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. Stable Marriage is a fundamental problem to both computer science and economics. Four well-known NP-hard optimization versions of this problem are the Sex-Equal Stable Marriage (SESM), Balanced Stable Marriage (BSM), max-Stable Marriage with Ties (max-SMT) and min-Stable Marriage with Ties (min-SMT) problems. In this paper, we analyze these problems from the viewpoint of Parameterized Complexity. We conduct the first study of these problems with respect to the parameter treewidth. First, we study the treewidth @math of the primal graph. We establish that all four problems are W[1]-hard. In particular, while it is easy to show that all four problems admit algorithms that run in time @math , we prove that all of these algorithms are likely to be essentially optimal. Next, we study the treewidth @math of the rotation digraph. In this context, the max-SMT and min-SMT are not defined. For both SESM and BSM, we design (non-trivial) algorithms that run in time @math . Then, for both SESM and BSM, we also prove that unless SETH is false, algorithms that run in time @math do not exist for any fixed @math . We thus present a comprehensive, complete picture of the behavior of central optimization versions of Stable Marriage with respect to treewidth. We prove the equivalence, up to a small polynomial approximation factor @math , of the lattice problems uSVP (unique Shortest Vector Problem), BDD (Bounded Distance Decoding) and GapSVP (the decision version of the Shortest Vector Problem). This resolves a long-standing open problem about the relationship between uSVP and the more standard GapSVP, as well the BDD problem commonly used in coding theory. The main cryptographic application of our work is the proof that the Ajtai-Dwork ([2]) and the Regev ([33]) cryptosystems, which were previously only known to be based on the hardness of uSVP, can be equivalently based on the hardness of worst-case GapSVP @math and GapSVP @math , respectively. Also, in the case of uSVP and BDD, our connection is very tight, establishing the equivalence (within a small constant approximation factor) between the two most central problems used in lattice based public key cryptography and coding theory. The Degree Anonymity problem arises in the context of combinatorial graph anonymization. It asks, given a graph (G ) and two integers (k ) and (s ), whether (G ) can be made k-anonymous with at most (s ) modifications. Here, a graph is k-anonymous if the graph contains for every vertex at least (k-1 ) other vertices of the same degree. Complementing recent investigations on its computational complexity, we show that this problem is very hard when studied from the viewpoints of approximation as well as parameterized approximation. In particular, for the optimization variant where one wants to minimize the number of either edge or vertex deletions there is no factor- (n^ 1- ) approximation running in polynomial time unless P = NP, for any constant (0 < 1 ). For the variant where one wants to maximize (k ) and the number (s ) of either edge or vertex deletions is given, there is no factor- (n^ 1 2 - ) approximation running in time (f(s) n^ O(1) ) unless W[1] = FPT, for any constant (0 < 1 2 ) and any function (f ). On the positive side, we classify the general decision version as fixed-parameter tractable with respect to the combined parameter solution size (s ) and maximum degree. We study undirected networks with edge costs that satisfy the triangle inequality. Let @math denote the number of nodes. We present an @math -approximation algorithm for a generalization of the metric-cost subset @math -node-connectivity problem. Our approximation guarantee is proved via lower bounds that apply to the simple edge-connectivity version of the problem, where the requirements are for edge-disjoint paths rather than for openly node-disjoint paths. A corollary is that, for metric costs and for each @math , there exists a @math -node connected graph whose cost is within a factor of @math of the cost of any simple @math -edge connected graph. Based on our @math -approximation algorithm, we present an @math -approximation algorithm for the metric-cost node-connectivity survivable network design problem, where @math denotes the maximum requirement over all pairs of nodes. Our results contrast with the case of edge costs of 0 or 1, where Kortsarz, Krauthgamer, and Lee. [SIAM J. Comput., 33 (2004), pp. 704-720] recently proved, assuming NP @math DTIME( @math ), a hardness-of-approximation lower bound of @math for the subset @math -node-connectivity problem, where @math denotes a small positive number. We develop a parameterized complexity theory for counting problems. As the basis of this theory, we introduce a hierarchy of parameterized counting complexity classes #W @math , for @math , that corresponds to Downey and Fellows's W-hierarchy [R. G. Downey and M. R. Fellows, Parameterized Complexity, Springer-Verlag, New York, 1999] and we show that a few central W-completeness results for decision problems translate to #W-completeness results for the corresponding counting problems. Counting complexity gets interesting with problems whose decision version is tractable, but whose counting version is hard. Our main result states that counting cycles and paths of length k in both directed and undirected graphs, parameterized by k, is #W @math f(k) n^c @math f: N N @math 2^ O(k) n^ 2.376 $ algorithm for finding a cycle or path of length k [N. Alon, R. Yuster, and U. Zwick, J. ACM, 42 (1995), pp. 844--856]. Given two graphs @math and @math , the Subgraph Isomorphism problem asks if @math is isomorphic to a subgraph of @math . While NP-hard in general, algorithms exist for various parameterized versions of the problem: for example, the problem can be solved (1) in time @math using the color-coding technique of Alon, Yuster, and Zwick; (2) in time @math using Courcelle's Theorem; (3) in time @math using a result on first-order model checking by Frick and Grohe; or (4) in time @math for connected @math using the algorithm of Matou s ek and Thomas. Already this small sample of results shows that the way an algorithm can depend on the parameters is highly nontrivial and subtle. We develop a framework involving 10 relevant parameters for each of @math and @math (such