Complexity and Approximability of Hyperplane
Covering Problems
Michael Khachay
Krasovsky Institute of Mathematics and Mechanics, UB RAS,
16 S. Kovalevskoy st., Ekaterinburg, Russia
Ural Federal University, 19 Mira st., Ekaterinburg, Russia
Omsk State Technical University, 11 Mira ave, Omsk, Russia
Abstract. The well known N.Megiddo complexity result for Point Cover Prob-
lem on the plane is extended onto d-dimensional space (for any fixed d > 1).
It is proved that Min-dPC problem is L-reducible to Min-(d + 1)PC problem,
therefore for any xed d > 1 there is no PTAS for Min-dPC problem, unless
P = NP.
1 Introduction
Settings of geometric covering problem and related problems are usual in various oper-
ations research domains [1-3]: optimal facility location theory, cluster analysis, pattern
recognition, etc. Mathematically, family of such problems can be partition into two
classes.
The first one contains special cases and modifications of well-known abstract Set
Cover problem. The main general feature shared by these problems is the niteness of
the initial family of subsets, for which it is required to find a subfamily (or just prove its
existence) covering some target set and satisfying given optimality conditions. There
are many papers studying problems from this class (see survey at [4]). The classical
papers [5-7] seem to be the most important among them. First two papers contain in-
tractability proof of Set Cover problem and two main design patterns for constructing
approximationalgorithms for this problem. The last paper proves the optimality of these
patterns, unless P = NP.
The second class consists of problems without the mentioned above finiteness con-
straint. Usually, the initial family of subsets is given here implicitly in terms of some
geometric property characterizing its elements. For instance, for a givenset it is required
to find a minimal cardinality cover by straight lines, circles of a given radii, etc.
2 Point Cover (2PC) Problem
In the paper, a series of hyperplane covering problems for given finite sets in finite-
dimensional vector spaces of fixed dimension d > 1 is considered. The first element of
This research was partially supported by RFBR, grants no. 13-07-00181 and 13-01-210, and
Ural Branch of RAS, grants no. 12-P-1-1016 and 12-S-1017/1
Khachay M..
Complexity and Approximability of Hyperplane Covering Problems.
DOI: 10.5220/0004394601090113
In Proceedings of the 4th International Workshop on Image Mining. Theory and Applications (IMTA-4-2013), pages 109-113
ISBN: 978-989-8565-50-1
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
this family (for d = 2), also known as Point Covering on the plane (2PC) problem was
studied by N.Megiddo and A.Tamir [8] who proved its intractability in the strong sense.
We extend this result on to the case of appropriate fixed dimensionality d > 1
and prove that all these problems are Max-SNP-hard and consequently have no PTAS,
unless P = NP .
Problem 1. ‘Point covering by lines on the plane’ (2PC). A finite subset P = {p
1
, . . . ,
p
n
} Z
2
and natural number B are given. Is there exists a finite family C of straight
lines covering P such that |C| B?
Obviously, in the particular case when the set P is in the general position, i.e. each
triple of its points does not belong to the same straight line, the 2PC problem has a
trivial solution (’Yes’ whether B ⌈|P |/2 and ’No’ otherwise), which can be found
in a polynomial time. But in the general case this problem is intractable.
Theorem 1 ([8]). The 2PC problem is NP-complete in the strong sense.
Note that Theorem 1 applies that 2PC problem could not be solved by only polynomial
time algorithms, but also pseudo-polynomial time.
3 Hyperplanes Covering Problems
Let us consider the more general problem settings.
Problem 2. ‘Hyperplane covering in d-dimensional space’ (dPC). For a fixed d > 1, a
finite subset P = {p
1
, . . . , p
n
} Z
d
and natural number B are given. Is there exists a
cover C of P by hyperplanes such that |C| B?
Problem 3. ‘Minimal hyperplane covering in d-dimensional space (Min-dPC). Let a
finite subset P = {p
1
, . . . , p
n
} Z
d
be given. It is required to find a minimum cardi-
nality partition J
1
, . . . , J
L
of a set N
n
= {1, . . . , n} such that for each i N
L
there is
a hyperplane H
i
and
{p
j
P : j J
i
} H
i
.
We extend the result of Theorem 1 onto the case of d-dimensional space for any
fixed d > 1. We start with construction of polynomial-time reduction of (d 1)PC to
dPC problem. Let an instance of (d 1)PC be given by subset P = {p
1
, . . . , p
n
}
N
d1
M
and B N. We use a natural isomorphic embedding of (d 1)-dimensional into
d-dimensional vector space:
x R
d1
7→ [x, 0] R
d
.
Map any point p
i
P into couple of points in Z
d
by the formula
¯p
2i1
= [p
i
, w
i
], ¯p
2i
= [p
i
, w
i
],
where
w
i
= (K + 2)
i1
and K =
l
(d 1)
d1
2
(M 1)
d1
m
.
