The Approximability and Integrality Gap of Interval Stabbing and Independence Problems

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1 CCCG 01, Charlottetown, P.E.I., August 8 10, 01 The Approximability an Integrality Gap of Interval Stabbing an Inepenence Problems Shalev Ben-Davi Elyot Grant Will Ma Malcolm Sharpe Abstract Motivate by problems such as rectangle stabbing in the plane, we stuy the minimum hitting set an maximum inepenent set problems for families of -intervals an -union-intervals. We obtain the following: (1) constructions yieling asymptotically tight lower bouns on the integrality gaps of the associate natural linear programming relaxations; () an LP-relative - approximation for the hitting set problem on -intervals; (3) a proof that the approximation ratios for inepenent set on families of -intervals an -union-intervals can be improve to match tight uality gap lower bouns obtaine via topological arguments, if one has access to an oracle for a PPAD-complete problem relate to fining Borsuk-Ulam fixe-points. 1 Introuction In this work, we examine a family of NP-har packing an covering problems. Our stuy is motivate by the minimum rectangle stabbing problem, in which we are given a family H of axis-aligne rectangles in the plane, an the goal is to fin a minimum-carinality family of horizontal an vertical lines that intersect (or stab ) each rectangle in H at least once. Viewing this as a geometric covering problem, we also consier the relate ual geometric packing problem of fining a maximum conflict-free subset, where the goal is to fin a maximum subset of H containing no pair of rectangles that can be stabbe by a single horizontal or vertical line. The rectangle stabbing an conflict-free subset problems have many applications. The rectangles themselves can be the bouning boxes of arbitrary connecte objects in the plane, so applications nee not be limite to problems involving rectangles. The rectangle stabbing problem can irectly encoe the problem of optimally subiviing the plane into a gri of axis-aligne cells so as to separate a given family of points, with applications to fault-tolerant sensor networks [3] an resource allocation in parallel processing systems [7]. The maximum Comp. Sci. an A.I. Lab, MIT, shalev@mit.eu Comp. Sci. an A.I. Lab, MIT, elyot@mit.eu Operations Research Center, MIT, willma@mit.eu Supporte in part by ONR Grant N Department of Combinatorics an Optimization, University of Waterloo, sharpe.malcolm@gmail.com conflict-free subset problem, an its higher imensional analogues, are relevant to areas such as resource allocation, scheuling, an computational biology []. The properties of certain rectangle stabbing instances are also of theoretical interest in combinatorics [19]. Given a family H of axis-aligne rectangles, we write for the minimum carinality of a family of lines stabbing it, an α(h) for the maximum size of a conflictfree subset. It is clear that α(h). As in many geometric packing-covering ual problems, there is a boun in the other irection. In 1994, Taros prove that α(h), which is easily seen to be tight [18]. However, all known proofs of Taros s result rely on topological fixe-point theorems, an consequently o not seem to lea to polynomial-time approximations. In fact, only a 4-approximation is known for the maximum conflict-free subset problem [], espite the fact that we can establish the optimal objective value to within a factor of by solving a linear program. Improving upon this remains an important open problem. In this paper, we obtain results for generalize versions of the stabbing an conflict-free subset problems. We examine the stanar linear programming relaxations for hitting set an inepenent set problems involving -intervals, an establish asymptotically tight upper an lower bouns on their integrality gaps. These bouns imply that no LP-relative approximation algorithm can obtain a factor below for either the rectangle stabbing or the maximum conflict-free subset problems. Aitionally, we establish some interesting theoretical consequences of topological methos such as Taros s. For example, we show that the maximum conflict-free subset problem amits a -approximation if one has access to an oracle for a PPAD-complete problem. This article procees as follows: in the current section, we efine the generalize problems that we stuy, explain the current state of the art, an escribe our contribution. Section contains our integrality gap upper an lower bouns, an Section 3 contains algorithmic results that epen on PPAD oracles. 1.1 Preliminaries We begin by efining generalize versions of the rectangle stabbing an conflict-free subset problems. For N, a -interval I is a union of non-empty com-

2 4 th Canaian Conference on Computational Geometry, 01 pact intervals I 1,..., I R. The input to all problems we consier will be a finite collection H of -intervals, represente explicitly. We say that a subset of H is inepenent if its members are pairwise isjoint, an a set X R is a hitting set for H if it intersects every member of H. We efine the hypergraph G H = (V, E) where V = H an E consists of all subsets I H such that there is a point p R that hits exactly the intervals in I. Such a point p shall be calle a representative of the hyperege I E, an we let P (H) enote a set containing an arbitrary representative for each istinct ege in G H. Note that P (H) H as there are at most H interval enpoints. We call G H a -interval hypergraph, an observe that -interval hypergraphs generalize -regular hypergraphs (which are obtaine when each -interval in H is simply points). We enote by α(h) the maximum size of an inepenent set in H, an enote by the minimum size of a hitting set for H, in analogy with the usual notation of α(g) an ρ(g) for the maximum inepenent set size an minimum ege cover size of a hypergraph G. Special cases such as the rectangle stabbing problem arise when we impose structural restrictions on H. If {J i } i=1 is a family of isjoint intervals an each - interval I = i=1 Ii in H satisfies I i J i for all i, then H is known as a collection of -union-intervals, an G H is known as a -union hypergraph. The term -trackinterval is sometimes use for the same concept, with the iea that each -interval contains a piece from one of ifferent tracks, each of which is a isjoint copy of R. The rectangle stabbing an minimum conflict-free subset problems correspon precisely to the minimum hitting set an maximum inepenent set problems on -union-intervals, but with each track mappe onto a separate Eucliean imension. In general, one can think of the hitting set problem for -union-intervals as the problem of hitting a family of -imensional boxes using a minimum number of walls, each of which is orthogonal to one of the coorinate axes. For 1-interval hypergraphs (which are the same as 1- union hypergraphs), the inepenent set an ege cover problems can both be solve in polynomial time via simple greey algorithms that perform a left-to-right sweep across the intervals. However, even for -union hypergraphs, the inepenent set an ege cover problems are both APX-har. Nagashima an Yamazaki, an inepenently Bar-Yehua et al., have shown the conflictfree subset problem to be APX-har [, 14], even when the rectangles are all unit squares with integer vertices. Kovaleva an Spieksma show that the rectangle stabbing problem is APX-har even when each rectangle is of the form [x, x + 1] [y, y] for integers x an y [11]. For a hypergraph G, the relations α(g) ρ(g) an ρ(g) O(log V ) α(g) are well known, with the latter being tight for general hypergraphs. However, using methos of topological combinatorics, Kaiser proves that ρ(g H) α(g H ) is upper boune by +1 for -interval hypergraphs an for -union hypergraphs (for ), a boun inepenent of V [10]. His result improves upon that of Taros, who originally establishe a tight upper boun for the = case [18]. In a onepage paper, Alon shows that an upper boun of can be establishe without topological methos by applying Turán s theorem [1]. The best known lower bouns for large are Ω( log ) an Ω( ) for -interval an - log union hypergraphs respectively [13]. 1. Overview of Results We use the term uality gap to enote the quantity sup H α(h), where H ranges over a collection of - intervals. In an effort to stuy various uality gaps, we examine stanar linear programming relaxations for the hitting set an inepenent set problems. The stanar LP relaxation for the maximum inepenent set problem correspons to the maximum fractional inepenent set problem, an can be written as follows: max x I I I s.t. x I 1 I p x I 0 p P (H) I H (1) A corresponing ual linear program for the minimum fractional hitting set problem is as follows: min s.t. p P (H) y p y p 1 p I y p 0 I H p P (H) () If α (H) is the optimal objective value for (1) an ρ (H) is the optimal objective value for (), then we have α(h) α (H) = ρ (H) an can write sup H α(h) sup H ρ (H) sup α (H) H α(h). The quantity sup H ρ (H) is calle the integrality gap of the minimum hitting set problem (for -intervals or - α union-intervals). Similarly, sup (H) H α(h) is the integrality gap of the maximum inepenent set problem. Since ρ (H) an α (H) α(h) are always at least 1, both integrality gaps are a lower boun on the uality gap. For the case of 1-intervals, we actually have α(h) = ; both linear programs have an integrality gap of 1 because the incience matrix of G H exhibits the consecutive ones property an is thus totally unimoular.

3 CCCG 01, Charlottetown, P.E.I., August 8 10, 01 Often, upper bouns on integrality gaps for packing an covering problems come alongsie LP-relative approximation algorithms. Bar-Yehua et al. employ the local ratio technique to obtain a polynomial-time LPrelative -approximation algorithm for the -interval maximum inepenent set problem, proving that the integrality gap of maximum inepenent set for - intervals is at most. Their result carries over to the version in which each element in H has a positive weight an a maximum weight inepenent set is esire. In Section, we show that their boun is tight up to an aitive constant by establishing the following: Theorem 1 For any > 0, there exists a collection H of -intervals (respectively, -union-intervals) for which 1 (respectively, ). α (H) α(h) Our constructions generalize examples from [5] an [8] but employ a novel amplification trick. For both the -interval an -union-interval hitting set problems, we are able to prove that the integrality gap is exactly. We show the following: Theorem There exists a polynomial-time LPrelative -approximation for the -interval hitting set problem. Accoringly, for any collection H of -intervals, ρ (H). Theorem 3 For any > 0, there exists a collection H of -union-intervals for which ρ (H). Theorem uses stanar techniques to generalize a - approximation algorithm for rectangle stabbing ue to [7], but Theorem 3 employs a novel construction. The table below summarizes the known integrality an uality gap bouns for large : -Interval Lower Boun Upper Boun Duality Gap Ω( log ) [13] + 1 [10] Max-IS Integ. Gap 1 [] Min-HS Integ. Gap -Union Duality Gap Ω( log ) [13] [10] Max-IS Integ. Gap [] Min-HS Integ. Gap We note that for =, Kaiser s topology-base uality gap upper bouns of + 1 an match our inepenent set integrality gap lower bouns of 1 an, but are tighter than the constructive integrality gap upper bouns of ue to Bar-Yehua et al. Hence, espite knowing that α(h) is boune above by 3 an for families of -intervals an -unionintervals respectively, no polynomial-time approximation factor below 4 is known for the maximum inepenent set problem on -union-intervals. We observe, however, that Kaiser s proof can be turne into a - approximation if one has access to an oracle to solve the topological subproblems that arise. The particular topological problems in question are closely relate to fining Borsuk-Ulam fixe-points. Unfortunately, the problem of fining Borsuk-Ulam fixe-points is PPADcomplete [15] an thus seems unlikely to amit polynomial algorithms unless a major breakthrough occurs. Nevertheless, we establish the following in Section 3: Theorem 4 There exists an algorithm for the maximum inepenent set problem on -intervals (respectively, -union-intervals) returning a solution of size at least α(h) 3 (respectively α(h) ), requiring O(log(α(H))) calls to an oracle for a PPAD-complete fixe-point problem, an polynomial time for all other computations. Despite the fact that Theorem 4 is likely not of practical value, we fin it interesting because it implies that the -imensional maximum conflict-free subset problem is a natural APX-har geometric optimization problem whose best known approximability appears to improve in the presence of a PPAD oracle. It remains an open problem to fin an alternative metho of achieving the approximation ratios of Theorem 4 while bypassing the nee for a PPAD oracle (of course, this may very well be impossible, but proving so woul separate P from PPAD, resolving a longstaning open problem). 1.3 Relate Work Many variations an special cases of -interval stabbing an inepenence problems have been stuie in a variety of contexts. Kovalena an Spieksma have examine the special case of the rectangle stabbing problem in which each rectangle is a horizontal line segment e [11, 1]. They obtain an LP-relative e 1 -approximation for this case, alongsie an example showing that the integrality gap is precisely e e 1. Even et al. explore weighte an capacitate variations of -union-interval hitting set [6]. Their results inclue a 3-approximation for a variant in which each point may only be use to hit a specifie number of -intervals, but may be purchase multiple times. Spieksma consiers the version of maximum - interval inepenent set where the goal is to select a single interval from each -interval such that none intersect [17]. It is shown that a straightforwar greey proceure yiels a -approximation. Some aitional harness results are also known. Even et al. show that there is a constant c > 0 such that it is NP-har to approximate the -interval hitting set problem to within c log [6]. Dom et al. show that rectangle stabbing is W [1]-har, even when the input consists of squares of the same size, implying that the problem is unlikely to be fixe-parameter tractable in the optimal objective value [4].

4 4 th Canaian Conference on Computational Geometry, 01 Integrality Gap Bouns To prove Theorem 1 an establish tight lower bouns on the integrality gap of the inepenent set problem, we rely on an amplification lemma. We shall refer to the iniviual intervals composing a -interval as its pieces, an call a piece inert if it is a point. We shall call H a clique if α(h) = 1, an write r(h) for the rank of G H the maximum number of -intervals intersecte by any point in R. We observe that if H is a clique an r(h) = p, then a fractional inepenent set of value H p can be obtaine by simply putting weight 1 p on each -interval in H. In some situations, we can o better: Lemma 5 Suppose that H is a clique, an that H contains no two inert pieces that intersect an no - intervals consisting entirely of inert pieces. Furthermore, suppose that r(h ) = q, where H is a moifie version of H obtaine by eleting all inert pieces. Then for any N N, there is a clique H of -intervals amitting a fractional inepenent set of value N H Nq+1. Proof. We construct H by making N copies of each -interval in H, an then perturbing all inert pieces in the resulting family of -intervals such that no two inert pieces intersect, while preserving intersections of inert pieces with non-inert pieces. It is immeiate that H is still a clique; note that copies of the same -interval in H must intersect in H because no -interval consists entirely of inert pieces. Moreover, r(h ) Nq + 1, so 1 we can place a weight of Nq+1 on each -interval in H, yieling a fractional inepenent set of value N H Nq+1. By taking the limit as N, Lemma 5 yiels an integrality gap lower boun of H q given a -interval graph satisfying the necessary requirements. We note that the amplification in Lemma 5 also works for -unionintervals. We procee with the proof of Theorem 1: Proof of Theorem 1. For the case of -interval graphs, we exhibit a clique H satisfying the conitions of Lemma 5 with H = 1 an q = 1. We label the -intervals {a 0, a 1,..., a }. Each -interval will have exactly one non-inert piece (a close interval in R) an 1 inert pieces. We position the non-inert pieces such that no two intersect, which ensures that q = 1. Then for all 0 i, we position the remaining 1 inert pieces of a i (each of which is a point) on the non-inert pieces of -intervals {a i+1,..., a i+( 1) }, where the aition is moulo 1. This ensures that an inert piece of a i intersects non-inert pieces in {a i+1,..., a i+( 1) }, hence ensuring that inert pieces of {a i 1,..., a i ( 1) } all intersect the non-inert piece of interval a i. This proves that the construction yiels a clique, from which it follows that the inepenent set problem in -interval graphs has an integrality gap of H q = 1. An example of the construction for = 3 is shown (intervals are vertically separate for clarity): a 0 a 1 a a 3 a 4 a 3 a 4 a 4 a 0 a 0 For the case of -union-intervals, we exhibit a clique H of size 4 4 satisfying the conitions of Lemma 5 with q =. This yiels an integrality gap H q =. We label the 4 4 -union-intervals by {a i 1, a i, a i 3, a i 4} 1 i=1. We shall say that each interval has its k th piece in the k th track, where each track is a copy of R. We first explain what happens in tracks 1 through 1, an then explain what happens in the final track, which is treate ifferently. For 1 i 1, all of the pieces in track i are inert (single points) except for the i th pieces of a i 1, a i, a i 3, an a i 4, which are arrange as follows: a 1 a 1 a i 1 a i 3 a a a i a i 4 Now, for all j i, the i th pieces of {a j 1, aj, aj 3, aj 4 } are positione accoring to the following rules: If j < i, put a j 1, aj in ai 1 a i ; put a j 3, aj 4 in ai 3 a i 4 If j > i, put a j 3, aj 4 in ai 1 a i ; put a j 1, aj in ai 3 a i 4 In the last track, none of the intervals nee to be inert. For all 1 i 1, the th pieces of {a i 1, a i, a i 3, a i 4} are positione similarly to the iagram above, but are permute to inuce the remaining three epenencies among the -intervals. Figure 1 illustrates this an provies an example of the entire construction for = 4. Observe that any two -union-intervals with the same superscript i must be ajacent in either track i or track. For 1 i < j 1, we check that all 16 epenencies between a i 1, a i, a i 3, a i 4 an a j 1, aj, aj 3, aj 4 are accounte for: In track i, a j 3 an aj 4 intersect ai 1 a i ; a j 1 an a j intersect ai 3 a i 4. In track j, a i 1 an a i intersect a j 1 aj ; ai 3 an a i 4 intersect a j 3 aj 4. Thus H is a clique. It is easy to verify that the other conitions of Lemma 5 are satisfie with q =, so the proof is complete. Next, we provie a polynomial algorithm yieling an upper boun of for the integrality gap of the general -interval hitting set problem: Proof of Theorem. Let H be a collection of - intervals, an let {y p : p P (H)} be an optimal fractional hitting set of weight ρ (H) obtaine by solving linear program (). We emonstrate how to roun {y p} to an integral solution of weight at most ρ (H). For a -interval I in H, let I I be any piece of I that is hit by weight at least 1 uner {y p} (one must exist by the a 3

5 CCCG 01, Charlottetown, P.E.I., August 8 10, 01 a 1 1 a 1 3 a 1 a 3 a 3 1 a 3 3 a 1 3 a 1 4 a 1 3 a 1 4 a 1 3 a 1 4 a 1 a 1 4 a a 3 a 4 a 3 3 a 3 4 a 1 a a 3 1 a 3 a 3 3 a 3 4 a 1 1 a 1 a 3 1 a 3 a 1 3 a 1 4 a 1 1 a 1 a 1 a a 1 3 a 1 4 a 3 a 4 a 4 a 3 a 3 4 a 1 1 a 1 a 1 1 a 1 a 1 1 a 1 Figure 1: A clique of 1 4-union-intervals satisfying the conitions of Lemma 5 with q = pigeonhole principle). Then the set C = {I : I H} is a set of intervals in R that are each hit by weight at least 1 uner {y p}. By multiplying solution {yp} by, we obtain a new fractional hitting set {yp} of weight ρ (H) that hits, with weight at least 1, all elements of C. However, the incience matrix for the hitting set problem on 1- intervals is totally unimoular, so there must exist an integral hitting set Q of weight at most ρ (H) that hits all of C one can be foun by simply solving linear program () again for C instea of H. Of course, Q is also a hitting set for H, from which it follows that ρ (H). By simply returning Q, we obtain a polynomial-time LP-relative -approximation for the -interval hitting set problem, completing the proof. We note that the above algorithm also works for the weighte variant of the minimum hitting set problem, in which each point p P (H) is given a positive cost, an the goal is to compute a minimum cost hitting set. Finally, we establish Theorem 3 by giving a set of - union-intervals with a hitting set integrality gap of : Proof of Theorem 3. Fix > 0. Choose any integer t an any integer n t. Fix some small δ, say δ = 0.1. Here, we regar the tracks {J 1,..., J } as isjoint copies of R. A -union-interval I is calle aligne if, for all 1 k, the piece of I in J k has the form [i k + δ, j k δ] for some integers 0 i k < j k n. In other wors, a -union-interval is aligne if the enpoints of all of its pieces each barely miss an integer point between 0 an n. Let H be the collection of all aligne -union-intervals I such that the total length of all pieces in I is exactly t δ. Note that H is finite. Let P contain all points of the form i for i {0, 1,..., n 1} in each of the tracks, for a total of n points. Each -union-interval in H must contain at least t points in P, so we can obtain a fractional hitting set of total weight n t by placing a value of 1 t at each point in P. This shows that ρ (H) n t. Let Q be any feasible integral hitting set for H. We Q wish to show that ρ (H), so we may assume that Q < n. Let b i be the number of points of Q in J i. By the pigeonhole principle, there must exist an open interval K i [0, n] in track J i that has integer enpoints, has length at least n bi b i+1, an contains no points in Q. Consequently, there is a -union-interval K = i=1 Ki having total length n b i i=1 b i+1 that has integer enpoints an is misse by Q in all tracks. However, Q hits all aligne -union-intervals having total length t δ, an K is misse by Q, so we must have i=1 n b i b i + 1 < t. By rearranging this, we obtain ( i=1 ) 1 1 (n + 1) > b i + 1 t +. The left sie of the above equation is a harmonic mean. Since an arithmetic mean is always greater than or equal to the corresponing harmonic mean, we get 1 (b i + 1) > i=1 (n + 1) t + an hence Q = i=1 b i > (n+1) t+. Diviing by the upper boun we ha for ρ (H) gives t(n + 1) ρ > (H) n(t + ) t ( n > 1 ) t t + n, where the last inequality is ue to n+1 n chose t an n such that t ) ( ρ (H) > 1 + t t completing the proof of Theorem. 3 Topology-base algorithms an n t > 1. Since we, we get = + >, In this section, we sketch a proof of Theorem 4, illustrating how to obtain a 3-approximation (respectively, a -approximation) for the inepenent set problem on -intervals (respectively, -union-intervals), supposing one has access to oracles for PPAD-complete topological subproblems. We assume familiarity with the complexity class PPAD an its connection to topological fixepoint theorems; see [15] for backgroun information. Our approach follows Kaiser s uality gap upper boun proof [10], which we outline here. We first consier the case of -union-intervals. For concreteness, we consier a family H of axis-aligne rectangles in the plane. Let n be an arbitrary positive integer. Kaiser consiers the space S n S n (where S n is the bounary of an (n + 1)-imensional unit ball), an associates each

6 4 th Canaian Conference on Computational Geometry, 01 point x S n S n to a set of n horizontal lines an n vertical lines in the plane. Kaiser then constructs a family of n + real-value functions h H 1,..., h H n+ on S n S n having the following properties: 1. If h H i (x) = 0 for all i, then x correspons to a set of lines that intersect all rectangles in H.. If h H 1 (x) = h H (x) =... = h H n+(x) 0 for all i, then x correspons to a set of n horizontal lines an n vertical lines efining a gri from which we can easily fin a conflict-free set of rectangles of size n + 1 in polynomial time. Kaiser then establishes that, for topological reasons, there must exist a point x S n S n such that h H 1 (x) = h H (x) =... = h H n+(x) an thus a point x must exist satisfying item 1 or item above. Taros s result that α(h) follows immeiately by setting n = α(h), since then a point where h H i (x) 0 for all i cannot exist, an thus a stabbing consisting of α(h) horizontal lines an α(h) vertical lines must exist. Kaiser s proof can easily be aapte to yiel a proceure P that, given an integer n, fins either a stabbing of size n or a conflict-free subset of size n + 1, using polynomial time plus a single call to an oracle for a topological fixe-point problem. To obtain a -approximation for the maximum conflict-free subset problem, it then suffices to fin a cutoff point t N such that P returns a conflict-free subset S of size t when run with n = t 1, but returns a stabbing of size t when run with n = t (if this happens, we have S α(h) ). Although there may be many cutoff points t, one must exist in the interval [, α(h)]. One can be foun using only