Approximation Hardness of Optimization Problems in Intersection Graphs of D-Dimensional Boxes

Approximation Hardness of Optimization Problems in Intersection Graphs of d-dimensional Boxes

Miroslav Chleb´ık∗ Janka Chleb´ıkov´a †

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 ∈ 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 {S , v ∈ V } 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 † d Faculty of Mathematics, Physics and Informatics, Come- required nO(k ) time to guarantee an approximation fac- nius University, Mlynská 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 • 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 reduced 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. All graphs in this paper • Many optimization problems are APX-hard even are simple. For a graph G = (V,E) and a nonnegative when restricted to suitable subdivisions of graphs, integer k, a k-subdivision of an edge e ∈ E is the e.g., for a fixed k ≥ 0, in graphs obtained by operation replacing the edge e = {u, v} by a path 2k (respectively, 3k) subdivision of each edge from with endvertices u, v, and k new internal vertices. A k-subdivision of G, denoted by divk(G), is a graph (Min-Maxl-Match) asks to find a maximal match- obtained from G by k-subdividing of each edge e of ing of minimum cardinality in G. A maximal match- G. For an integer B ≥ 3, let GB denote the set of ing corresponds an independent edge dominating set, graphs of maximum degree at most B (without isolated hence Min-Maxl-Match corresponds to Minimum In- vertices). Notice that a graph G = (V,E) from GB dependent Edge Dominating Set. The objective |V | of the Maximum Induced Matching problem (Max- has |E| ≤ 2 B edges, its k-subdivision divk(G) has |E|k + |V | ≤ |V | (Bk + 2) vertices, and |E|(k + 1) ≤ Induced-Match), resp. Maximum Total Matching 2 (Max-Total-Match), is to find a maximum induced |V | B(k + 1) edges. Moreover, equalities hold for B- 2 matching in G, resp. a maximum total matching in G. regular graphs. Let im(G) denote the cardinality of a maximum induced A graph G = (V,E) is a d-box graph if it is matching in G. an intersection graph of a family of axis-parallel d- Given a spanning forest F for G, an edge {u, v} of dimensional boxes (so called d-boxes). It means, a d- F is a pendant edge for F if the degree of u or v in box R can be assigned for each vertex v ∈ V such that v F is 1. The goal of the Maximum Spanning Forest {u, v} ∈ E if and only if the boxes R and R intersect. u v problem (Max-SF) is to find a spanning forest of G In a graph G = (V,E) a vertex v ∈ V is said to cover with the maximum number of pendant edges among all itself, all edges incident with v, and all vertices adjacent spanning forests in G. to v. An edge {u, v} is said to cover itself, vertices u and For the basic optimization terminology we refer the v, and all edges incident with u or v.A vertex cover is reader to Ausiello et al. [3]. For any NPO optimization a subset C of V that covers E, an edge cover is a subset problem P , I is the set of instances of P , sol (x) is S of E that covers V , a dominating set is a subset D P P the set of feasible solutions of x ∈ I , and m (x, y) is of V that covers V , and an edge dominating set is a P P the value of feasible solution y, for every pair x ∈ I subset S of E that covers E. Two elements of V ∪E are P and y ∈ sol (x). Let P and Q be two NPO problems independent if neither covers the other. A subset I of V P and f be a polynomial time computable function that is a strong independent set in G, if for u, v ∈ I, u 6= v, maps instances of P to instances of Q. Then f implies dist (u, v) > 2. A strong independent set is also G is said to be an L-reduction from P to Q, if there called 2-packing, or 2-independent set by some authors. are constants α, β ∈ (0, ∞) and a polynomial time The goal of the Maximum Independent Set computable function g such that for every x ∈ I problem, resp. Maximum Strong Independent Set P (1) OPT (f(x)) ≤ αOPT (x), (2) for every y0 ∈ (Max-SIS), is to find an independent set, resp. a strong Q P sol (f(x)), g(x, y0) ∈ sol (x) so that |OPT (x) − independent set, of maximum cardinality in G; let is(G), Q P P m (x, g(x, y0))| ≤ β|OPT (f(x)) − m (f(x), y0)|. To resp. sis(G), denote its cardinality. The Minimum Ver- P Q Q show APX-completeness of a problem P ∈ APX it tex Cover problem (Min-VC), resp. Maximum Min- is enough to show that there is an L-reduction from imal Vertex Cover (Max-Minl-VC), is looking for some APX-complete problem to P . This enables us to a vertex cover of minimum cardinality, resp. a mini- prove APX-completeness by means of the easier-to-use mal vertex cover of maximum cardinality, in G. De- L-reducibility. note vc(G) the optimum value for Min-VC. Similarly, Maximum Minimal Edge Cover (Max-Minl-EC) 2 Intersection Graphs of Axis-Parallel Boxes is looking for a minimal edge cover of maximum cardinality in G. The problems Minimum Dominating Set Roberts [37] proved