the Quadratic Assignment Problem (see, e.g., [8]), which is one of the computationally most challenging combinatorial optimisation problems. Some branch-and-bound algorithms and heuristics for the QMSTP were presented, e.g., in [1,9,4]. However, not much is known about the structure of the polytope that arises after a linearisation of x(e)x(f ), e, f ∈ E , e ̸= f , by introducing new variables y(e, f ), e, f ∈ E , e ̸= f . In order to better

702

A. Fischer, F. Fischer / Operations Research Letters 41 (2013) 701–705

understand the polyhedral structure of the QMSTP and of combinatorial optimisation problems with a quadratic objective function in general, Buchheim and Klein [3,2] suggested considering the special case of the QMSTP resp. of a combinatorial optimisation problem with exactly one quadratic term in the objective function. Because the MSTP is polynomially-solvable QMSTP-1 (QMSTP with one quadratic term) can be solved in polynomial time, too, and by the well-known ‘‘optimisation equals separation’’ result [6], we can hope to fully characterise the polytope of the linearised version of QMSTP-1. Furthermore, the separation algorithms for valid inequalities or facets of QMSTP-1 may also be useful for solving the general QMSTP because valid inequalities of the onemonomial-case remain valid for the general case. First computational experiments in [3,2] also indicate this behaviour. QMSTP-1 can be formally described as follows. Let u1 , v1 , u2 , v2 ∈ V with u1 v1 , u2 v2 ∈ E, u1 v1 ̸= u2 v2 , either {u1 , v1 } ∩ {u2 , v2 } = ∅ or v1 = v2 , u1 ̸= u2 , and c¯ ∈ R be the monomial weight. Then QMSTP-1 reads minimise q(T ) := c (T ) +

c¯ , 0,



u1 v 1 , u2 v 2 ∈ T , otherwise,

−x(E (S )) − y ≥ 1 − |S |,

S ⊂ V , u1 , u2 ∈ S , v1 = v2 ̸∈ S ,

for a complete description. If we can show that (QP) is a complete description this conjecture follows because then {(S , S ′ ): S , S ′ ⊂ V , S ∩ S ′ = ∅, u1 , v2 ∈ S , u2 , v1 ∈ S ′ } = ∅ and inequalities (5) with (S , S ′ ) ∈ {(S , S ′ ): S , S ′ ⊂ V , S ∩ S ′ = ∅, u1 , u2 ∈ S , v1 = v2 ∈ S ′ , |S ′ | > 1} are implied by (5) with (S , S ′ ) ∈ {(S , S ′ ): S , S ′ ⊂ V , S ∩ S ′ = ∅, u1 , u2 ∈ S , S ′ = {v1 }} and (2). Note that in the meantime Buchheim and Klein independently proved the abovementioned conjectures. A complete proof for the connected case can be found in [2]. 2. Notation and previous results In the following we write [k] instead of {1, . . . , k}, k ∈ N. We denote the objective value of a spanning tree T w. r. t. c˜ : E → R by

vLP (˜c , T ) =



c˜ (e)

and

vDP (z ) :=

e∈E (T )

T ⊆ G is a spanning tree.

subject to

QMSTP-1. In the connected case, their conjecture looks a bit different. It says that apart from the standard linearisation (6)–(8) and the formulation of the MSTP (1)–(2), x ≥ 0 one only needs

In [3] the case v1 = v2 is called the connected case because the two edges u1 v1 and u2 v2 share a common node, otherwise it is called the unconnected case. In most parts we will not distinguish between the two cases. In this article we will prove that the following equations and inequalities are a complete description of the integer polytope if we linearise the monomial x(u1 v1 ) · x(u2 v2 ) by introducing a new variable y. Let



(1 − |S |)zS

S:∅̸=S ⊆V

denotes the value of a solution z of (DP) with z = (zS )S:∅̸=S ⊆V . The following result follows from [5] and can, e.g., be found in [7] (proof of Theorem 6.13).

