Approximation Hardness of Optimization Problems in Intersection Graphs of d-dimensional Boxes Miroslav Chleb´ık¤ Janka Chleb´ıkov´a y Abstract Maximum Independent Set, in d-box intersection The Maximum Independent Set problem in d-box graphs, graphs for any fixed d ¸ 3. i.e., in the intersection graphs of axis-parallel rectangles in Rd, is a challenge open problem. For any fixed d ¸ 2 Overview. Many optimization problems like Max- the problem is NP-hard and no approximation algorithm imum Clique, Maximum Independent Set, and with ratio o(logd¡1 n) is known. In some restricted cases, Minimum (Vertex) Coloring are NP-hard for gen- e.g., for d-boxes with bounded aspect ratio, a PTAS exists eral graphs but solvable in polynomial time for interval [17]. In this paper we prove APX-hardness (and hence non- graphs [20]. Many of them are known to be NP-hard existence of a PTAS, unless P = NP), of the Maximum already in 2-dimensional models of geometric intersec- Independent Set problem in d-box graphs for any fixed tion graphs (e.g., in unit disk graphs). In most cases the 443 geometric restrictions allow us to obtain better approxi- d ¸ 3. We state also first explicit lower bound 442 on efficient approximability in such case. Additionally, we mation algorithms (or even in polynomial time solvabil- provide a generic method how to prove APX-hardness for ity) for problems that are in general graphs extremely many NP-hard graph optimization problems in d-box graphs hard to approximate. On the other hand, these geomet- for any fixed d ¸ 3. In 2-dimensional case we give a generic ric restrictions make the task to achieve some hardness approach to NP-hardness results for these problems in highly results more difficult. restricted intersection graphs of axis-parallel unit squares Among basic NP-hard graph optimization problems (alternatively, in unit disk graphs). in d-box graphs (d ¸ 2), only Maximum Clique is known to be exactly solvable in polynomial time ([5], 1 Introduction [29], [35]). Some optimization problems are known to be NP-hard in d-box intersection graphs for any fixed The intersection graph of a family of sets S , v 2 V , is v d ¸ 2, for example Maximum Independent Set [18], a graph with vertex set V such that u is adjacent to v [25], or Minimum Coloring [30]. if and only if S \ S 6= ;. The family fS ; v 2 V g is an u v v The challenging open problem, the Maximum In- intersection representation of this graph. Geometrical dependent Set problem (shortly, Max-IS), in d-box models of intersection graphs deal with families of intersection graphs (d ¸ 2) can be formulated as follows: subsets of Rd with some geometric properties. In this for a given set R of n axis-parallel d-dimensional boxes, paper we are mainly interested in families of axis- find a maximum cardinality subset R¤ ⊆ R of pairwise parallel d-dimensional boxes. Their intersection graphs disjoint boxes. The problem has attracted an attention are called d-box intersection graphs, or simply d-box of many researchers ([1], [11], [12], [17], [23], [26], [33]) graphs. Recall that a d-dimensional box (d-box) is a due to its applications in map labeling, data mining, subset of Rd that is a Cartesian product of d intervals VLSI design, image processing, and point location in in R. For convenience, terms an interval and a rectangle d-dimensional Euclidean space. As the problem is NP- are used for 1-box and 2-box, respectively. hard for any fixed d ¸ 2 ([18], [25]), its approximability In this paper we describe a generic approach to is intensively studied. approximation hardness results for many graph opti- Let us describe briefly known approximability re- mization problems as, e.g., Minimum Vertex Cover, sults for it, see [12] for more detailed overview. The earliest result was the shifting grid method based PTAS by Hochbaum and Maass [23] for the case of unit d- ¤Max Planck Institute for Mathematics in the Sciences, Insel- cubes. This method works for any collection of ‘fat’ straße 22-26, D-04103 Leipzig, Germany, [email protected] objects in Rd of roughly the same size. Their scheme y d Faculty of Mathematics, Physics and Informatics, Come- required nO(k ) time to guarantee an approximation fac- nius University, Mlynsk´a dolina, 842 48 Bratislava, Slovakia, 1 [email protected] tor of (1+ k ). By applying dynamic programming along one of the dimensions, running time has been reduced graphs of maximum degree 3 (Section 3). d¡1 to nO(k ) and generalized to objects, not necessarily ² Each graph obtained from another one by at least fat, but whose projections to the last (d¡1) coordinates 3-subdivision of each edge is an intersection graph are fat and of roughly the same size. This was essen- of d-boxes for any fixed d ¸ 3. Moreover, its tially established by Agarwal et al. [1] in their work on representation can be provided in time polynomial 1 in its size. (Section 2). unit-height rectangles in the plane. To achieve (1 + k )- approximation they need time O(n2k¡1 + n log n). Both these results are interesting for their general- Generalizing in another direction, Erlebach et al. ity and can be of independent interest. The same ap- [17] obtained a PTAS for fat objects of possibly varying plies to a newly introduced notion of a shift-reduction sizes, such as arbitrary d-cubes or bounded aspect ratio (Section 3) used in the proof of APX-hardness for many 4 d-boxes. The running time of their algorithm (nO(k ) graph optimization problems in d-boxes (d ¸ 3), e.g., d¡1 for d = 2) has been improved to nO(k ) by Chan [12]. Minimum Vertex Cover, Minimum (Independent) For arbitrary d-boxes, even for d = 2, the existence Dominating Set, and Maximum Induced Match- of a PTAS or a constant factor approximation algo- ing. The methods also allow to provide explicit lower rithm, is largely open problem. As has been observed bounds on efficient approximability. This is demon- in several papers ([1], [26]), a logarithmic approxima- strated on the Maximum Independent Set prob- tion factor is possible in this case. For example, Agar- lem in d-box graphs (d ¸ 3), for which we prove NP- 1 wal et al. [1] described O(n log n)-time algorithm with hardness to achieve approximation factor of 1 + 442 . d¡1 Usually better approximation algorithms for optimiza- factor at most dlog2 ne. This generalizes to dlog2 ne - d¡1 tion problems in d-box graphs need a representation of approximation algorithms with O(n log2 n) running time, for general d. Nielsen [33] independently de- an input graph by d-boxes, not merely a graph. Our scribed algorithm with optimum-sensitive approxima- hardness results apply to this setting as well. Moreover, tion factor (1 + log (is(R)))d¡1, where is(R) is the they apply to the highly restricted case, in which no 2 d maximum number of independent boxes of R. Cur- point of R is simultaneously covered by more than two rently, no polynomial time algorithm is known with d-boxes, and each d-box intersects at most 3 others. o(logd¡1 n)-approximation factor, although Berman et Our methods do not apply to the planar case in d¡1 which there is still hope for a PTAS. All problems al. [11] have observed that a log2 n bound can be re- duced by arbitrary multiple constant. More precisely, studied in the paper are NP-hard in 2-dimensions as d¡1 O(bd¡1) well, even in very restricted setting of intersection a factor dlogb ne can be achieved in n time, for any fixed integer b ¸ 2. As it was already men- graphs of axis-parallel unit squares (or unit disks). tioned in [11], it remains open to understand the limits We provide a generic approach to NP-hardness results on the approximability of the Maximum Independent for many graph optimization problems in intersection Set problem in intersection graphs of d-box graphs for graphs of axis-parallel unit squares (alternatively, in d ¸ 2. unit disk graphs). We also contribute to another independent set Our results. In this paper we present the proof of problem connected with sets of d-boxes, introduced APX-hardness (with some explicit lower bounds) for the by Bafna et al. [4]. The problem is the Maximum Maximum Independent Set problem in axis-parallel Weighted Independent Set problem (Max-w-IS) d-dimensional boxes for any fixed d ¸ 3. It follows, in in proper d-union graphs, i.e., in graphs that can be particular, that the existence of a PTAS for the problem expressed as the union of d proper interval graphs. restricted to d-boxes with bounded aspect ratio [17] For d = 2 the problem is quite well understood. For cannot be generalized (unless P = NP) to arbitrary d- d ¸ 3 we improve significantly the upper bounds boxes for any fixed d ¸ 3. on approximability of this problem (see [4], [9], [10], We provide also a generic method how to prove [6]). Namely, the Max-w-IS problem in proper d-union 1 approximation hardness results in d-box graphs that graphs, d ¸ 2, can be approximated within (d + 2 ) in is applied simultaneously to several graph optimization polynomial time, and within (2d ¡ 1) in nearly linear problems such as covering, domination, and matching time. problems. The idea of our APX-hardness results is based on the proof of the following two facts: Definitions and notations.