that every graph can be realized (Min-DS), Minimum Independent Dominating Set as an intersection graph of axis-parallel d-dimensional (Min-IDS), resp. Minimum Edge Dominating Set boxes for some d depending on the graph. For any (Min-EDS), ask for a dominating set, an independent fixed d ≥ 2 the recognition of d-box graphs is NP- dominating set, resp. an edge dominating set, of mini- hard ([28], [39]), and hence the reconstruction of their mum size in G. Let ds(G), ids(G), and eds(G) stand representation by d-boxes as well. In this section for the corresponding minima for Min-DS, Min-IDS, we prove that highly non-trivial subclasses of general and Min-EDS in G. graphs are d-box graphs for d = 3 (and hence for any A matching is a subset M of E whose elements d ≥ 3). are pairwise independent. A matching M is induced if for each edge e = {u, v} ∈ E, u, v ∈ V (M) im- Theorem 2.1. Let us given a graph G = (V,E) and for plies e ∈ M. A subset T of V ∪ E whose elements each edge e ∈ E an integer s(e) ≥ 3. Let G0 denote a are pairwise independent is said to be a total matching graph obtained from G by s(e)-subdivision of each edge for G. The Minimum Maximal Matching problem e, i.e., replacing e = {u, v} by a path with endvertices u, v, and s(e) new internal vertices (all paths are pairwise internally disjoint). For any fixed integer d ≥ 3, the ered by more than two boxes. In planar case, a 4K- graph G0 can be realized as the intersection graph of a approximation algorithm is known for Max-w-IS in set of axis-parallel d-dimensional boxes. Moreover, such such case, where K is the maximum number of rect- representationP can be done in time polynomial in size of angles that simultaneously cover a point in plane [31]. G and e s(e). 2.1 Intersection Graphs of Sets of Axis Parallel Sketch of the proof. We can assumeP that V = {1, Lines. For d ≥ 3, d-box intersection graphs are from 2, . . . , n} and denote N = |V | + e s(e). Any edge topological point of view as complex as general graphs. e = {i, j} (with i < j for the definiteness) is replaced Topological obstructions do not allow us to apply similar in G0 by a path i, A1, A2,..., As(e), j. We suppose e e e constructions in planar case. Also the complexity of 0 that d = 3 and describe the representation of G as intersection graphs of axis-parallel lines in dimensions an intersection graph of a set {R1,R2,...,RN } of axis- 2 and 3 significantly differs one from another. In R2 3 parallel boxes in R . For any d > 3 one can take simply these graphs are exactly the complete bipartite graphs, d−3 {Ri × [0, 1] : i = 1, 2,...,N} as the corresponding on which classical optimization problems are easily d 0 set of boxes in R , representing G . solvable. On the other hand, for intersection graphs First, we fix a sufficiently large integer C and put of sets of axis-parallel lines in R3 the situation is the I := [0,C]. (The value of C can be easily given same as in case of axis-parallel boxes. It can be proved explicitly, along with the other constants chosen later similarly as in Theorem 2.1 that any higher subdivisions in the proof, depending polynomially on N.) The box of general graphs (at least 5 subdivision of each edge) 3 I will contain all boxes which we will construct. can be realized as intersection graphs of sets of axis- Next, we choose integers 0 < a1 < b1 < a2 < parallel lines in R3. b2 < ··· < an < bn < C that are well separated one from another (say, nearly equidistributed). The box 3 APX-hardness of Graph Problems in Certain 3 Ri := [ai, bi] will be our representation of the vertex Subdivisions of Graphs i ∈ V , for each i ∈ {1, 2, . . . , n}. Let N(i) denote the Let P be a set of the problems Max-IS, Min-VC, Min- set of neighbors of a vertex i in G, i ∈ {1, 2, . . . , n}. The DS, Min-IDS, Min-EDS, Max-Induced-Match, and notation U(x) is used for the unit interval [x, x + 1]. Max-SIS. Our aim now is to show APX-completeness Now we describe boxes representing new vertices, for each problem from P when restricted to suitable i.e., vertices of G0 that are not vertices of G. For subdivisions of graphs. First, we show that certain each fixed vertex i ∈ V we choose in [a , b ) well i i subdivision operations are in fact L-reductions that separated integers a for each j ∈ N(i). Let us consider i,j self-reduce any problem from P when restricted to G now one fixed edge e = {i, j} ∈ E with i < j and B (except Min-IDS). 