S := {(S , S ′ ): S , S ′ ⊂ V , S ∩ S ′ = ∅, u1 v1 , u2 v2 ∈ E (S , S ′ )}

Lemma 1 ([5,7]). Let T be a minimum spanning tree (MST) in G w. r. t. c˜ : E → R computed by the greedy algorithm. Let f1 , . . . , f|V |−1 be the edges selected by the (best-in) greedy algorithm in order and denote by Xk ⊆ V , k ∈ [|V | − 1], the nodes of the connected component of (V , {f1 , . . . , fk }) that contains fk . Furthermore, let s(k) ∈ [|V | − 1], k ∈ [|V | − 2], denote the smallest index greater than k so that fs(k) ∩ Xk ̸= ∅. Then the dual solution

with E (X , Y ) := {e = {u, v} ∈ E: u ∈ X , v ∈ Y }. Then (QP) reads

z ∗ (˜c , T ) = (zS )S:∅̸=S ⊆V ,



minimise

c (e) · x(e) + c¯ · y

with

e∈E

−x(E (S ) ∪ E (S ′ )) − y ≥ 2 − |S | − |S ′ |, xui vi − y ≥ 0, i ∈ {1, 2},

(S , S ′ ) ∈ S,

(5) (6)

y − xu1 v1 − xu2 v2 ≥ −1,

(7)

y ≥ 0.

(8)

Let F bea family of sets, then we write z (F) = F ∈F zF and ¯F , respectively. So the dual problem (DQP) is z¯ (F) = F ∈F z





maximise

(1 − |S |)zS



(2 − |S | − |S ′ |)¯z(S ,S ′ ) − ζy

(9)

(S ,S ′ )∈S

−

subject to



zS −

S:e⊆S ⊆V



−

(10)

zS + ζui vi − ζy ≤ c (ui vi ),

i ∈ {1, 2},

(11)

ζu1 v1 , ζu2 v2 , ζy ≥ 0.

vQP (˜c , T ) =



c˜ (e) +

vDQP (z , z¯ , ζ ) =



z¯(S ,S ′ ) ≥ 0,

(12)

(S , S ) ∈ S, ′



c¯ , 0,

u1 v1 , u2 v2 ∈ T , otherwise,

zV free,

(1 − |S |)zS

S:∅̸=S ⊆V

+

−¯z (S) − ζu1 v1 − ζu2 v2 + ζy ≤ c¯ , ∅ ̸= S ( V ,

Remark 2. Note that we may assume, w.l.o.g., that each variable z{u} , u ∈ V , of the solution z ∗ (˜c , T ) has an arbitrarily large value, because these variables do not contribute to the objective value and do not appear in any constraint except for z{u} ≥ 0. We will make use of this property later in Corollary 5.

and the value of a solution (z , z¯ , ζ ) of (DQP) with z = (zS )S :∅̸=S ⊆V , z¯ = (¯z(S ,S ′ ) )(S ,S ′ )∈S and ζ = (ζy , ζu1 v1 , ζu2 v2 ) by

(S ,S ′ )∈S: e∈E (S )∪E (S ′ )

S:ui vi ⊆S ⊆V

zS ≥ 0,

is an optimal solution of (DP). In particular, for any edge e ∈ E there holds lhs(z , e) := − S:e⊆S ⊆V zS = c˜ (fi ), where i ∈ [|V | − 1] is the smallest index so that e ⊆ Xi .

e∈E (T )

z¯(S ,S ′ ) ≤ c (e),

e ∈ E \ {u1 v1 , u2 v2 },



S = Xk , k < |V | − 1, S = X|V |−1 = V , otherwise,

We denote the value of spanning tree T w. r. t. c˜ : E → R and weight c¯ by

∅̸=S ⊆V

+

zS :=

(1), (2), x ≥ 0

subject to

c˜ (fs(k) ) − c˜ (fk ), −˜c (f|V |−1 ), 0,





(2 − |S | − |S ′ |)¯z(S ,S ′ ) − ζy .

(S ,S ′ )∈S

(13) (14)

Indeed, Buchheim and Klein conjectured that in the unconnected case the model (QP) above provides a complete description of

3. Complete description In this section we will prove that (QP) is indeed a complete description of the integer polytope for QMSTP-1. We start with a

A. Fischer, F. Fischer / Operations Research Letters 41 (2013) 701–705

703

Fig. 1. Visualisation of assumption (A). The solid edges are contained in the forest T˜ .

lemma about the structure of optimal spanning trees if we adapt the coefficients of the edges u1 v1 , u2 v2 . Lemma 3. Let G = (V , E ) be a complete undirected graph and c: E → R a weight function. Then there exist a forest T˜ ⊆ G and four edges ei , e′i ∈ E \ E (T˜ ), i = 1, 2, with {e1 , e2 } ⊆ {e′1 , e′2 }, e′1 ̸= e′2 , c (e′1 ) ≤ c (e′2 ) and either e1 ̸= e2 or e1 = e2 = e′1 , so that the four graphs T0 := T˜ + e′1 + e′2 , T1 := T˜ + u1 v1 + e2 , T2 := T˜ + u2 v2 + e1 ,

and T12 := T˜ + u1 v1 + u2 v2 are spanning trees, and for each weight function c˜ : E → R with c˜ (e) = c (e) for all e ∈ E \ {u1 v1 , u2 v2 } one of these four trees is an MST in G w. r. t. c˜ . Proof. Consider the tree T generated by the greedy algorithm, but starting with the forest containing the two edges u1 v1 , u2 v2 . Let f1 , . . . , f|E |−2 be the sequence of edges E \ {u1 v1 , u2 v2 } in the order they have been considered by the greedy algorithm, in particular c (f1 ) ≤ · · · ≤ c (f|E |−2 ). We set T˜ := T − u1 v1 − u2 v2 , and choose ei := fki , ki = min j ∈ [|E | − 2]: fj ̸∈ T , T − ui vi + fj is a tree ,





i = 1, 2.

Throughout the rest of the article we will assume, w.l.o.g. (otherwise rename the nodes): If e1 = e2 = e′1 , then e′1 lies on the path between u1 and u2 not using v1 , v2 on the cycle in T12 + e′1 .

(A)

Note that (A) automatically holds in the connected case, see also Fig. 1. Corollary 5. Let T ∈ {T0 , T1 , T2 , T12 } be an MST w. r. t. c˜ : E → R with c˜ (e) = c (e) for all e ∈ E \ {u1 v1 , u2 v2 } and z = z ∗ (˜c , T ). We consider the following condition e1 ̸= e2

c (e′1 ) ≥ min{˜c (u1 v1 ), c˜ (u2 v2 )}.

or

(C)

1. If (C) holds, then lhs(z , u1 v1 ) + lhs(z , u2 v2 ) = c˜ (T ) − c (T˜ ).

(15)

2. Otherwise, if (C) does not hold and assuming (A), then T ̸= T12 and lhs(z , u1 v1 ) + lhs(z , u2 v2 )

First assume e1 ̸= e2 and set {e′1 , e′2 } := {e1 , e2 }. Then by the choice

of e1 , e2 the graph T˜ +e′1 +e′2 = T˜ +e1 +e2 = T −u1 v1 +e1 −u2 v2 +e2 forms a tree, too. Next consider the greedy algorithm for c˜ . We can assume that all edges in E \ {u1 v1 , u2 v2 } are considered in the

same order because their costs remain unchanged. Then the greedy algorithm selects all edges E (T ) \ {u1 v1 , u2 v2 }, and the first of u1 v1 and e1 , and the first of u2 v2 and e2 by the choice of e1 , e2 . So the greedy algorithm for c˜ will generate one of the trees T0 , T1 , T2 , T12 . Now assume e1 = e2 . Then set e′1 := e1 = e2 , choose m = min i ∈ [|E | − 2]: fi ̸∈ T , T − u1 v1



 − u2 v2 + e′1 + fi is a tree , and set e′2 := fm . We show that k1 = k2 < m. Let X , Y , Z be

the three components of T˜ and assume, w.l.o.g., that u1 v1 connects X and Y and u2 v2 connects X and Z . Because T˜ + u1 v1 + e′1 and