110
Such a way, we construct the subset
¯
P Z
d
and the setting (
¯
P , B) of the dPC problem.
It is evident, that any hyperplane cover of P induces the equivalent cover (with the
same number of hyperplanes) of
¯
P in R
d
. The converse statement should be proved.
Denote by π
0
the hyperplane {[x, 0] : x R
d1
}. Let Pr
π
0
Q be an orthogonal
projection of the subset Q R
d
onto π
0
.
Lemma 1. Let subsets Q P and
¯
Q
¯
P be related by Q = Pr
π
0
¯
Q and the following
inequalities be valid
|
¯
Q| d + 1,
dim aff
¯
Q d 1.
Then dim affQ d 2.
Lemma 2. Let
¯
Π = {¯π
1
, . . . , ¯π
t
} be a hyperplane cover of subset
¯
P . The subset P
also has a hyperplane cover Π such that |Π| t.
Lemma 3. The described above reduction (d1)PC to dPC can be done in polynomial
time of Length((d 1)P C).
On the basis of these lemmas we can prove the following
Theorem 2. For an arbitrary fixed d > 1, the dPC problem is NP-complete (and the
Min-dPC problem is NP-hard) in the strong sense.
Now we show that the supposed above (d 1)PC to dPC reduction can be reformu-
lated as L-reduction [9] from Min-(d 1)PC to Min-dPC problem.
Definition 1. Let sets I and S, set-valued map F : I 2
S
and some target function
c :
S
II
F (I) R
+
be given. The quadruple A = (I, S, F, c), where each I I is
mapped to optimization problem
min{c(s) : s F (I)},
is called a combinatorial minimization problem.
W.o.l.g., any I I is called an instance of the problem A and its optimum value is
denoted by OP T (I).
Definition 2. Consider problems A and B of combinatorial minimization. It is called,
that there is an L-reduction from A into B, if there are two LSPACE-computable func-
tions R and S and positive constants α and β such that the following conditions are
valid:
1. for each instance I of the problem A, R(I) is an instance of B and
OP T (R(I)) αOP T (I);
111
2. for each feasible solution z of R(I), S(z) is a feasible solution of I such that
c
A
(S(z)) OP T (I) β(c
B
(z) OP T (R(I))),
where c
A
, c
B
are target functions of A and B correspondingly.
Now we are ready to formulate a recurrent L-reduction of problems in question.
Theorem 3. For each fixed d > 2, there is an L-reduction of Min-(d 1)PC to Min-
dPC problem.
Taking into account the following known result
Theorem 4 ([11]). Min-2PC problem is Max-SNP-hard.
one can formulate the last
Theorem 5. For each fixed d > 1, the Min-dPC problem is Max-SNP-hard.
Consequently, Min-dPC problem has no polynomial-time approximation schema
(PTAS) for each fixed d > 1, unless P = NP .
4 Conclusions
We show that Hyperplane covering problem remains intractable and poorly approximat-
able even in fixed dimension spaces (for any d > 1). This result extends the well known
Point Cover intractability result obtained by N. Megiddo and A. Tamir. Obviously, Min-
dPC problem can be trivially approximated in polynomial time within O(n/d) approxi-
mation guarantee. But the question on the existence of polynomial time algorithms with
lower (e.g. fixed) approximation guarantee is still open.
References
1. Agarwal P.K. and Procopiuc C.M. Exact and approximation algorithms for clustering, Al-
gorithmica, No. 33, 201–206. (2002).
2. Langerman S. and P. Morin. Covering things with things, Discrete Computat. Geom., 717–
729, (2005).
3. Khachai M. Computational complexity of recognition learning procedures in the class of
piecewise-linear committee decision rules, Automation and Remote Control, 71, No. 3, 528–
539, (2010).
4. Vazirany V. Approximation algorithms, Springer (2001).
5. Johnson D. Approximation algorithms for combinatorial problems, Journal of Computer and
System Sciences, 9, No. 3, 256–278, (1974).
6. Lov´asz L. On the ratio of integer and fractional covers, Discrete Mathematics. No. 13, 383–
390, (1975).
7. Feige U. A Threshold of ln n for Approximating Set Cover, Journal of the ACM, 45, No. 4,
634–652, (1998).
8. Megiddo N. and Tamir A. On the complexity of locating linear facilities in the plane, Oper-
ations research letters. 1, No. 5, 194–197, (1982).
112
9. Papadimitriou C. and Yannakakis M. Optimization, approximation, and complexity classes,
J. Comput. System Sci., 43, No. 3, 425–440, (1991).
10. Papadimitriou C. Computational Complexity, Addison-Wesley, (1995).
11. Khachai M. and Poberii M.Computational complexity of combinatorial problems related to
piecewise linear committee pattern recognition learning procedures, Pattern Recognition and
Image Analysis. 22, No. 2, 278–290, (2012).
113