1 2 s(e) define boxes Re, Re,..., Re representing vertices 1 2 s(e) Lemma 3.1. Let a graph G = (V,E) and an edge e ∈ E Ae, Ae,..., Ae , respectively. Let us define first 0 1 s(e) be given. Let G be a graph obtained from G by 2- Re := I ×U(ai,j)×U(ai,j), Re := U(aj,i)×I ×U(aj,i), and consider a box S := U(a ) × U(a ) × I. subdivision of the edge e, i.e., replacing e = {u, v} by e j,i i,j 0 0 0 0 If s(e) = 3, one can take R2 := S to represent the a path u, u , v , v with new vertices u and v . Then e e 0 0 2 1 2 3 vc(G ) = vc(G) + 1 and is(G ) = is(G) + 1. vertex Ae of the path i, Ae, Ae, Ae, j. If s(e) ≥ 4, 2 3 s(e)−1 then the vertices Ae, Ae,..., Ae will be realized Applying iteratively the steps of the proof of 2 3 s(e)−1 by subboxes Re, Re,..., Re of Se of the form Lemma 3.1 we obtain the following l R := U(aj,i) × U(ai,j) × [cl, dl], l = 2, 3, . . . , s(e) − 1, e Theorem 3.1. Let us given a graph G = (V,E) and where c2 = 0, ds(e)−1 = C, ae + 1 < c3 < d2, cs(e)−1 < d < a and, if s(e) ≥ 5, c < d < c for each edge e ∈ E, let s(e) be a nonnegative integer. s(e)−2 j,i l+1 l l+2 0 whenever 2 ≤ l ≤ s(e) − 3. One can easily check Let G be a graph obtained from G by 2s(e)-subdivision of each edge e, i.e., replacing e = {u, v} by a path that the intersection graph of the set {R1,R2,...,Rn}∪ S with endvertices u, v, and 2s(e) new vertices (the paths {R1,R2,...,Rs(e)} of axis-parallel boxes in R3 e∈E e e e are pairwise internally disjoint). Let P be either the is (isomorphic to) G0. Moreover, the construction problem Min-VC, or Max-IS. Then described above has time complexity polynomial in size P P of G and s(e). ¤ 0 e (A) OPTP (G ) = OPTP (G) + e s(e); 0 Remark 2.1. The graph G from the Theorem 2.1 is of (B) every y ∈ solP (G) can be transformed in polynomial 0 P 0 0 girth at least 9. In any representation of G by axis- time (in size of G and e s(e)) to y ∈ solP (G ) d 0 P parallel d-boxes no point of R is simultaneously cov- such that |y | = |y| + e s(e); 0 0 (C) every y ∈ solP (G ) can be transformed in polyno- Similarly, we can prove that div2k, resp. div3k, when 0 P mial time to y ∈ solP (G) such that |y |− e s(e) ≤ restricted to GB, is an L-reduction that self-reduces 0 |Py| if P is Max problem; respectively |y| ≤ |y | − Min-VC, resp. Min-DS, Min-EDS, Max-Induced- e s(e) if P is Min problem. Match, and Max-SIS. Moreover, div3k for k > 0, when restricted to GB, reduces Min-DS to Min-IDS in the Similarly as is and vc, several other graph parame- same way as well. The second condition from the defini- ters behave well under suitable subdivision operations. tion of an L-reduction is satisfied with β = 1 by Theo- We will demonstrate that at least for ds, eds, im, and rems 3.1 and 3.2. The existence of α for an L-reduction sis. is again based on the lower bound OPTP (G) ≥ c|VG| (with a positive constant c depending on the problem Lemma 3.2. Let a graph G = (V,E) and an edge e ∈ E P and on B, but independent of G within the class be given. Let G0 be a graph obtained from G by 3- G ) works for each problem from P. To comment subdivision of the edge e, i.e., replacing e = {u, v} by B briefly on those lower bounds, let G = (V,E) ∈ G a path u, u0, w, v0, v with new vertices u0, w, and v0. B be fixed. Easy counting arguments show that is(G) ≥ Then (i) ds(G0) = ds(G) + 1, (ii) eds(G0) = eds(G) + 1, ids(G) ≥ ds(G) ≥ |V | , vc(G) ≥ |V | , eds(G) ≥ |V | , (iii) im(G0) = im(G) + 1, and (iv) sis(G0) = sis(G) + 1. B+1 B+1 2B |V | |V | sis(G) ≥ B2+1 , and im(G) ≥ 4B2−4B+2 . (Let us note Using the steps of the proof of previous lemma we that for some of these results our restriction to graphs obtain the following GB without isolated vertices is crucial.) One can use these lower bounds to prove that the Theorem 3.2. Let us given a graph G = (V,E) and for problems are in APX when restricted to G . For any of each edge e ∈ E, let s(e) be a nonnegative integer. Let B minimization problems above any feasible solution ap- G0 be a graph obtained from G by a 3s(e)-subdivision proximates within a constant. For maximization prob- of each edge e, i.e., replacing e = {u, v} by a path lems above, the lower bounds given apply to any inclu- with endvertices u, v, and 3s(e) new vertices (the sionwise maximal independent set (strong independent paths are pairwise internally disjoint). If P is one set, and induced matching, respectively), that provide of the problems Min-DS, Min-EDS, Max-Induced- a constant factor approximation in this case. Simple Match, or Max-SIS, then the conditions (A)–(C) greedy methods in bounded degree graphs provide con- from Theorem 3.1 hold. Moreover, if s(e) > 0 for each stant factor approximations. E.g., any inclusionwise e ∈ E, then ids(G0) = ds(G0) and every dominating maximal independent set is a (B + 1)-approximation set D in G can be transformed in polynomial time to for any of problems Max-IS, Min-DS, or Min-IDS, an independent dominating set D0 in G0 with |D0| = P when restricted to graphs of degree at most B. Ad- |D| + s(e). e ditionally, each problem from P is well known to be Remark 3.1. If s(e) is an odd integer for each edge APX-hard when restricted to G3 (graphs of degree at e ∈ E in Theorem 3.2, then the graph G0 is bipartite. most 3), in most cases a proof for 3-regular graphs is given as well (see [2], [16], [19], [21], [32], and ref- Now let us firstly consider the problem Max-IS. For erence therein). Moreover, explicit lower bounds on any graph G = (V,E) and fixed integer k ≥ 0 we obtain their efficient approximability are known for several of by Theorem 3.1 them ([13], [14], [15]). Hence, the considered problems are APX-complete when restricted to G3 (or even in 3- (3.1) is(div2k(G)) = is(G) + |E|k. regular graphs). Due to properties of L-reduction the following theorems hold To see that div2k is an L-reduction for Max-IS restricted to the set GB, we have to check that for some Theorem 3.3. For any fixed integer k ≥ 0 the re- constant α we have is(div2k(G)) ≤ αis(G) for every strictions of problems Max-IS and Min-VC to 2k- G ∈ GB. The key point is that for some positive con- subdivisions of 3-regular graphs are APX-complete. stant c (depending on B) one can easily prove the lower bound for the optimum value of the form is(G) ≥ c|V | Theorem 3.4. For any fixed integer k ≥ 0 the restric- |V | tions of problems Min-DS, Min-EDS, Max-Induced- for every G = (V,E) ∈ GB. Namely, is(G) ≥ . B+1 Match, Max-SIS, and Min-IDS to 3k-subdivisions of Now, recalling that |E| ≤ |V | B, it can be easy verified 2 graphs of maximum degree 3 are APX-complete. k that the choice α := 1 + B 2c will do. The second condition from the definition of L-reduction is satisfied with Remark 3.2. If k is odd then Theorem 3.4 claims β = 1 by Theorem 3.1. Hence div2k is an L-reduction APX-completeness results in bipartite graphs of maxi- that self-reduces Max-IS restricted to GB. mum degree 3 and girth at least 9k + 3. 0 For later applications to the Max-IS problem in (2 ) |OPTQ(f(x)) − mQ(f(x), h(x, y))| ≤ |OPTP (x) − d-box graphs we now formulate some explicit NP-hard mP (x, y)|, x ∈ IP , y ∈ solP (x), gap type results for this problem restricted to certain (30) |OPT (x) − m (x, g(x, y0))| ≤ |OPT (f(x)) − subdivisions of graphs. We will use the corresponding P P Q m (f(x), y0)|, x ∈ I , y0 ∈ sol (f(x)). NP-hard gap results for Max-IS in B-regular graphs, Q P Q B = 3, 4 proved in [14]. For any ε > 0 it is NP-hard Observation. Shift-reductions compose as follows: If to decide for a B-regular graph G = (V,E) of whether ³ ´ ³ ´ f1 is a shift for (P,Q) with difference ϕ1, and f2 is a |V | |V | is(G) < 2 1 − 3δB − ε or is(G) > 2 1 − 2δB + ε shift for (Q, R) with difference ϕ2, then f2 ◦ f1 is a shift where δ3 ≈ 0.0103305 and δ4 ≈ 0.020242915. Using for (P,R) with difference ϕ = ϕ1 + ϕ2 ◦ f1. the formula (3.1) we see that this translates to the Many reductions between pairs of optimization following NP-hardness result for 4-subdivision of B- problems in the literature are shift-reductions, and regular graphs: for any ε > 0 it is NP-hard to decide proofs given there show exactly the properties (2) and |V | of whether is(div4(G)) < 2 (1 + 2B − 3δB − ε), or (3) from Definition 3.1 for them. Here we are focused |V | is(div4(G)) > 2 (1 + 2B − 2δB + ε). Consequently, on the fact that for a lot of graph problems certain sub- approximation within any constant smaller than 1 + division operations on graphs are shifts that reduce the δB is NP-hard. In particular, we have proved problem to itself, or to another known problem. 1+2B−3δB the following Theorem 3.1 (resp., Theorem 3.2) says, in particular, that operation div (resp., div ) is a shift for Theorem 3.5. It is NP-hard to approximate Max-IS 2k 3k all problems considered in these theorems. However, in in 4-subdivision of 3-regular graphs within 1 + 1 , and 675 some applications it is useful to consider subdivisions in 4-subdivision of 4-regular graphs within 1 + 1 . 442 that are not uniform, but rather variable. Let us describe such more general subdivision operations more 3.1 Extension of Hardness Results to Other formally. Problems. In order to prove the version of Theo- Let G stand in what follows for a set of graphs rems 3.1 and 3.2 for some other problems, we introduce recognizable in polynomial time. For any graph G = the notion of a shift-reduction. First, for any NPO prob- (V,E) from G let s := s denote an edge function lem P we define the sign σ of P to be +1 if the goal G P s : E → Z+ that is polynomial time computable. of P is maximum, and −1 if the goal is minimum. 0 In this setting divs(e) will denote a (polynomial time Definition 3.1. Let P and Q be two NPO problems computable) transformation that maps G ∈ G to a and f be a polynomial time computable function that graph G0 obtained from G by s(e)-subdivision of each maps instances of P to instances of Q. We say that f edge e ∈ E. Then Theorem 3.1 says that for any edge is a shift reduction from P to Q if there are polynomial function s the operator div2s(e) is a shift for the Max-IS time computable functions ϕ, h, and g such that problem (respectively,P for Min-VCP ) restricted to G with difference s (e) (resp., − s (e)). Similarly, (1) ϕ : I → R, e G e G P Theorem 3.2 says that if P is any of problems Min-