T˜ + u2 v2 + e′1 are both trees, e′1 must connect Y and Z . Furthermore,

because T˜ + e′1 + e′2 is a tree, e′2 must connect X with either Y or Z . Consequently, either T − u1 v1 + e′2 or T − u2 v2 + e′2 is a tree, proving k1 = k2 < m. Regarding the greedy algorithm with c˜ , similar to above, depending on the values of c˜ (u1 v1 ) and c˜ (u2 v2 ), the algorithm generates one of the four trees T0 , T1 , T2 , T12 .  Note that we will use the notation T , T˜ , e1 , e2 , e′1 , e′2 throughout the article. The previous lemma implies the following. Corollary 4. The objective value of an MST in G w. r. t. c˜ : E → R with c˜ (e) = c (e) for all e ∈ E \ {u1 v1 , u2 v2 } will be min{˜c (T0 ), c˜ (T1 ), c˜ (T2 ), c˜ (T12 )}

= c (T˜ ) + min{c (e1 ) + c (e2 ), c˜ (u1 v1 ) + c (e2 ), c (e1 ) + c˜ (u2 v2 ), c˜ (u1 v1 ) + c˜ (u2 v2 )}. ′

′

= 2 · min{˜c (u1 v1 ), c˜ (u2 v2 ), c (e′2 )},

(16) ′

and there exist two families of sets Sp , p ∈ [k], Sq , q ∈ [k′ ], satisfying the following conditions u1 , u2 ∈

 p∈[k]

Sp ,

v1 , v2 ∈



Sq′ ,

q∈[k′ ]

∀ p ∈ [k], q ∈ [k′ ] : Sp ∩ Sq′ = ∅,

(S)

so that z ({Sp : p ∈ [k]}) = min{˜c (u1 v1 ), c˜ (u2 v2 ), c (e2 )} − c (e′1 ) ≤ z ({Sq′ : q ∈ [k′ ]}). ′

Proof. First observe that ui vi ∈ T for some i ∈ {1, 2} implies lhs(z , ui vi ) = c˜ (ui vi ) by Lemma 1. Assume (C) holds. Let e1 ̸= e2 and, w.l.o.g., u1 v1 ̸∈ T then e1 ∈ T and lhs(z , u1 v1 ) = c (e1 ). This proves (15) for all cases if e1 ̸= e2 . Otherwise, if e1 = e2 and c (e′1 ) ≥ min{˜c (u1 v1 ), c˜ (u2 v2 )}, then, w.l.o.g., u1 v1 ∈ T . If u2 v2 ∈ T then lhs(z , ui vi ) = c˜ (ui vi ), i = 1, 2, as above and (15) follows. Because e′1 = e2 ∈ T we know that e′1 is the most expensive edge on the cycle in T + u2 v2 except for u2 v2 , hence e′1 is the edge that connects u2 and v2 first in the greedy algorithm, so lhs(z , u2 v2 ) = c (e′1 ). This implies again (15). Next, assume (C) does not hold, i.e., e1 = e2 = e′1 and c (e′1 ) < c˜ (ui vi), i = 1, 2. In this case e′1 ∈ T and f ∈ T where f ∈ Arg min c˜ (e): e ∈ {u1 v1 , u2 v2 , e′2 } . So by (A), u1 , u2 are connected for the first time when e′1 is selected, hence ui , vi , i = 1, 2, are connected for the first time when f is selected. Consequently lhs(z , ui vi ) = c˜ (f ), i = 1, 2, proving (16). It remains to prove the existence of sets satisfying (S) and the last condition. For this assume first v1 ̸= v2 and let Xij , j ∈ [k], k ∈ N, be the components in order that contain u1 , u2 but not v1 , v2 , and Xi′ , j ∈ [k′ ], k′ ∈ N, the components that contain j

v1 , v2 but not u1 , u2 according to Lemma 1. Note that i1 > i′1 and fi1 = e′1 (because e′1 is considered before u1 v1 , u2 v2 and e′2 and because of assumption (A)), and with ik+1 = i′k′ +1 defined so that fik+1

