The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+eps)-approximations in f(k,eps)n^ O(1) time where k is some parameter of the input. We obtain the following results on parameterized approximability: 1) In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA'05]. The best-known polynomial-time approximation factor is O(loglog n) [Chalermsook and Chuzhoy, SODA'09] and it admits a QPTAS [Adamaszek and Wiese, FOCS'13; Chuzhoy and Ene, FOCS'16]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant eps>0 and integer k>0, in time f(k,eps)n^ g(eps) , either outputs a solution of size at least k (1+eps), or declares that the optimum solution has size less than k. 2) In the (2-dimensional) geometric knapsack problem (TDK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of TDK with rotations (TDKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factors are 558 325+eps and 4 3+eps, resp. [, FOCS'17]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA'15]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for TDKR. For all considered problems, getting time f(k,eps)n^ O(1) , rather than f(k,eps)n^ g(eps) , would give FPT time f'(k)n^ O(1) exact algorithms using eps=1 (k+1), contradicting W[1]-hardness.
One of the first fruitful connections between parameterized complexity and approximability was observed independently by Bazgan @cite_23 and Cesati and Trevisan @cite_29 : They showed that EPTASs, i.e., @math -approximation algorithms with @math time, imply fixed-parameter tractability for the decision version. Thus, proofs for 1 -hardness of the decision version became a strong tool for ruling out improvements of PTASs, with running time @math , to EPTASs. More recently, @cite_25 improved this approach by directly proving 1 -hardness of obtaining a @math -approximation, thus bypassing the requirement of a 1 -hard decision version (see also @cite_33 ).
@cite_29 @cite_33 @cite_25 @cite_23
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par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . We consider the @math -Center problem and some generalizations. For @math -Center a set of @math center vertices needs to be found in a graph @math with edge lengths, such that the distance from any vertex of @math to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by [SODA 2010]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations, where the two paradigms are combined. In particular, we prove that for any @math computing a @math -approximation is W[2]-hard for parameter @math and NP-hard for graphs with highway dimension @math . The latter does not rule out fixed-parameter @math -approximations for the highway dimension parameter @math , but implies that such an algorithm must have at least doubly exponential running time in @math if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor of @math by combining the parameters @math and @math : we develop a fixed-parameter @math -approximation with running time @math . Additionally we prove that, unless P=NP, our techniques cannot be used to compute fixed-parameter @math -approximations for only the parameter @math . We also provide similar fixed-parameter approximations for the weighted @math -Center and @math -Partition problems, which generalize @math -Center. A Fixed-Parameter Tractable ( ) @math -approximation algorithm for a minimization (resp. maximization) parameterized problem @math is an FPT algorithm that, given an instance @math computes a solution of cost at most @math (resp. @math ) if a solution of cost at most (resp. at least) @math exists; otherwise the output can be arbitrary. For well-known intractable problems such as the W[1]-hard Clique and W[2]-hard Set Cover problems, the natural question is whether we can get any -approximation. It is widely believed that both Clique and Set-Cover admit no FPT @math -approximation algorithm, for any increasing function @math . Assuming standard conjectures such as the Exponential Time Hypothesis (ETH) eth-paturi and the Projection Games Conjecture (PGC) r3 , we make the first progress towards proving this conjecture by showing that 1. Under the ETH and PGC, there exist constants @math such that the Set Cover problem does not admit an FPT approximation algorithm with ratio @math in @math time, where @math is the size of the universe and @math is the number of sets. 2. Unless @math , for every @math there exists a constant @math such that Clique has no FPT cost approximation with ratio @math in @math time, where @math is the number of vertices in the graph. In the second part of the paper we consider various W[1]-hard problems such as , , Directed Steiner Network and . For all these problem we give polynomial time @math -approximation algorithms for some small function @math (the largest approximation ratio we give is @math ). We consider the k-Center problem and some generalizations. For k-Center a set of kcenter vertices needs to be found in a graph G with edge lengths, such that the distance from any vertex ofi¾?G to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by [ICALP 2011]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations. In particular, we prove that for any @math computing a @math -approximation is W[2]-hard for parameter k, and NP-hard for graphs with highway dimension @math . The latter does not rule out fixed-parameter @math -approximations for the highway dimension parameteri¾?h, but implies that such an algorithm must have at least doubly exponential running time in h if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor ofi¾?2 by combining the parameters k andi¾?h: we develop a fixed-parameter 3 2-approximation with running time @math . We also provide similar fixed-parameter approximations for the weightedk-Center and @math -Partition problems, which generalize k-Center.