(2) for every x ∈ IP and every y ∈ solP (x), DS, Min-EDS, Max-Induced-Match, or Max-SIS, then for any edge function s, div is a shift for P h(x, y) ∈ solQ(f(x)) so that σQmQ(f(x), h(x, y)) − P3s(e) σP mP (x, y) ≥ ϕ(x), restricted to G with difference σP e s(e). The problem Min-IDS is not self-reducible in this way but Min- 0 (3) for every x ∈ IP and every y ∈ solQ(f(x)), DS reduces to it (assuming s (e) > 0 for each edge 0 0 G g(x, y ) ∈ solP (x) so that σQmQ(f(x), y ) − e ∈ E and G = (V,E) ∈ G), namely div is a shift 0 ¯ 3s(e)P σP mP (x, g(x, y )) ≤ ϕ(x). ¯ for (Min-DS G, Min-IDS) with difference − e sG(e). In such case f is called a shift for the ordered Some graph problems are known to be related to those pair (P,Q) with a difference ϕ. If both P and Q are studied in Theorem 3.2. Translated to our terminology restrictions of the same problem P0 to instance sets G these relations can be stated as follows: and G0, respectively, we say simply that f is a shift for Lemma 3.3. The identity map is a shift for the follow- P restricted to G. 0 ing pairs of graph problems: Clearly, (2) implies σ OPT (f(x)) − Q Q (i) for (Min-IDS, Max-Minl-VC) with differ- σ OPT (x) ≥ ϕ(x), and (3) implies the opposite in- P P ence |V |, equality. Hence σQOPTQ(f(x)) − σP OPTP (x) = ϕ(x), and with this equality in hand one can rewrite (ii) for (Min-EDS, Min-Maxl-Match) with differ- inequalities in (2) and (3) in the form ence 0, (iii) for (Min-Maxl-Match, Max-Total-Match) For a fixed k ≥ 0 and B ≥ 3 differences of with difference |V |, and assuming that G is a set our uniform shift reductions div3k among problems of graphs without isolated vertices, restricted to GB clearly satisfy (ii) from Lemma 3.4 due ¯ ¯ to bounds |ϕ(G)| ≤ |V | + 2k|E| ≤ (1 + kB)|V |. As ¯ ¯ (iv) for (Min-DS G, Max-Minl-EC G) with differ- we have already mentioned, (i) holds (in graphs GB) for ence |V |, any of problems we reduce from, namely Min-DS and ¯ ¯ Min-EDS. As all studied problems are clearly in APX ¯ ¯ (v) for (Max-Minl-EC G, Max-SF G) with differ- when restricted to GB, the following theorem follows ence 0. Theorem 3.6. For any fixed integer k ≥ 0, each Obviously, the identity is a shift for (P,Q) of problems Max-Minl-VC, Max-Minl-EC, Min- with difference ϕ if and only if it is a shift for Maxl-Match, Max-Total-Match, and Max-SF (Q, P ) with difference −ϕ. Hence combining The- when restricted to 3k-subdivisions of graphs of maximum orem 3.2 with Lemma 3.3 we can obtain the fol- degree 3, is APX-complete. lowing results: if P is one of the problems Min- Maxl-Match, Max-Total-Match, Max-Minl-EC, 4 Approximation Hardness Results in d-box or Max-SF, then for any edge function s div3s(e) Intersection Graphs is a shift forP the problemP P restrictedP to G with Our Theorem 2.1 shows that any graph obtained by difference − s (e), 2 s (e), 2 s (e), and P e G e G e G at least 3-subdivision of each edge from another (arbi- 2 s (e), respectively. Moreover, div is a shift e G P 3s(e) trary) graph is a d-box intersection graph for any fixed with difference |V | + 2 s (e) for any of pairs ¯ G e G ¯ d ≥ 3. This immediately shows that many optimization (Min-DS¯ , Max-Minl-EC), (Min-DS¯ , Max-SF), G ¯ G problems in intersection graphs are as hard to approx- ¯ and (Min-EDS G, Max-Total-Match). The prob- imate as in general graphs. It is rather obvious to see lem Max-Minl-VC (similarly as Min-IDS) is not for such problems as Minimum Steiner Tree or Min- self-reducible in this way but Min-DS reduces to it imum Traveling Salesman for which replacing edges (assuming sG(e) > 0 for each edge e ∈ E and by pairwise internally disjoint paths (and splitting edge G = (V,E¯ ) ∈ G), namely div3s(e) is a shift for weights properly) cannot make the problem easier to (Min-DS¯ , Max-Minl-VC) with difference |V | + approximate. P G G 2 e sG(e). On the other hand, many problems are known to be easier to approximate in d-box graphs than in Remark 3.3. One can also easily prove that div1 is a general ones, and no approximation hardness results shift (in both directions) between Min-DS and Max- are known for them in such graphs. We provide first Induced-Match, resp. Min-EDS and Max-SIS, with inapproximability results for some of them, namely, difference |VG|, resp. −|VG|, for vice versa shifts (see APX-hardness and hence non-existence of PTAS (unless [24], [27]). Moreover, it is also a shift from Max- P = NP). Moreover, all our hardness results apply Induced-Match to Min-IDS with difference −|VG|. even to the setting when the representation is given, not Using composition of div1 with div3s(e) above we can merely its intersection graph. This makes our hardness obtain that, in general, a subdivision operation of the results stronger, as the reconstruction problem is known form div1+3s(e) shares these properties with div1 (with to be NP-hard. Rather straightforward application of an appropriate change of difference of the shift after Theorems 2.1, 3.3, 3.4, and 3.6 yields in the following composition). We omit the details, as that kind of subdivisions will not be used in what follows. Theorem 4.1. Let d ≥ 3 be a fixed integer. Each of the problems Max-IS, Min-VC, Min-DS, Min- One can observe the following EDS, Max-Induced-Match, Max-SIS, Min-IDS, Max-Minl-VC, Max-Minl-EC, Min-Maxl-Match, Lemma 