704

A. Fischer, F. Fischer / Operations Research Letters 41 (2013) 701–705

k ˜ (fip+1 ) − = fi′ ′ = f it follows that z ({Xip : p ∈ [k]}) = p=1 c k +1 c˜ (fip ) = c˜ (f )− c˜ (e′1 ), as well as z ({Xi′q : q ∈ [k′ ]}) = c˜ (f )− c˜ (fi′ ). Be1 cause i1 > i′1 we know that c˜ (e′1 ) ≥ c˜ (fi′ ) and because of Xip ∩ Xi′q = 1 ∅ for all p ∈ [k], q ∈ [k′ ], we can choose Sp := Xip and Sq′ := Xi′q . Finally, if v1 = v2 we may simply use Sp := Xip as before and S1′ := {v1 }, k′ = 1, by Remark 2.  Corollary 6. Let (z , z¯ , ζ ) be a point satisfying constraints (10), (11) and (13), (14) of (DQP). Assume that there exist two families of sets Sp ⊆ V , p ∈ [k], and Sq′ ⊆ V , q ∈ [k′ ], satisfying (S) with z ({Sp : p ∈ [k]}) ≤ z ({Sq′ : q ∈ [k′ ]}). Then there exists a point (z ′ , z¯ ′ , ζ ′ ) with ζ ′ = ζ also satisfying constraints (10), (11) and (13), (14) with vDQP (z , z¯ , ζ ) = vDQP (z ′ , z¯ ′ , ζ ′ ) and z¯ ′ (S) ≥ z ({Sp : p ∈ [k]}). Proof. If zSp = 0 for all p ∈ [k] then (z ′ , z¯ ′ , ζ ′ ) = (z , z¯ , ζ ) satisfies the required conditions because z¯ ≥ 0 by (13), so we may assume that there is at least one i ∈ [k] with zSi > 0 and, consequently, a j ∈ [k′ ] with zS ′ > 0. Let d := min{zSi , zS ′ }. We define ζ (1) := ζ j

j

and (1)

zS

z S − d, zS ,

 :=



(1)

z¯(S ,S ′ ) :=

S ∈ {Si , Sj′ }, otherwise,

z¯(S ,S ′ ) + d, z¯(S ,S ′ ) ,

and

(S , S ′ ) = (Si , Sj′ ), otherwise. (1)

(1)

Note that at least one of zSi , zS ′ is zero, so the number of non-zero j

coefficients decreased. Because zSi and zS ′ do not appear in the leftj

hand side of constraints (11), these are satisfied by (z (1) , z¯ (1) , ζ (1) ), too, and constraints (13) and (14) trivially hold. Furthermore it is z ({Sp : p ∈ [k]}) + z¯ (S) = z (1) ({Sp : p ∈ [k]}) + z¯ (1) (S).

(17)

For (10) let e ∈ E \ {u1 v1 , u2 v2 } be an arbitrary edge. If e ̸⊆ Si and e ̸⊆ Sj′ then the left-hand side of (10) cannot be increased, so assume, w.l.o.g., e ⊆ Si . Then e ̸⊆ Sj′ by assumption and direct computation shows that the left-hand side of (10) for (z (1) , z¯ (1) , ζ (1) ) remains the same. Similarly, direct computation shows vDQP (z , z¯ , ζ ) = vDQP (z (1) , z¯ (1) , ζ (1) ). Iterating the argument we get a feasible solution (z ′ , z¯ ′ , ζ ′ ) := (z (r ) , z¯ (r ) , ζ (r ) ), r ∈ N, with (r ) zSp = 0 for all p ∈ [k]. Hence, by assumption and (17) there holds z¯ ′ (S) = z ′ ({Sp : p ∈ [k]}) + z¯ ′ (S) = z ({Sp : p ∈ [k]}) + z¯ (S)

≥ z ({Sp : p ∈ [k]}). This completes the proof.



