Complexity

As mentioned in Section 2.2, the linear problems MSF and MST are efficiently solvable and MST polynomially reduces to MSF by a simple change of the cost function. The idea of reduction can be carried over to the quadratic versions, only that the linear costs need to be reduced by a modified value M which has to be big enough such that the objective value decreases each time an edge is added to a forest with less than |V | − 1 egdes.

Lemma 4.1.1.

QMST polynomially reduces to QMSF.

Proof. Consider a QMST instance on a connected graph G = (V, E) with objective func- tion z(x) = c⊤x + x⊤Qx. With cmax:= max{ce | e ∈ E} and qmax := max{qef | e, f ∈ E} de-

fine M := max{cmax, qmax, 0}+1 and new linear costs ˜ce:= ce−M (|V |−1). By this construction,

the objective values z(x) and ˜z(x) := ˜c⊤x + x⊤Qx of each forest F differ by |E(F )| · M (|V | − 1). Furthermore, any optimal forest with respect to ˜z is connected, since adding an arbitrary edge e ∈ E\E(F ) to a forest F with less than |V | − 1 edges changes the objective value by the additional term

˜ ce+ X f ∈E(F ) qef = ( ce |{z} <M −M (|V | − 1) ) + X f ∈E(F ) qef | {z } <(|V |−2)M < 0

4.1. PROPERTIES AND ALGORITHMS 41 and thus improves the solution. Therefore, an optimal spanning forest F∗ with respect to ˜z always is a tree, which furthermore is optimal with respect to the original objective function z and has the value z(x∗) = ˜z(x∗) − (|V | − 1)M .

The complexity of QMSF and QMST in turn rises significantly compared to the linear versions as the NP-hardness of general BQP is carried over. There are only few special cases which have been shown to be tractable, e. g., the case of a multiplicative objective function with positive factors [153]. Even the special case of the adjacent-only versions become NP-complete. This property can be shown by a polynomial transformation of the Hamiltonian path (HP) problem to AQMST [9]. A HP is a path which passes each vertex of a given graph exactly once, and the HP problem is the corresponding decision problem and proven to be NP-complete (see, e. g. [77]). Theorem 4.1.1.

AQMSF and AQMST are NP-complete.

Proof. For a given connected graph G = (V, E) define ce= 0 for all e ∈ E, qef = 1 if edges e and f

are adjacent, and qef = 0 else. Let T∗ be an optimal spanning tree with incidence vector x∗,

moreover let dv := |δ(v)|. Then, for each vertex v ∈ V there are dv(dv− 1) ordered edge pairs

in E(T∗) which share vertex v. With qef = 1 for all these pairs, they contribute dv(dv− 1) to

the objective function, such that

z(x∗) =X

v∈V

dv(dv− 1).

The sum is minimal if the tree is a path, say from u to w, such that du = dw= 1 and dv = 2 for

all v ∈ V \{u, w}. Therefore, z(x) ≥ 2(n − 2) for any incidence vector x of a tree, and equality holds if and only if the tree is an Hamiltonian path in G. By this construction the existence of a HP can be proven by solving an AQMST problem. By Lemma 4.1.1 AQMSF is NP-complete, too.

Since the adjacent-only problems are special cases, QMSF and QMST with arbitrary quadratic costs are at least as hard as the adjacent-only problems. The alternative proof for the NP- hardness shows that actually one of the practically hardest problems in quadratic optimization, the NP-complete quadratic assignment problem, which searches for an assignment minimizing a quadratic objective function and which we investigate in Chapter 6, is reducable to QMST. Theorem 4.1.2.

QMSF and QMST are NP-complete.

Proof. Consider a QAP instance on a bipartite graph G = (Va˙∪Vb, E) with disjoint vertex

sets Va = {a1, . . . , an} and Vb = {b1, . . . , bn} and with linear costs ce for e ∈ E and quadratic

costs qef for non-adjacent edges e, f ∈ E. Extend G to a non-bipartite graph ˜G = (V, E ∪ ˜E)

with auxiliary edge set ˜E = {{bi, bi+1} | i ∈ {1, . . . , n−1}}, see Figure 4.1, and define the QMST

costs analogous to the QAP costs plus qef = ∞ for adjacent edges e, f ∈ E and qef = 0 for

the remaining pairs of edges, i. e., for all e ∈ ˜E and f ∈ E ∪ ˜E. Then, any assignment A in G can be extended to a spanning tree T in ˜G by adding the n − 1 auxiliary edges in ˜E, where the QAP costs of A and the QMST costs of T are equal by construction. Conversely, each spanning tree T with an objective value less than ∞ contains only edges e ∈ E which are pairwise not adjacent plus the n−1 edges in ˜E. The edges in E then define an assignment in G with the same objective value as T . Thus, an optimal solution of QMST with finite objective value directly yields an optimal QAP solution.

42 CHAPTER 4. QUADRATIC SPANNING FORESTS AND TREES a1 a2 a3 .. . an b1 b2 b3 .. . bn G a1 a2 a3 .. . an b1 b2 b3 .. . bn ˜ E ˜ G

Figure 4.1: The original bipartite graph G and its extension ˜G.