3.4. Let f be a shift with difference ϕ for a pair Max-Total-Match, and Max-SF restricted to (in- (P,Q) of NPO problems. Assume further that there is tersection graphs of) sets of axis-parallel d-dimensional a positive constant C such that for every x ∈ IP it holds boxes, is APX-hard and does not admit PTAS (unless |ϕ(x)| ≤ C · OPTP (x). Then f is an L-reduction from P = NP). These hardness results apply also to instances P to Q with α = σQσP + C and β = 1. This is in whose the intersection graph is simultaneously of maxi- K particular true (with C := c ) if for a positive constants mum degree at most 3, of girth at least k (for any pre- c, K ∈ (0, ∞) and for every x ∈ IP (i) OPTP (x) ≥ c|x|, scribed constant k), and, except Max-IS and Min-VC, (ii) |ϕ(x)| ≤ K|x|. bipartite as well. These results could be stated as explicit NP-hard hardness result for any of studied problems restricted gap type results and provide explicit lower bounds on to unit square graphs (alternatively, unit disk graphs). their approximability. We will demonstrate it on the Some of these results were well studied and are known main problem of our interest, Maximum Independent for unit disk graphs. Set. Theorem 4.3. Each of the problems Max-IS, Min- Theorem 4.2. Let d ≥ 3 be a fixed integer and R be VC, Min-DS, Min-EDS, Max-Induced-Match, a set of d-boxes such that the intersection graph GR = Max-SIS, Min-IDS, Max-Minl-VC, Max-Minl- (VR,ER) of R is the 4-subdivision graph of a 4-regular EC, Min-Maxl-Match, Max-Total-Match, and graph. Then the following decision problem is NP-hard: Max-SF is NP-hard when restricted to (intersection to decide whether the maximum number of pairwise graphs of) axis-parallel unit squares (alternatively, unit disjoint d-boxes of R is less than 0.4966261808|VR| or disks) in the plane. These hardness results apply also greater than 0.4977507872|VR| (under promise that one to instances whose intersection graph is simultaneously of these two cases occurs). Consequently, it is NP- of maximum degree 3, of girth at least k (for any pre- hard to approximate Max-IS problem in d-boxes within scribed constant k), and, except Max-IS and Min-VC, 1 bipartite as well. 1 + 442 . Remark 4.1. The same approximation hardness re- Remark 4.2. The described method can be applied to sults as we obtained in Theorems 4.1 and 4.2 for (inter- more general geometric graphs, e.g., to contact graphs section graphs of) sets of d-boxes easily follow for (in- of unit discs, for which we can obtain the same NP- tersection graphs of) sets of axis-parallel lines, for any hardness results. Recall that contact graphs are a special fixed d ≥ 3, with slightly worse explicit lower bounds as kind of intersection graphs of geometrical objects in in Theorem 4.2. which we do not allow the objects to cross but only to touch one another. 4.1 Rectangle Intersection Graphs. Topological obstructions prevent to obtain similar inapproximability 5 Maximum Weighted Projection-Independent results in 2-dimensional case. However, we can use some Set of d-boxes of ideas above to give a generic approach to NP-hardness In this section we discuss another (weighted) Max-IS results for the same kind of graph optimization problems problem that is related to geometric graphs generated in rectangle intersection graphs, or even in unit square by sets of d-boxes. The problem was introduced by graphs (alternatively, in unit disk graphs). Bafna et al. as the Independent Subset of Rect- First, we can prove easily planar variant of our angles (shortly, IR) problem. The research was mo- structural Theorem 2.1. If G is a planar graph and G0 tivated by a fundamental problem in computational is a graph obtained from G by at least 1-subdivision molecular biology (see [4] for more details). of every edge of G, then G0 is a rectangle intersec- Recall that a graph is an interval graph if its ver- tion graph. Moreover, its rectangle representation can tices can be assigned to intervals on the real line so that be constructed in linear time. It easily follows from vertices are adjacent iff the corresponding intervals in- known algorithms on weak visibility representation of tersect. Such assignment is called the interval represen- planar graphs [38]. Such algorithm draws a planar tation of the interval graph. In case of the proper inter- graph G = (V,E) so that each vertex of G is represented val representation, no interval properly contains another by a horizontal segment, each edge is represented by a one. A graph is a proper interval graph if it admits a vertical segment, and incidence between vertices and proper interval representation. Define two d-boxes to be edges translates to intersections of horizontal and verti- projection-independent if for each axis their projections cal segments. Replacing segments properly by thin axis- on this axis are disjoint. parallel rectangles we obtain a set of rectangles whose The IR problem was formulated as follows: given a intersection graph