Kovalyov, M. Y.

Ryzhikov, A.

BSU Digital Library

INTERVAL SELECTION PROBLEMS WITH LIMITED OVERLAP
Andrew Ryzhikov, Mikhail Y. Kovalyov
United Institute of Informatics Problems, National Academy of Sciences, Belarus
6 Surganov str., 220072, Minsk, Belarus
kovalyov_my@yahoo.co.uk, ryzhikov.andrew@gmail.com
1. Introduction. Let I be a ﬁnite set of intervals on the real line. We assume that the
endpoints of all intervals have rational coordinates. Every element I 2 I has a non-negative
rational weight w(I). A set of intervals is called independent if no two intervals in this set have a
common interior point, and k-independent if it is a union of k pairwise disjoint independent sets.
Here k is a non-negative integer number. In the problem Weighted k-Independent Set of
Intervals the objective is to ﬁnd a k-independent subset of I with maximum total weight. This
problem is widely studied and has a lot of applications in interval scheduling, resource allocation,
etc. For more details see surveys by Kovalyov et al. [1] and Kolen et al. [2]. It can be solved in
polynomial time (Bouzina and Emmons [3]). We consider a generalization of this problem where
the selected set of intervals must not be k-independent, but some overlap measure (which we call
composite u-redundancy) of this set must be limited by a given number R. We deﬁne the composite
u-redundancy in Section 2.
A special case of theWeighted 1-Independent Set of Intervals problem is the problem
where the weight of every interval is equal to its length. For arbitrary positive k we generalize
this problem in the following way.There is a set I of intervals on the real line and the objective
is to ﬁnd a k-independent subset of I with maximum measure of the union. We call this problem
Maximum Coverage by k-independent Set of Intervals. We study this problem and
also its generalization where the selected set must not be k-independent, but the composite u-
redundancy of this set must be limited by a given number R.
2. Main deﬁnitions. Let u be a non-negative integer number and J be a subset of I. We
deﬁne the set of projective u-redundancy of J to be the set of such points on the real line that
belong to at least u + 1 intervals from J. The measure of this set, that is, the total length of
intervals in it, is called the projective u-redundancy of J and is denoted by P (J; u).
Further, let x1; : : : ; xm be all the distinct left and right endpoints of intervals from J sorted
in the increasing order. Let sj be the number of intervals in J containing the interval [xj ; xj+1],
1  j  m  1. We deﬁne the total u-redundancy of the set J as
T (J; u) =
m 1X
j=1
maxf(xj+1   xj)  (sj   u); 0g:
Thus, in the projective u-redundancy, only one excessive interval of the intersection contributes
to the redundancy value, and in the total u-redundancy all the excessive intervals of the intersection
contribute to the redundancy value.
Let p and t be non-negative rational numbers such that p + t > 0. We deﬁne the composite
u-redundancy of a set J to be the value p  P (J; u) + t  T (J; u).
Both projective and total u-redundancy can be viewed as the measures that indicate the extent
to which the set of intervals is not u-independent. In particular, the following lemma is true.
Lemma 1. A ﬁnite set of intervals is k-independent if and only if its projective (or total)
k-redundancy is equal to zero.
3. Maximum weight selection problem. In the problem MaxWeight we are given
three integer numbers u, p, and t, an upper bound R on the composite u-redundancy and a
ground set I = fI1; : : : ; Ing of intervals. Each interval is associated with a non-negative rational
weight. The objective is to select a subset J  I of the maximum total weight, provided that its
composite u-redundancy does not exceed R. For R = 0 this problem is precisely the Weighted
u-Independent Set of Intervals problem.
The complexity of the MaxWeight problem is characterized by the following theorem.
Theorem 1. The MaxWeight problem is NP-hard (in the ordinary sense) for any ﬁxed u,
p, t even if the weight of every interval is equal to its length and all endpoints of the intervals have
integer coordinates.
The next two theorems show that for any ﬁxed u two restricted cases of the considered problem
can be solved by pseudo-polynomial algorithms.
Theorem 2. Let W be the total weight of all intervals in I. There exists a pseudo-polynomial
dynamic programming algorithm with running time O(u2Wnu+2) for the case where the weights
of all intevals are integer numbers.
Theorem 3. There exists a pseudo-polynomial dynamic programming algorithm with running
time O(u2(R+ 1)nu+2) for the case where the endpoints of all intervals have integer coordinates.
It is an open question whether the general MaxWeight problem is strongly NP-hard or
pseudo-polynomially solvable.
4. Maximum coverage selection problem. The problem MaxCoverage diﬀers from the
MaxWeight problem in that the criterion is to maximize the measure of the union of the selected
intervals, that is, the total length of the intervals of this union. For R = 0 this problem is precisely
the Maximum Coverage by u-independent Set of Intervals problem. The complexity of
the MaxCoverage problem is characterized by the following theorem.
Theorem 4. The MaxCoverage problem is NP-hard (in the ordinary sense) for arbitrary
ﬁxed non-negative rational numbers p and t, and for both u = 0 and u = 1.
One special case of this problem can be solved in pseudo-polynomial time.
Theorem 5. Let L be the union measure of all intervals in I. There exists a pseudo-polynomial
dynamic programming algorithm with running time O(Lnu+2) for the case where u 2 f0; 1g and
the endpoints of all intervals have integer coordinates.
Theorem 6. There exists a 12 -approximation algorithm with running time O(n log n) for the
MaxCoverage problem with u = 1.
We also prove that for the developed algorithm the number 12 in this bound cannot be replaced
with a larger constant.
According to the following theorem, the case u  2 is much simpler.
Theorem 7. There exists an algorithm with running time O(n log n) that ﬁnds a 2-independent
subset J of intervals such that the union of the intervals in J coincides with the union of the
intervals in I.
This implies that the problemsMaxCoverage andMaximum Coverage by u-independent
Set of Intervals are solvable in O(n log n) time for u  2.
The work is partially supported by the Belarusian Republican Foundation for Fundamental
Research under the grant number Ф15СО-043.
References
1. Kovalyov M.Y., Ng C.T., Cheng T.C.E. Fixed interval scheduling: Models, applications, computational
complexity and algorithms // European Journal of Operational Research. 2007. V. 178. No. 2. P. 331–342.
2. Kolen A.W.J., Lenstra J.K., Papadimitriou C.H., Spieksma F.C.R. Interval scheduling: A survey //
Naval Research Logistics. 2007. V. 54. No. 5. P. 530–543.
3. Bouzina K.I., Emmons H. Interval Scheduling on identical machines // Journal of Global Optimization.
1996. V. 9. No. 3–4. P. 379–393.


Interval selection problems with limited overlap

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Interval selection problems with limited overlap

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