Parameterized Complexity of Stabbing Rectangles and Squares in the Plane

Abstract

Abstract. The NP-complete geometric covering problem Rectangle Stabbing is defined as follows: Given a set of horizontal and vertical lines in the plane, a set of rectangles in the plane, and a positive integer k, select at most k of the lines such that every rectangle is intersected by at least one of the selected lines. While it is known that the problem can be approximated in polynomial time with a factor of two, its parameterized complexity with respect to the parameter k was open so far-only its generalization to three or more dimensions was known to be W[1]-hard. Giving two fixed-parameter reductions, one from the W[1]-complete problem Multicolored Clique and one to the W[1]-complete problem Short Turing Machine Acceptance, we prove that Rectangle Stabbing is W[1]-complete with respect to the parameter k, which in particular means that there is no hope for fixed-parameter tractability with respect to the parameter k. Our reductions show also the W[1]-completeness of the more general problem Set Cover on instances that "almost have the consecutiveones property", that is, on instances whose matrix representation has at most two blocks of 1s per row. For the special case of Rectangle Stabbing where all rectangles are squares of the same size we can also show W[1]-hardness, while the parameterized complexity of the special case where the input consists of rectangles that do not overlap is open. By giving an algorithm running in (4k + 1) k · n O(1) time, we show that Rectangle Stabbing is fixedparameter tractable in the still NP-hard case where both these restrictions apply.

Fixed-Parameter Algorithms for Cochromatic Number and Disjoint Rectangle Stabbing

“…Using this new combination, we are also able to resolve another open question [11,12], namely whether Disjoint Rectangle Stabbing is fixed parameter tractable.…”

“…In [12], the authors prove a W[1]-hardness result for a higher dimensional version of the Rectangle Stabbing problem and show several restrictions of this two dimensional version fixed-parameter tractable. Recently, Dom et al [11] and Giannopoulos et al [22] independently considered the general two dimensional version, the Rectangle Stabbing problem, and showed it to be complete for the parameterized complexity class W [1]. Dom et al [11] also showed that the restriction of Disjoint Rectangle Stabbing to b-Disjoint Square Stabbing, when all the rectangles are squares of the size b × b, is NP-complete for any fixed b ≥ 2.…”

“…Recently, Dom et al [11] and Giannopoulos et al [22] independently considered the general two dimensional version, the Rectangle Stabbing problem, and showed it to be complete for the parameterized complexity class W [1]. Dom et al [11] also showed that the restriction of Disjoint Rectangle Stabbing to b-Disjoint Square Stabbing, when all the rectangles are squares of the size b × b, is NP-complete for any fixed b ≥ 2. Furthermore, both Dom et al [11] and Giannopoulos et al [22] considered the b-Disjoint Square Stabbing problem for a fixed b and show it to be fixed parameter tractable.…”

“…Dom et al [11] also showed that the restriction of Disjoint Rectangle Stabbing to b-Disjoint Square Stabbing, when all the rectangles are squares of the size b × b, is NP-complete for any fixed b ≥ 2. Furthermore, both Dom et al [11] and Giannopoulos et al [22] considered the b-Disjoint Square Stabbing problem for a fixed b and show it to be fixed parameter tractable. All these papers leave the parameterized complexity of the Disjoint Rectangle Stabbing problem open.…”

“…In [11], it is observed that the version of the Rectangle Stabbing problem, where we are given only the set of rectangles and we need to find at most k horizontal and vertical lines to stab them, polynomially reduces to the version where we are also given a set of horizontal and vertical lines. Hence this version of the Disjoint Rectangle Stabbing problem (where we are not given a set of lines) is also fixed parameter tractable.…”