Now we prove our main result. Theorem 7. (QP) is a formulation of QMSTP-1. Proof. Let T be an MST for (LP) w. r. t. c; by Lemma 3 this is one of the trees T0 , T1 , T2 , T12 . In the whole proof T˜ denotes the subgraph as defined in Lemma 3. c¯ ≥ 0 and T ∈ {T0 , T1 , T2 }: Then T fulfils vLP (c , T ) = vQP (c , T ) and an optimal solution z = z ∗ (c , T ) of (DP) can be extended to a feasible solution (z , z¯ , ζ ) of (DQP) by setting z¯ = 0 and ζu1 v1 = ζu2 v2 = ζy = 0 with vDQP (z , z¯ , ζ ) = vDP (z ) = vLP (c , T ) = vQP (c , T ), so T is an optimal solution of (QP). c¯ ≥ 0 and T = T12 : By Corollary 4 there holds c (u1 v1 ) ≤ c (e1 ) and c (u2 v2 ) ≤ c (e2 ). Assume, w.l.o.g., c (e1 ) − c (u1 v1 ) ≤ c (e2 ) − c (u2 v2 ). Set ε := min{c (e1 )− c (u1 v1 ), c¯ }, then ε ≤ c (e2 )− c (u2 v2 ), and set cˆ (e) :=

c (e) + ε, c (e),



e ∈ {u1 v1 , u2 v2 }, otherwise.

Let T ′ be an MST w. r. t. cˆ and let z = z ∗ (ˆc , T ′ ) be an optimal solution of (DP). Direct computation using Corollary 4 shows that either T12 (if ε = c¯ ) or T2 (if ε < c¯ ) is an MST w. r. t. cˆ , and in both cases vQP (c , T ′ ) = vLP (ˆc , T ′ ) − ε . Furthermore, because z is feasible for (DP) with cˆ there holds lhs(z , e) ≤ cˆ (e) = c (e), e ∈ E \{u1 v1 , u2 v2 }, as well as lhs(z , e) ≤ cˆ (e) = c (e) + ε for e ∈ {u1 v1 , u2 v2 }. Hence, with z¯ = 0, ζu1 v1 = ζu2 v2 = 0 and ζy = ε , solution (z , z¯ , ζ ) satisfies (10) and (11), and because c¯ ≥ 0 constraint (12) as well, so it is feasible for (DQP). Furthermore the objective value fulfils vDQP (z , z¯ , ζ ) = vDP (z ) − ζy = vQP (c , T ′ ) proving the optimality of T ′ . c¯ < 0 and c (u1 v1 ) + c (u2 v2 ) + c¯ ≥ c (T ) − c (T˜ ): Note that in this case T ̸= T12 (otherwise c (u1 v1 ) + c (u2 v2 ) + c¯ < c (u1 v1 ) + c (u2 v2 ) = c (T12 ) − c (T˜ )). Let z = z ∗ (c , T ) be an optimal solution of (DP) and set z¯ = 0, ζy = 0 and ζui vi = c (ui vi ) − lhs(z , ui vi ), i = 1, 2. Then (z , z¯ , ζ ) satisfies (10) and (11). To check (12) we consider two cases. 1. Condition (C) holds for c˜ = c. Corollary 4 and (15) imply lhs(z , u1 v1 )+ lhs(z , u2 v2 ) = c (T )− c (T˜ ) ≤ c (u1 v1 )+ c (u2 v2 )+ c¯ . Thus, −ζu1 v1 − ζu2 v2 ≤ c¯ and (12) is satisfied. Because vDQP (z , z¯ , ζ ) = vDP (z ) = vLP (c , T ) = vQP (c , T ) tree T is optimal for (QP). 2. Condition (C) does not hold for c˜ = c. Then by Corollaries 5 and 6 there exist a solution (z ′ , z¯ ′ , ζ ′ ), ζ ′ = ζ , satisfying (10) and (11) and vDQP (z ′ , z¯ ′ , ζ ′ ) = vDQP (z , z¯ , ζ ), and two families Sp , p ∈ [k], and Sq′ , q ∈ [k′ ], such that z¯ ′ (S) ≥ z ({Sp : p ∈ [k]}) = min{c (u1 v1 ), c (u2 v2 ), c (e′2 )}− c (e′1 ). Consequently, direct computation with (16) shows that (12) is satisfied by (z ′ , z¯ ′ , ζ ′ ) for T ∈ {T0 , T1 , T2 } and because T ̸= T12 it is vDQP (z ′ , z¯ ′ , ζ ′ ) = vDQP (z , z¯ , ζ ) = vDP (z ) = vLP (c , T ) = vQP (c , T ), hence T is optimal. c¯ < 0 and c (u1 v1 ) + c (u2 v2 ) + c¯ ≤ c (T ) − c (T˜ ): Because T is optimal w. r. t. c, Corollary 4 implies c (u1 v1 ) + c (u2 v2 ) + c¯ ≤ min{c (e1 ) + c (u2 v2 ), c (e′1 ) + c (e′2 ), c (u1 v1 ) + c (e2 )}.

(18)

Assume, w.l.o.g., c (u1 v1 ) ≤ c (u2 v2 ). We consider two cases. 1. Condition (C) holds for c˜ = c. We set

 c (u v ) + c (u2 v2 ) + c¯ − min{c (u2 v2 ), c (e2 )},   1 1 e = u1 v1 , cˆ (e) := min{c (u2 v2 ), c (e2 )}, e = u2 v2 ,   c (e), otherwise. Then cˆ (u1 v1 ) ≤ min{c (e1 ) + c (u2 v2 ), c (e′1 ) + c (e′2 ), c (u1 v1 ) + c (e2 )} − min{c (u2 v2 ), c (e2 )} ≤ c (e1 ), where the last inequality follows from the first term in the min in (18) if c (u2 v2 ) ≤ c (e2 ), from the second if e1 ̸= e2 or the third if c (u1 v1 ) ≤ c (e′1 ) ≤ c (e1 ) (because (C) holds this is exhaustive). By definition cˆ (u2 v2 ) ≤ c (e2 ), so T12 is an MST w. r. t. cˆ with vLP (ˆc , T12 ) = c (T˜ )+c (u1 v1 )+c (u2 v2 )+¯c . In particular, condition (C) holds for c˜ = cˆ . Let z = z ∗ (ˆc , T12 ) and set z¯ = 0, ζy = 0, ζui vi = c (ui vi ) − lhs(z , ui vi ), i = 1, 2. Because lhs(z , u1 v1 ) ≤ cˆ (u1 v1 ) ≤ c (u1 v1 ) by c¯ < 0 and (18) as well as lhs(z , u2 v2 ) ≤ cˆ (u2 v2 ) ≤ c (u2 v2 ) we have ζui vi ≥ 0 for i = 1, 2. Hence, (z , z¯ , ζ ) satisfies (10) and (11) and, because −ζu1 v1 − ζu2 v2 ≤ c¯ by (15), constraint (12) is satisfied as well. So, (z , z¯ , ζ ) is feasible for (DQP) and vQP (c , T12 ) = vLP (ˆc , T12 ) = vDP (z ) = vDQP (z , z¯ , ζ ) proves optimality of T12 for (QP). 2. Condition (C) does not hold for c˜ = c, in particular, e′1 = e1 = e2 and T ̸= T12 . We set cˆ (e) :=



c (u1 v1 ) + c (u2 v2 ) + c¯ − c (e2 ), c (e),

e = u1 v 1 , otherwise.

