Download Approximating the Maximum Quadratic Assignment Problem1 1

Approximating the Maximum Quadratic Assignment Problem
1
Esther M. Arkin2 Refael Hassin3 Maxim Sviridenko4
Keywords: Approximation algorithm; quadratic assignment problem
1 Introduction
In the maximum quadratic assignment problem three n n nonnegative symmetric matrices
A = (aij), B = (bij ), and
C = (cij ) are given and the objective is to compute a permutation of
V = f1; : : :; ng so that P = a(i);(j) bi;j + Pi2V ci;(i) is maximized.
The problem is NP-hard and generalizes many NP-hard problems such as max clustering
with given sizes (see denition below). An indication to the hardness of approximating the
problem is that the best known approximation factors for two special cases of max
clustering
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1
?
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with given sizes are c , when all sizes are equal to a constant c [7], and (n) when all sizes
except for one are 1 [5].
In this note we provide an approximation algorithm with a constant performance guarantee, 41 ,
under the assumption that the weights in B satisfy the triangle inequality (TI), bi;j bi;k + bk;j , for
all i; j; k 2 V . For maximum linear arrangement (see denition below) the bound guaranteed
by our algorithm is 12 , which slightly improves a result of [15].
i;j 2V
i6 j
2 Special cases and related problems
There are many interesting special cases, some of which are `graphic', that is, A is a 0/1 incidence
matrix of a graph. In this case, the problem is to compute in B a subgraph isomorphic to A of
maximum total weight. In some of these applications, the role of C may be to represent the eect
of a partial solution which has already been determined, so that the weights of C represent the total
weight incurred by i if it is permuted to (i) through interactions with the imposed partial solution
(see, for example, [23]). We will describe below the meaning of A and B in these applications.
In max weight perfect matching A consists of n2 disjoint edges. This problem can be solved
in polynomial time.
In max clustering with given sizes A is the union of vertex disjoint cliques. Assuming TI
[7] and [13] provide algorithms with bound 21 if all clusters have the same size and [13] gives a 2p1 2
bound if the sizes dier. In a special case, max dispersion, A consists of a clique of p vertices
and n ? p isolated vertices. 14 and 12 approximation algorithms were developed in [20] and [16],
respectively.
A shortened preliminary version of this paper appeared in Proc. Eleventh Annual ACM-SIAM Symposium on
Discrete Algorithms (SODA '00), 889-890, January, 2000.
2
Department of Applied Mathematics and Statistics, SUNY Stony Brook, Stony Brook, NY 11794-3600,
[email protected]. Partially supported by NSF (CCR9732220).
3 Department of Statistics and Operations Research,
Tel-Aviv University, Tel-Aviv 69978, Israel,
[email protected]
4
Basic Research Institute in Computer Science (BRICS), University of Aarhus, Denmark, [email protected]. Research
of the third author was partially supported by the Russian Foundation for Basic Research, grants 97-01-00890, 9901-00601, 99-01-00510
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In max cut with given sizes A consists of a complete bipartite graph. [1] and [8] give 21 and
0.651 approximations for the general and for the equal parts case (graph bisection), respectively,
without TI. Both results are slightly improved in [6].
Constant factor approximations, without TI, when A consists of a collection of vertex disjoint
paths are described in [12]. In particular, the bound for packing of 3-edge paths is 43 .
In max TSP A consists of a cycle. With TI, a 65 bound is given in [17]. Without TI, a
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deterministic algorithm with bound 43 and a randomized algorithm with bound 33
are given in [21]
and [14], respectively.
In max star packing A consists of a collection of vertex disjoint stars. A 2p1 2 -approximation
assuming TI is given in [13].
In maximum latency TSP A consists of a Hamiltonian path with weights 1; 2; :::; n ? 1 in this
order. With TI, a 12 factor is given in [4], while [14] obtains a bound of 21 time that of max TSP,
without TI.
In maximum linear arrangement A is a complete graph with nonnegative weights and
bi;j = jj ? ij. [15] contains a randomized asymptotic 12 -approximation algorithm for this problem.
It also contains a 31 -approximation for generalized maximum linear arrangement in which
a vector x1; :::; xn is given and bi;j = jxj ? xi j.
In contrast, [22] show that no constant factor approximation exists for minimum quadratic
assignment unless P=NP. In fact, Queyranne showed that approximating minimum quadratic
assignment with the triangle inequality within any constant factor in polynomial time implies
P=NP even for the problem with matrix B introducing a line metric [19]. Special graphic cases in
which A consists of p vertex disjoint paths, (cycles, cliques) have constant factor approximations
under restrictions: For p xed see [10, 11], and for equal-sized sets see [9]. However, these special
cases with general p and unequal sized sets are open.
Notice also that the maximum quadratic assignment problem with triangle inequality
is MAXSNP-hard since it contains the maximum f1; 2g-TSP as a special case. The MAXSNPhardness of the later problem can be simply derived from the work [18].
3 Algorithm
Denote by apx the expected weight returned by Quadratic Assignment of Figure 1. Let opt
denote the weight of an optimal solution.