4.1.3 Lower bounds

Due to the high complexity of the QMSF and the QMST problem few literature on exact algorithms exists. Indeed, current B&B approaches can only solve dense QMST instances up to 15 and sparse instances up to 20 vertices although much effort is put into the calculation of good bounds. Of course all lower bounds mentioned in Section 3.3 are applicable, and especially two of them are used in performance studies, the Gilmore-Lawler bound and the bound proposed by Assad and Xu, which in the case of a QMST problem both have to solve several linear MST problems. A combination of several approaches leads to another lower bound for QMST and is presented in [149]. Here, a partial application of the first RLT level for a QMST formulation replaces the quadratic terms xexf by auxiliary variables yef but instead of the constraints of

the standard linearization the following constraints are added. X e∈E yef = (|V | − 1) xf ∀ f ∈ E X e∈E(G[S]) yef ≤ (|S| − 1) xf ∀ ∅ 6= S ⊆ V and ∀ f ∈ E yee= xe ∀ e ∈ E yef = yf e ∀ e 6= f ∈ E yef ≥ 0 ∀ e 6= f ∈ E

A Lagrangean relaxation of the equation yef = yf e then leads to the lower bound, which for a

Lagrangean multiplier λ = 0 provides the Gilmore-Lawler bound and improves with increasing λ. Cordone and Passeri proposed in [49] to calculate a set of lower bounds, one for each branching step in the B&B tree. In each step, the quadratic objective function is relaxed by an under- estimating linear function, which contains the original linear costs, the quadratic costs of all edges fixed in the tree, and the cheapest quadratic costs for a suitable set of edges which are not fixed so far. For this, let X1 be the set of all variables fixed to one and let X0 be the set of

all variables fixed to zero in the current branching step. For an edge e ∈ E\{X1∪ X0} let F be

set of edges whose adding to X1∪ {e} does not close a cycle. Within F choose k edges f having

minimal quadratic costs qef with respect to e and define the resulting set with Fe,Xk 1,X0. Then,

the relaxed objective is defined as

minX

e∈E

˜ cexe

4.1. PROPERTIES AND ALGORITHMS 43 with the approximate costs

˜ ce := ce+ X f ∈X1 qef+ X f ∈F_e,X1,X0n−2−|X1| qef

yielding a lower bound for the current subproblem. This bound can be strenghtened by con- straining the k edges from F not to close a cycle with each other, too. The bound becomes even stronger by additionally applying the idea of Assad and Xu for lower bounds: the original objec- tive function does not change by a cost replacement ce(u) := ce−(n−2)ueand qef(u) := qef+uf

for a vector u ∈Rn_{. However the presented lower bound is affected and can be strengthened by}

a good choice of u. A leveling procedure proposed in [149] yields the best u and the strongest bound.

Apparently it is necessary for all kinds of practical applications to solve problems of much higher dimensions than n = 15 or n = 20. For this purpose, Assad and Xu already proposed two heuristics for the QMST [9]. The first of them, called the Average Contribution Method, estimates the average contribution

pe= ce+ n − 2 |E| − 1 X f 6=e qef

of each edge e ∈ E to the objective function and then solves the linear MST problem with respect to the costs pe. The second heuristic is called Sequential Fixing Method and improves

the calculation of the average contribution by also considering the influence of already fixed edges such that the average contribution is calculated as

pe = ce+ n − 2 − |U | |F | − 1 X f ∈F qef,

where U ⊆ E is the set of fixed edges and F ⊆ E\U the set of free edges, which are those edges which are neither fixed nor closing a cycle with the edges in U . The running times of both heuristics are quadratic in the number of edges and by this they outperform all subsequent heuristics in terms of running times.

In terms of accuracy in turn, the more recent heuristics have much better performances. Com- mon heuristics are the Random Local Search with Tabu Thresholding (RLS-TT) algorithm of

¨

Oncan and Punnen [138], and the Tabu Search (TS) and the Variable Neighborhood Search (VNS) algorithms of Cordone and Passeri [49], which all search on similar neighborhoods. The considered k-neighborhood structures consist of spanning trees with exactly k different edges compared to the current solution. For reasons of running times, k needs to be very small which makes the algorithm vulnerable for getting stuck in a local optimum. To avoid this, RLS-TT includes random moves within the k-neighborhood, VNS applies a shaking procedure and TS allows slight worsenings of solutions. Out of the three, TS seems to outperform the others both in terms of running time and accuracy [49]. Heuristics based on artificial intelligence such as the Artificial Bee Colony (ABC) heuristic of Sundar and Singh [164] and the Edge-Window-Decoder strategy of Soak et al. [161] lead to competitive results with RLS-TT [138] and TS [49]. More heuristics for the QMST problem are presented, e. g., in [144, 174] and [130].

44 CHAPTER 4. QUADRATIC SPANNING FORESTS AND TREES