is exactly div1(G). To obtain more set R of positively weighted axis-parallel d-boxes such general subdivisions of G, just replace the “vertical rect- that for each axis, the projection of a d-box on this axis angle” corresponding to an edge e ∈ E by more rectan- does not contain another one. The goal of IR is to find gles, creating a path of required length. As problems a maximum weighted subset R∗ ⊆ R of d-boxes that studied in this paper are known to be NP-hard in pla- are pairwise projection-independent. Equivalently, it nar graphs (even in planar graphs of maximum degree is Max-w-IS in the corresponding project-intersection 3), NP-hardness of the restriction of any of them to rect- graph GR, where vertex set of GR is R and two d- angle intersection graphs easily follows from Theorems boxes of R are adjacent by an edge in GR iff their 3.1, 3.2, and Lemma 3.3. Even more, we can prove NP- projections on (at least) one axis intersect. Graphs GR (for arbitrary set R) are termed as d-union interval Theorem 5.2. For any fixed d ≥ 2 the Max-w-IS graphs, as it can be expressed as the union of d interval problem in proper d-union graphs (with their representa- graphs ([6]). Hence the IR problem yields to Max-w- tion given) can be approximated within 2d − 1 in nearly IS in proper d-union graphs. The Max-w-IS problem linear time. in proper d-union graphs was studied in more details for d = 2. Bafna et al. [4] showed that in this case the corresponding graphs are 5-claw free and using known References result of Berman for Max-w-IS in t-claw free graphs ([8]), the current best approximation for Max-w-IS in [1] P. K. Agarwal, M. van Kreveld, and S. Suri, Label proper 2-union graphs has a factor of 5 . 2 placement by maximum independent set in rectangles, In [4] it is noted that the projection-intersection Comput. Geom. Theory Appl. 11(3-4) (1998), 209– graph GR for a set R satisfying assumptions of the 218. d IR problem is (2 + 1)-claw free for d ≥ 3. Best [2] P. Alimonti and V. Kann, Some APX-completeness known algorithms for Max-w-IS in t-claw free graphs results for cubic graphs, Theoretical Computer Science provide approximation factor for this problem that is 237 (2000), 123–134. exponential in d (as repeated as well in [6]). Also in [3] G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, [10] (and [9]) it is repeated the question of an existence A. Marchetti-Spaccamela, and M. Protasi, Complexity of efficient algorithms for d-dimensional version with and approximation, Springer, 1999. approximation factor increasing less drastically (e.g., [4] V. Bafna, B. Narayanan, and R. Ravi, Nonoverlapping linearly in d). However, it is rather easy to prove that local alignments (weighted independent sets of axis- parallel rectangles), Discrete Applied Mathematics 71 proper d-union graphs are in fact (2d + 1)-claw free, so (1996), 41–53. the answer is straightforward. [5] E. Balas and C.-S.Yu, On graphs with polynomially Lemma 5.1. Proper d-union graphs are (2d + 1)-claw solvable maximum-weight clique problem, Networks 19 (1989), 247–253. free. [6] R. Bar-Yehuda, M. M. Halldórsson, J. Naor, Proof. Let G = (V,E) be a proper d-union graph. H. Schachnai, and I. Shapira, Scheduling split intervals, Hence there are d proper interval graphs G = (V,E ), Proceedings of the 13th Annual ACM-SIAM Sympo- i i sium on Discrete Algorithms, SODA 2002, pp. 742– i = 1, 2, . . . , d such that E = ∪d E . Assume that G i=1 i 751. contains induced (2d+1)-claw, and let L be the set of its [7] G. Di Battista, P. Eades, R. Tamassia, and I. G. Pd (2d + 1) edges. Clearly, (2d + 1) = |L| ≤ i=1 |L ∩ Ei|, Tollis, Graph drawing: Algorithms for the visualization hence there is an index i0 for which |L ∩ Ei0 | > 2. of graphs, Prentice Hall, 1999. d Consequently, Gi0 contains induced 3-claw. However, [8] P. Berman, A 2 -approximation for maximum weight independent set in d-claw free graphs, Nordic Journal this contradicts with the fact that Gi0 is a proper interval graph, because an interval graph is a proper of Computing 7 (2000), 178–184. interval graph if and only if it is 3-claw free [36]. This [9] P. Berman and B. DasGupta, A simple approxima- contradiction shows that G is (2d + 1)-claw free.¤ tion algorithm for nonoverlapping local alignments (weighted independent sets of axis-parallel rectangles), Using results of [8] we immediately get the following Biocomputing (Panos M. Pardalos and Jose Proncipe, theorem. eds.), vol. 1, Kluwer Academic Publisher, 2002, pp. 129–138. Theorem 5.1. For any fixed d ≥ 2 the Max-w-IS [10] P. Berman, B. DasGupta, and S. Muthukrishnan, Sim- problem in proper d-union graphs can be approximated ple approximation algorithm for nonoverlapping local within (d + 1 ) in polynomial time. alignments, Proceedings of the 13th Annual ACM- 2 SIAM Symposium on Discrete Algorithms, SODA The best known algorithms for t-claw free graphs 2002, pp. 677–678. are rather impractical, hence Berman et al. [9], [10] [11] P. Berman, B. DasGupta, S. Muthukrishnana, and showed a simple O(n log n)-time algorithm with approx- S. Ramaswami, Efficient approximation algorithms for imation factor 3 for 2-dimensional version. They ob- tiling and packing problems with rectangles, Journal of Algorithms 41 (2001), 443–470. serve that the algorithm can be extended to yield a d [12] T. M. Chan, Polynomial-time approximation schemes (2 − 1)-approximation for d-dimensional version. But for packing and piercing fat objects, Journal of Algo- the straightforward extension of their algorithm to d- rithms 46 (2003), 178–189. dimensional version gives (2d − 1)-approximation. Re- [13] M. Chleb´ıkand J. Chleb´ıková, Approximation hardness call that for more general d-interval graphs Bar-Yehuda of minimum edge dominating set and minimum max- et al. [6] achieved 2d-approximation algorithm. imal matching, Proceedings of the 14th International Symposium on Algorithms and Computation, ISAAC (1984), 2–6. 2003, LNCS 2906, 2003, pp. 415–424. [31] L. Lewin-Eytan, J. Naor, and A. Orda, Routing and ad- [14] , Inapproximability results for bounded variants mission control in networks with advance reservations, of optimization problems, Proceedings of the 14th In- Proceedings of the 5th International Workshop on Ap- ternational Symposium on Fundamentals of Computa- proximation Algorithms for Combinatorial Optimiza- tion Theory, FCT 2003, LNCS 2751, 2003, pp. 27–38. tion, APPROX 2002, LNCS 2462, 2002, pp. 215–228. [15] , Approximation hardness of dominating set [32] D. F. Manlove, On the algorithmic complexity of twelte problems, Proceedings of the 12th Annual European covering and independence parameters of graphs, Dis- Symposium on Algorithms, ESA 2004, LNCS 3221, crete Appl. Math. 91 (1999), 155–175. 2004, pp. 192–203. [33] F. Nielsen, Fast stabbing of boxes in high dimensions, [16] W. Duckworth, D. F. Manlove, and M. Zito, On Theoret. Comput. Sci. 246 (2000), 53–72. the approximability of the maximum induced matching [34] J. Nieminen, Two bounds for the domination number problem, to appear in Journal of Discrete Algorithms. of a graph, J. Inst. Math. Appl. 14 (1974), 183–187. [17] T. Erlebach, K. Jansen, and E. Seidel, Polynomial-time [35] E. Prisner, Graphs with few cliques, Graph Theory, approximation schemes for geometric graphs, Proceed- Combinatorics, and Applications: Proceedings of 7th ings of the 12th Annual ACM-SIAM Symposium on Conference on the Theory and Applications of Graphs Discrete Algorithms, SODA 2001, pp. 671–679. (Y. Alavi and A. Schwenk, eds.), John Wiley and Sons, [18] R. J. Fowler, M. S. Paterson, and S. L. Tanimoto, Inc., 1995, pp. 945–956. Optimal packing and covering in the plane are NP- [36] F. S. Roberts, Indifference graphs, Proof Techniques in complete, Information Processing Letters 12 (1981), Graph Theory (F. Harary, ed.), Academic Press, New 133–137. York, 1969, pp. 139–146. [19] F. Gavril and M. Yannakakis, Edge dominating sets in [37] , On boxicity and cubicity of a graph, Recent graphs, SIAM J. Appl. Math. 38 (1980), 364–372. Progress in Combinatorics (W. T. Tutte, ed.), Aca- [20] U. I. Gupta, D. T. Lee, and J.Y.-T.Leung, Efficient demic Press, 1969, pp. 301–310. algorithms for interval graphs and circular-arc graphs, [38] P. Rosenstieh and R. E. Tarjan, Rectilinear planar lay- Networks 12 (1982), 459–467. outs and bipolar orientations of planar graphs, Discrete [21] M. Halldórsson, R. W. Irwing, K. Iwama, D. F. Comp. Geometry 1 (1986), 343–353. Manlove, S. Miyazaki, Y. Morita, and S. Scott, Ap- [39] M. Yannakakis, The complexity of the partial order di- proximability results for stable marriage problem with mension problem, SIAM Journal on Algebraic Discrete ties, Theoretical Computer Science 306 (2003), 431– Methods 3 (1982), 351–358. 447. [22] S. T. Hedetniemi, A max-min relationship between matchings and domination in graphs, Congr. Numer. 40 (1983), 23–34. [23] D. S. Hochbaum and W. Maass, Approximating schemes for covering and packing problems in image processing and VLSI, Journal of ACM 32 (1985), 130– 136. [24] J. D. Horton and K. Kilakos, Minimum edge dominating sets, SIAM J. Discrete Math. 6 (1993), 375–387. [25] H. Imai and T. Asano, Finding the connected compo- nents and a maximum clique of an intersection graph of rectangles in the plane, Journal of Algorithms 4 (1983), 310–323. [26] S. Khanna, S. Muthukrishnan, and M. Paterson, On approximate rectangle tiling and packing, Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1998, pp. 384–393. [27] C. W. Ko and F. B. Shepherd, Bipartite domination and simultaneous matroid covers, SIAM J. Discrete Math 16 (2003), 517–523. [28] J. Kratochv´ıl, A special planar satisfiability problem and a consequence of its NP-completeness, Discrete Applied Mathematics 52 (1994), 233–252. [29] D. T. Lee, Maximum clique problem of rectangle graphs, Advances in Computing Research 1 (1983), 91–107. [30] D. T. Lee and J. Y.-T. Leung, On the 2-dimensional channel assignment problem, IEEE Trans. Comput. 33