Theorem 1 apx opt.
Proof: Since MB is a maximum matching with respect to the weights b, it follows that for every
p; q 2 V such that (p; q) 2= MB but (p; p0) 2 MB and (q; q 0) 2 MB
bp;q bp;p + bq;q :
(1)
Also, if v is not incident to any edges in MB then for i = 1; :::; l bv;w ; bv;z bw ;z . Let be an
optimal permutation. Summing over all i; j 2 V i =
6 j we get
1
4
0
0
i
opt =
Xa
i;j 2V
i6=j
i ; j bi;j +
( )
( )
Xc
i2V
i;(i)
Xl b
i=1
w ;z
i
i
(degA(w ) + degA(z ) ) +
i
i
Xc
i2V
i
i(i) =
i
Xf
i2V
i
i;(i) Xf
i2V
i;^(i) :
We now consider the approximate solution returned by Quadratic Assignment. The algorithm sets ( (wi); (zi)) to (^ (wi); ^ (zi )) or to (^(zi ); ^ (wi)) each with probability 12 . Therefore,
2
Quadratic Assignment
input
1. n n nonnegative matrices A = (aij ) and C = (cij ).
2. n n nonnegative matrix B = (bij ), satisfying the triangle inequality.
P
returns
P
1. Permutation of V = f1; :::; ng of weight i;j 2V a(i);(j )bi;j + i2V ci;(i):
begin
Compute a maximum weight matching MB in V with respect to weights fbij g.
Assume that MB = f(w1; z1); :::; (wl; zl )g where l = b n2 c
for i 2 V
if i 2 fwk ; zk g for some k 2 f1; :::; lg
then b(i) := bw ;z .
else b(i) := 0 (there is at most one such vertex).
end if
end for
for i 2 V
degiA := j2V nfig aij .
end for
for i; j 2 V
k
k
P
fij := b(i)degjA + cij .
^ := an optimal solution to a linear assignment problem max Pi fi; i .
for i = 1; :::; l
(wi) := ^ (wi) and (zi) := ^ (zi) with probability .
(wi) := ^ (zi) and (zi) := ^(wi), otherwise.
end for
( )
1
2
end for
if v is a vertex which is free with respect to
return .
end Quadratic Assignment
MB , then (v) := ^(v).
Figure 1: Algorithm Quadratic Assignment
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a p ; q for (p; q) 2= MB is multiplied in a random approximate solution by one of the numbers
bpq ; bpq ; bp q ; bp q , each with probability where (p; p0) 2 MB and (q; q 0) 2 MB . Hence, the expected
contribution of pair ^ (p); ^ (q ) is equal to
^( ) ^ ( )
0
0
1
4
0 0
1a
1a
1a
^ (p);
^ (q ) (bpq + bpq + bp q + bp q ) ^ (p);
^ (q ) 2 maxfbpp ; bqq g 4
4
4 ^(p);^(q) (bpp + bqq )
where the rst inequality follows from the triangle inequality. If (p; q ) 2 MB then a^(p);^(q) is always
multiplied by bpq . Summing over all p 6= wi and q 6= zi we get that the contribution of the edge
(wi; zi) to apx is at least 41 (degA^(w ) + degA^(z ) )bw ;z . Notice also that the contribution of ci(i) in
apx is at least 21 ci^(i) since i doesn't change its assignment with probability 21 . We obtain
X
X
X
apx 14 b(i)degA^(i) + 21 ci;^(i) 14 fi;^(i)
i2V
i2V
i2V
0
0
0 0
i
0
i
i
0
0
0
i
so that the claim of the theorem follows.
The running time of the algorithm is O(n3) due to the matching and assignment steps it uses.
We note that the maximum weight matching can be replaced by a `greedy' matching (include
vertex-disjoint edges in non-increasing order of weights). The only property required from the
matching is bp;q bp;p + bq;q which the greedy matching satises as well. A greedy matching can
be computed in O(n2 log n) time.
If C = 0 the assignment problem in the algorithm has a special `factored structure' and can
be solved as follows: Sort the b(i)'s and the degjA's in non-increasing orders i1; :::; in and j1; :::; jn,
respectively, and assign ir to jr for r = 1; :::; n (see for example Exercise 12.22a in [2]). Combined
with the previous observation, the running time of the algorithm in the C = 0 case is O(n2 log n).
The algorithm can be derandomized by the method of conditional probabilities [3].
Finally, we prove the following:
0
0
Theorem 2 For maximum linear arrangement, Quadratic Assignment is a -approximation
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2
algorithm.
Proof: Recall that in maximum linear arrangement bi;j = jj ? ij. The greedy matching (which
is one of numerous maximum weight matchings) contains the edges (1; n); (2; n ? 1); (3; n ? 2); ::: .
For any edge (p; q ) not in the greedy matching, Equation (1) can be replaced by
bp;q 12 (bp;p + bq;q ):
Consequently, the bound of Theorem 1 improves in this case to 21 .
0
0
Acknowledgement
We thank Alexander Ageev for helpful comments.
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