A. Fischer, F. Fischer / Operations Research Letters 41 (2013) 701–705

By direct computations and by assumption we get cˆ (u1 v1 ) ≤ c (u1 v1 ) ≤ c (u2 v2 ) = cˆ (u2 v2 ), cˆ (u1 v1 ) ≤ c (e′2 ), and, because T ̸= T12 it holds c (e′1 ) = c (e2 ) ≤ c (u2 v2 ) = cˆ (u2 v2 ). So Corollary 4 implies that T1 is an MST w. r. t. cˆ with vLP (ˆc , T1 ) = vQP (c , T12 ). Let z = z ∗ (ˆc , T1 ) and set z¯ = 0, ζy = 0, ζui vi = c (ui vi ) − lhs(z , ui vi ), i = 1, 2. Because z is feasible for (DP) w. r. t. cˆ we have lhs(z , ui vi ) ≤ cˆ (ui vi ) ≤ c (ui vi ), so ζui vi ≥ 0 for i = 1, 2, and (z , z¯ , ζ ) satisfies (10) and (11). (a) If (C) holds for c˜ = cˆ , then (15) implies lhs(z , u1 v1 ) + lhs(z , u2 v2 ) = cˆ (T1 ) − c (T˜ ) = cˆ (u1 v1 ) + c (e2 ) = c¯ + c (u1 v1 )+ c (u2 v2 ). Hence (12) is satisfied and (z , z¯ , ζ ) is feasible for (DQP) with vDQP (z , z¯ , ζ ) = vDP (z ) = vLP (ˆc , T1 ) = vQP (c , T12 ), so T12 is optimal for (QP). (b) If (C) does not hold for c˜ = cˆ , then by Corollaries 5 and 6 there exist a solution (z ′ , z¯ ′ , ζ ′ ), ζ ′ = ζ , satisfying (10) and (11) and vDQP (z ′ , z¯ ′ , ζ ′ ) = vDQP (z , z¯ , ζ ), and two families Sp , p ∈ [k] and Sq′ , q ∈ [k′ ], such that z¯ ′ (S) ≥ cˆ (u1 v1 ) − c (e′1 ). Direct computation with (16) shows that (z ′ , z¯ ′ , ζ ′ ) satisfies (12), so it is feasible for (DQP), and vDQP (z ′ , z¯ ′ , ζ ′ ) = vDQP (z , z¯ , ζ ) = vDP (z ) = vLP (ˆc , T1 ) = vQP (c , T12 ) proves optimality of T12 for (QP).  Remark 8. Note that in the construction of the optimal dual solution of (DQP) in the proof of Theorem 7 we only used non-zero values for the variables z¯(S ,S ′ ) , (S , S ′ ) ∈ S, if c¯ < 0 and if condition (C) does not hold. But this implies that the inequalities (5) in the primal problem (QP) are only required in these cases and can be dropped otherwise.

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So we have found a complete description of QMSTP-1. It remains for future work to consider further optimisation problems in the same manner or the QMSTP with two or more quadratic terms in the objective function. Furthermore, it might be interesting to look at higher degree polynomials. References [1] Arjang Assad, Weixuan Xu, The quadratic minimum spanning tree problem, Naval Research Logistics (NRL) 39 (3) (1992) 399–417. [2] Christoph Buchheim, Laura Klein, Combinatorial optimization with one quadratic term: spanning trees and forests, 2013, submitted for publication, available at http://www.optimization-online.org/DB_FILE/2013/07/3952.pdf. [3] Christoph Buchheim, Laura Klein, The spanning tree problem with one quadratic term, in: Kamiel Cornelissen, Ruben Hoeksma, Johann Hurink, and Bodo Manthey (Eds.), CTW, volume WP 13-01 of CTIT Workshop Proceedings, 2013, pp. 31–34. [4] Roberto Cordone, Gianluca Passeri, Solving the quadratic minimum spanning tree problem, Applied Mathematics and Computation 218 (23) (2012) 11597–11612. [5] Jack Edmonds, Matroids and the greedy algorithm, Mathematical Programming 1 (1) (1971) 127–136. [6] Martin Grötschel, Lászlo Lovász, Alexander Schrijver, Geometric Algorithms and Combinatorial Optimization, in: Algorithms and Combinatorics, vol. 2, Springer, 1993, second corrected edition. [7] Bernhard Korte, Jens Vygen, Combinatorial Optimization: Theory and Algorithms, fifth ed., Springer, 2012. [8] Eliane Maria Loiola, Nair Maria Maia de Abreu, Paulo Oswaldo BoaventuraNetto, Peter Hahn, Tania Querido, A survey for the quadratic assignment problem, European Journal of Operational Research 176 (2) (2007) 657–690. [9] Temel Öncan, Abraham P. Punnen, The quadratic minimum spanning tree problem: a lower bounding procedure and an efficient search algorithm, Computers & Operations Research 37 (10) (2010) 1762–1773.