O(log(α(H))) calls to P by running a galloping binary search that first tries t = 1, t =, t = 4,... until a stabbing of size t is returne, an then binary searches between t an t to fin a cutoff point. Kaiser s topological argument employs a result of Ramos that generalizes the Borsuk-Ulam theorem to cross proucts of spheres [16], so proceure P must invoke calls to an oracle for Ramos-style fixe-points. Ramos invokes a parity argument that can be aapte, in a straightforwar manner, to show that an appropriate computational version of the Ramos fixe-point problem lies in the complexity class PPAD. Inee, Ramos provies a searching algorithm to locate such fixe-points, although it may be exponential in the worst case. We also note that, by the iscreteness of our problem, the particular instances that we must solve can be efficiently represente an have rational solutions. This establishes Theorem 4 for -union-intervals. For the case of general -intervals, Kaiser provies a relate argument that can be aapte in the same manner to yiel a binary search algorithm. In this case, only an oracle for stanar Borsuk-Ulam fixe-points is require. However, ue to changes in how the functions h H i must be formulate, only a 3-approximation can be obtaine. Still, the integrality gap bouns imply that this is optimal among all LP-relative approximations. References [1] N. Alon. Piercing -intervals. Disc. Comp. Geom., 19(3, Special Issue): , [] R. Bar-Yehua, M. Hallórsson, J. Naor, H. Shachnai, an I. Shapira. Scheuling split intervals. SIAM J. Comput., 36(1):1 15 (electronic), 006. [3] G. Călinescu, A. Dumitrescu, H. Karloff, an P. Wan. Separating points by axis-parallel lines. Internat. J. Comp. Geom. Appl., 15(6): , 005. [4] M. Dom, M. Fellows, an F. Rosamon. Parameterize complexity of stabbing rectangles an squares in the plane. In Proc. WALCOM, LNCS 5341:98 309, 009. [5] A. Dumitrescu. On two lower boun constructions. In Proc. 11th Canaian Conf. Comp. Geom., [6] G. Even, R. Levi, D. Rawitz, S. Baruch, S. Shahar, an M. Svirienko. Algorithms for capacitate rectangle stabbing an lot sizing with joint set-up costs. ACM Trans. Algorithms, 4(3), 008. [7] D. Gaur, T. Ibaraki, an R. Krishnamurti. Constant ratio approximation algorithms for the rectangle stabbing problem an the rectilinear partitioning problem. J. Algorithms, 43(1): , 00. [8] A. Gyárfás an J. Lehel. A Helly-type problem in trees. Comb. Theory an its Appl., II, , [9] P.J-J. Herings an R. Peeters. Homotopy methos to compute equilibria in game theory. Economic Theory, 4(1): , 010. [10] T. Kaiser. Transversals of -intervals. Disc. Comp. Geom., 18():195 03, [11] S. Kovaleva an F. Spieksma. Primal-ual approximation algorithms for a packing-covering pair of problems. RAIRO Oper. Res., 36(1):53 71, 00. [1] S. Kovaleva an F. Spieksma. Approximation algorithms for rectangle stabbing an interval stabbing problems. SIAM J. Disc. Math., 0(3): , 006. [13] J. Matoušek. Lower bouns on the transversal numbers of -intervals. Disc. Comp. Geom., 6(3):83 87, 001. [14] H. Nagashima an K. Yamazaki. Harness of approximation for non-overlapping local alignments. Disc. Appl. Math., 137(3):93 309, 004. [15] C. Papaimitriou. On the complexity of the parity argument an other inefficient proofs of existence. J. Comput. System Sci., 48(3):498 53, [16] E. Ramos. Equipartition of mass istributions by hyperplanes. Disc. Comp. Geom., 15(): , [17] F. Spieksma. On the approximability of an interval scheuling problem. J. Sche., (5):15 7, [18] G. Taros. Transversals of -intervals, a topological approach. Combinatorica, 15(1):13 134, [19] V. Vatter. Small permutation classes. In Proc. Lon. Math. Soc. (3), 103(5):879-91, 011.