as treewidth, pathwidth, genus, maximum degree, number of vertices, number of components, etc.), and ask if an algorithm with running time [ f_1(p_1,p_2,..., p_ ) n^ f_2(p_ +1 ,..., p_k) ] exist, where each of @math is one of the 10 parameters depending only on @math or @math . We show that all the questions arising in this framework are answered by a set of 11 maximal positive results (algorithms) and a set of 17 maximal negative results (hardness proofs); some of these results already appear in the literature, while others are new in this paper. On the algorithmic side, our study reveals for example that an unexpected combination of bounded degree, genus, and feedback vertex set number of @math gives rise to a highly nontrivial algorithm for Subgraph Isomorphism. On the hardness side, we present W[1]-hardness proofs under extremely restricted conditions, such as when @math is a bounded-degree tree of constant pathwidth and @math is a planar graph of bounded pathwidth. We study the inapproximability of Vertex Cover and Independent Set on degree @math graphs. We prove that: Vertex Cover is Unique Games-hard to approximate to within a factor @math . This exactly matches the algorithmic result of Halperin halperin02improved up to the @math term. Independent Set is Unique Games-hard to approximate to within a factor @math . This improves the @math Unique Games hardness result of Samorodnitsky and Trevisan samorodnitsky06gowers . Additionally, our result does not rely on the construction of a query efficient PCP as in samorodnitsky06gowers . A set @math of points on a terrain, also known as an @math -monotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in @math . Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem asks for a minimum guarding set for the given input terrain. Using a reduction from PLANAR 3-SAT we prove that the decision version of this problem is NP-hard. This solves a significant open problem and complements recent positive approximability results for the optimization problem. We consider the over-time version of the Max-Min Fair Allocation problem. Given a time horizon @math , with at each time t a set of demands and a set of available resources that may change over the time defining instance @math , we seek a sequence of solutions @math that (1) are near-optimal at each time t , and (2) as stable as possible (inducing small modification costs). We focus on the impact of the knowledge of the future on the quality and the stability of the returned solutions by distinguishing three settings: the off-line setting where the whole set of instances through the time horizon is known in advance, the on-line setting where no future instance is known, and the k -lookahead setting where at time t , the instances at times @math are known. We first consider the case without restrictions where the set of resources and the set of agents are the same for all instances and where every resource can be allocated to any agent. For the off-line setting, we show that the over-time version of the problem is much harder than the static one, since it becomes @math -hard even for families of instances for which the static problem is trivial. Then, we provide a @math -approximation algorithm for the off-line setting using as subroutine a ρ-approximation algorithm for the static version. We also give a @math -competitive algorithm for the online setting using also as subroutine a ρ-approximation algorithm for the static version. Furthermore, for the case with restrictions, we show that in the off-line setting it is possible to get a polynomial-time algorithm with the same approximation ratio as in the case without restrictions. For the online setting, we prove that it is not possible to find an online algorithm with bounded competitive ratio. For the 1-lookahead setting however, we give a @math -approximation algorithm using as subroutine a ρ-approximation algorithm for the static version. We study the broadcast version of the CONGEST CLIQUE model of distributed computing. In this model, in each round, any node in a network of size @math can send the same message (i.e. broadcast a message) of limited size to every other node in the network. Nanongkai presented in [STOC'14] a randomized @math -approximation algorithm to compute all pairs shortest paths (APSP) in time @math on weighted graphs, where we use the convention that @math is essentially @math polylog @math and @math is essentially @math polylog @math . We complement this result by proving that any randomized @math -approximation of APSP and @math -approximation of the diameter of a graph takes @math time in the worst case. This demonstrates that getting a negligible improvement in the approximation factor requires significantly more time. Furthermore this bound implies that already computing a @math -approximation of all pairs shortest paths is among the hardest graph-problems in the broadcast-version of the CONGEST CLIQUE model and contrasts a recent @math -approximation for APSP that runs in time @math in the unicast version of the CONGEST CLIQUE model. On the positive side we provide a deterministic version of Nanongkai's @math -approximation algorithm for APSP. To do so we present a fast deterministic construction of small hitting sets. We also show how to replace another randomized part within Nanongkai's algorithm with a deterministic source-detection algorithm designed for the CONGEST model presented by Lenzen and Peleg at PODC'13. In this paper, we consider the problem of maximizing the spread of influence through a social network. Given a graph with a threshold value thr(v) attached to each vertex v, the spread of influence is modeled as follows: A vertex v becomes ''active'' (influenced) if at least thr(v) of its neighbors are active. In the corresponding optimization problem the objective is then to find a fixed number k of vertices to activate such that the number of activated vertices at the end of the propagation process is maximum. We show that this problem is strongly inapproximable in time f(k)@?n^O^(^1^), for some function f, even for very restrictive thresholds. In the case that the threshold of each vertex equals its degree, we prove that the problem is inapproximable in polynomial time and it becomes r(n)-approximable in time f(k)@?n^O^(^1^), for some function f, for any strictly increasing function r. Moreover, we show that the decision version parameterized by k is W[1]-hard but becomes fixed-parameter tractable on bounded degree graphs. Given a directed graph @math and a list @math of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of @math that contains a directed @math path for every @math . The special case Directed Steiner Tree (when we ask for paths from a root @math to terminals @math ) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every @math to every other @math ) is known to be W[1]-hard. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if @math is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list @math of requests form a directed graph that is a member of @math . Our main result is a complete characterization of the classes @math resulting in fixed-parameter tractable special cases: we show that if every pattern in @math has the combinatorial property of being "transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges," then the problem is FPT, and it is W[1]-hard for every recursively enumerable @math not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], @math -Root Steiner Tree is FPT for constant @math [Such 'y, WG 2016], Strongly Connected Steiner Subgraph is W[1]-hard [, SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before. We obtain a 1.5-approximation algorithm for the metric uncapacitated facility location problem (UFL), which improves on the previously best known 1.52-approximation algorithm by Mahdian, Ye and Zhang. Note, that the approximability lower bound by Guha and Khuller is 1.463. An algorithm is a ( @math , @math )-approximation algorithm if the solution it produces has total cost at most @math , where @math and @math are the facility and the connection cost of an optimal solution. Our new algorithm, which is a modification of the @math -approximation algorithm of Chudak and Shmoys, is a (1.6774,1.3738)-approximation algorithm for the UFL problem and is the first one that touches the approximability limit curve @math established by Jain, Mahdian and Saberi. As a consequence, we obtain the first optimal approximation algorithm for instances dominated by connection costs. When combined with a (1.11,1.7764)-approximation algorithm proposed by , and later analyzed by , we obtain the overall approximation guarantee of 1.5 for the metric UFL problem. We also describe how to use our algorithm to improve the approximation ratio for the 3-level version of UFL. We develop a parameterized complexity theory for counting problems. As the basis of this theory, we introduce a hierarchy of parameterized counting complexity classes #W[t], for t spl ges 1, that corresponds to Downey and Fellows' (1999) W-hierarchy and show that a few central W-completeness results for decision problems translate to #W-completeness results for the corresponding counting problems. Counting complexity gets interesting with problems whose decision version is tractable, but whose counting version is hard. Our main result states that counting cycles and paths of length k in both directed and undirected graphs, parameterized by k, are #W[1]-complete. This makes it highly unlikely that any of these problems is fixed-parameter tractable, even though their decision versions are. More explicitly, our result shows that most likely there is no f(k) spl middot n sup c -algorithm for counting cycles or paths of length k in a graph of size n for any computable function f: spl Nopf spl rarr spl Nopf and constant c, even though there is a 2 sup O(k) spl middot n sup 2.376 algorithm for finding a cycle or path of length k (2). In [Proceedings of the Second Symposium on Switching Circuit Theory and Logical Design (FOCS), 1961, pp. 34-38], Chow proved that every Boolean threshold function is uniquely determined by its degree-0 and degree-1 Fourier coefficients. These numbers became known as the Chow parameters. Providing an algorithmic version of Chow's theorem—i.e., efficiently constructing a representation of a threshold function given its Chow parameters—has remained open ever since. This problem has received significant study in the fields of circuit complexity, game theory and the design of voting systems, and learning theory. In this paper we effectively solve the problem, giving a randomized polynomial-time approximation scheme with the following behavior: Given the Chow parameters of a Boolean threshold function @math over @math bits and any constant @math , the algorithm runs in time @math and with high probability outputs a representation of a threshold function @math which is @math -close to @math . Along the way we prove several new results of independent interest about Boolean threshold functions. In addition to various structural results, these include @math -time learning algorithms for threshold functions under the uniform distribution in the following models: (i) the restricted focus of attention model, answering an open question of ; (ii) an agnostic-type model. This contrasts with recent results of Guruswami and Raghavendra who show NP-hardness for the problem under general distributions; (iii) the PAC model, with constant @math . Our @math -time algorithm substantially improves on the previous best known running time and nearly matches the @math bits of training data that any successful learning algorithm must use. Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer @math and arbitrary small @math , the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor @math even on graphs where the vertices can be almost partitioned into @math solutions. This gives a more structured and therefore stronger UGC-based hardness result for the Feedback Vertex Set problem that is also simpler (albeit using the "It Ain't Over Till It's Over" theorem) than the previous hardness result. In comparison to the classical Feedback Vertex Set problem, the DAG Vertex Deletion problem has received little attention and, although we think it is a natural and interesting problem, the main motivation for our inapproximability result stems from its relationship with the classical Discrete Time-Cost Tradeoff Problem. More specifically, our results imply that the deadline version is NP-hard to approximate within any constant assuming the Unique Games Conjecture. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations.