Every triangle-free planar graph on n vertices has an independent set of size at least (n+1) 3, and this lower bound is tight. We give an algorithm that, given a triangle-free planar graph G on n vertices and an integer k>=0, decides whether G has an independent set of size at least (n+k) 3, in time 2^ O(sqrt k ) n. Thus, the problem is fixed-parameter tractable when parameterized by k. Furthermore, as a corollary of the result used to prove the correctness of the algorithm, we show that there exists epsilon>0 such that every planar graph of girth at least five on n vertices has an independent set of size at least n (3-epsilon).
From the algorithmic side, the studying the complexity of maximization problems parameterized above polynomial-time computable lower bounds is an active area of research. Since the influental survey by @cite_2 , research in this area has led to development of many new algorithmic techniques for fixed-parameter algorithms: algebraic methods @cite_17 @cite_30 , probabilistic methods @cite_35 @cite_24 , combinatorial methods @cite_9 @cite_5 , and methods based on linear programming @cite_13 .
@cite_30 @cite_35 @cite_9 @cite_24 @cite_2 @cite_5 @cite_13 @cite_17
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We consider new parameterizations of NP-optimization problems that have nontrivial lower and or upper bounds on their optimum solution size. The natural parameter, we argue, is the quantity above the lower bound or below the upper bound. We show that for every problem in MAX SNP, the optimum value is bounded below by an unbounded function of the input-size, and that the above-guarantee parameterization with respect to this lower bound is fixed-parameter tractable. We also observe that approximation algorithms give nontrivial lower or upper bounds on the solution size and that the above or below guarantee question with respect to these bounds is fixed-parameter tractable for a subclass of NP-optimization problems. We then introduce the notion of 'tight' lower and upper bounds and exhibit a number of problems for which the above-guarantee and below-guarantee parameterizations with respect to a tight bound is fixed-parameter tractable or W-hard. We show that if we parameterize ''sufficiently'' above or below the tight bounds, then these parameterized versions are not fixed-parameter tractable unless P=NP, for a subclass of NP-optimization problems. We also list several directions to explore in this paradigm. We develop new techniques for deriving strong computational lower bounds for a class of well-known NP-hard problems. This class includes weighted satisfiability, dominating set, hitting set, set cover, clique, and independent set. For example, although a trivial enumeration can easily test in time O(n^k) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the problem is not solvable in time f(k)n^o^(^k^) for any function f, even if we restrict the parameter values to be bounded by an arbitrarily small function of n. Under the same assumption, we prove that even if we restrict the parameter values k to be of the order @Q(@m(n)) for any reasonable function @m, no algorithm of running time n^o^(^k^) can test if a graph of n vertices has a clique of size k. Similar strong lower bounds on the computational complexity are also derived for other NP-hard problems in the above class. Our techniques can be further extended to derive computational lower bounds on polynomial time approximation schemes for NP-hard optimization problems. For example, we prove that the NP-hard distinguishing substring selection problem, for which a polynomial time approximation scheme has been recently developed, has no polynomial time approximation schemes of running time f(1 @e)n^o^(^1^ ^@e^) for any function f unless an unlikely collapse occurs in parameterized complexity theory. We develop new techniques for deriving very strong computational lower bounds for a class of well-known NP-hard problems, including weighted satisfiability , dominating set , hitting set , set cover , clique , and independent set . For example, although a trivial enumeration can easily test in time O(nk) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the problem is not solvable in time f(k) no(k) for any function f, even if we restrict the parameter value k to be bounded by an arbitrarily small function of n. Under the same assumption, we prove that even if we restrict the parameter values k to be Θ(μ(n)) for any reasonable function μ, no algorithm of running time no(k) can test if a graph of n vertices has a clique of size k. Similar strong lower bounds are also derived for other problems in the above class. Our techniques can be extended to derive computational lower bounds on approximation algorithms for NP-hard optimization problems. For example, we prove that the NP-hard distinguishing substring selection problem, for which a polynomial time approximation scheme has been recently developed, has no polynomial time approximation schemes of running time f(1 e)no(1 e) for any function f unless an unlikely collapse occurs in parameterized complexity theory. We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2O(k2) whether a given set of m clauses of size r admits a truth assignment that satisfies at least ((2r - 1)m + k) 2r clauses. Thus Max-r-Sat is fixed-parameter tractable when parameterized by the number of satisfied clauses above the tight lower bound (1--2-r)m. This solves an open problem of Mahajan, Raman and Sikdar (J. Comput. System Sci., 75, 2009). Our algorithm is based on a polynomial-time data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with O(k2) variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size O(k2), then there is a truth assignment satisfying the required number of clauses. Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of Max-2-Sat with m clauses has at least 3k variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least (3m + k) 4 clauses. We also outline how the fixed-parameter tractability result on Max-r-Sat can be extended to a family of Boolean Constraint Satisfaction Problems. We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math . We study the boundary of tractability for the Max-Cut problem in graphs. Our main result shows that Max-Cut above the Edwards-Erd o s bound is fixed-parameter tractable: we give an algorithm that for any connected graph with n vertices and m edges finds a cut of size m 2 + (n-1) 4 + k in time 2^O(k)n^4, or decides that no such cut exists. This answers a long-standing open question from parameterized complexity that has been posed several times over the past 15 years. Our algorithm is asymptotically optimal, under the Exponential Time Hypothesis, and is strengthened by a polynomial-time computable kernel of polynomial size. We look at the hypothesis that all honest onto polynomial-time computable functions have a polynomial-time computable inverse. We show this hypothesis equivalent to several other complexity conjectures including: • In polynomial time, one can find accepting paths of nondeterministic polynomial-time Turing machines that accept Σ*. • Every total multivalued nondeterministic function has a polynomial-time computable refinement. • In polynomial time, one can compute satisfying assignments for any polynomial-time computable set of satisfiable formulae. • In polynomial time, one can convert the accepting computations of any nondeterministic Turing machine that accepts SAT to satisfying assignments.We compare these hypotheses with several other important complexity statements. We also examine the complexity of these statements where we only require a single bit instead of the entire inverse. We present an information-theoretic approach to lower bound the oracle complexity of nonsmooth black box convex optimization, unifying previous lower bounding techniques by identifying a combinatorial problem, namely string guessing, as a single source of hardness. As a measure of complexity, we use distributional oracle complexity, which subsumes randomized oracle complexity as well as worst case oracle complexity. We obtain strong lower bounds on distributional oracle complexity for the box @math , as well as for the @math -ball for @math (for both low-scale and large-scale regimes), matching worst case upper bounds, and hence we close the gap between distributional complexity, and in particular, randomized complexity and worst case complexity. Furthermore, the bounds remain essentially the same for high-probability and bounded-error oracle complexity, and even for combination of the two, i.e., bounded-error high-probability oracle complexity. This considerably extends the applicability of known bounds. Starting with Cook's pioneering work on NP-completeness in 1970, polynomial complexity theory, the study of polynomial-time com putability, has quickly emerged as the new foundation of algorithms. On the one hand, it bridges the gap between the abstract approach of recursive function theory and the concrete approach of analysis of algorithms. It extends the notions and tools of the theory of computability to provide a solid theoretical foundation for the study of computational complexity of practical problems. In addition, the theoretical studies of the notion of polynomial-time tractability some times also yield interesting new practical algorithms. A typical exam ple is the application of the ellipsoid algorithm to combinatorial op timization problems (see, for example, Lovasz [1986]). On the other hand, it has a strong influence on many different branches of mathe matics, including combinatorial optimization, graph theory, number theory and cryptography. As a consequence, many researchers have begun to re-examine various branches of classical mathematics from the complexity point of view. For a given nonconstructive existence theorem in classical mathematics, one would like to find a construc tive proof which admits a polynomial-time algorithm for the solution. One of the examples is the recent work on algorithmic theory of per mutation groups. In the area of numerical computation, there are also two tradi tionally independent approaches: recursive analysis and numerical analysis." We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning, operations research and pattern recognition. In the first class (Min ULR) one wishes, given a possibly infeasible system of linear relations, to find a solution that violates as few relations as possible while satisfying all the others. In the second class (Min RVLS) the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible. For both Min ULR and Min RVLS the four basic types of relational operators =, ⩾, > and ≠ are considered. While Min RVLS with equations was mentioned to be NP-hard in (Garey and Johnson, 1979), we established in (Amaldi; 1992; Amaldi and Kann, 1995) that min ULR with equalities and inequalities are NP-hard even when restricted to homogeneous systems with bipolar coefficients. The latter problems have been shown hard to approximate in (, 1993). In this paper we determine strong bounds on the approximability of various variants of Min RVLS and min ULR, including constrained ones where the variables are restricted to take binary values or where some relations are mandatory while others are optional. The various NP-hard versions turn out to have different approximability properties depending on the type of relations and the additional constraints, but none of them can be approximated within any constant factor, unless P = NP. Particular attention is devoted to two interesting special cases that occur in discriminant analysis and machine learning. In particular, we disprove a conjecture of van Horn and Martinez (1992) regarding the existence of a polynomial-time algorithm to design linear classifiers (or perceptrons) that involve a close-to-minimum number of features. This paper concerns the worst-case complexity of Gauss-Seidel method for solving a positive semi-definite linear system; or equivalently, that of cyclic coordinate descent (C-CD) for minimizing a convex quadratic function. The known provable complexity of C-CD can be @math times slower than gradient descent (GD) and @math times slower than randomized coordinate descent (R-CD). However, these gaps seem rather puzzling since so far they have not been observed in practice; in fact, C-CD usually converges much faster than GD and sometimes comparable to R-CD. Thus some researchers believe the gaps are due to the weakness of the proof, but not that of the C-CD algorithm itself. In this paper we show that the gaps indeed exist. We prove that there exists an example for which C-CD takes at least @math or @math operations, where @math is the condition number, @math is a well-studied quantity that determines the convergence rate of R-CD, and @math hides the dependency on @math . This implies that C-CD can indeed be @math times slower than GD, and @math times slower than R-CD in the worst case. Our result establishes one of the few examples in continuous optimization that demonstrates a large gap between a deterministic algorithm and its randomized counterpart. Based on the example, we establish several almost tight complexity bounds of C-CD for quadratic problems. One difficulty with the analysis of the constructed example is that the spectral radius of a non-symmetric iteration matrix does not necessarily constitutes a lower bound for the convergence rate. Recently, hardness results for problems in P were achieved using reasonable complexity-theoretic assumptions such as the Strong Exponential Time Hypothesis. According to these assumptions, many graph-theoretic problems do not admit truly subquadratic algorithms. A central technique used to tackle the difficulty of the above-mentioned problems is fixed-parameter algorithms with polynomial dependency in the fixed parameter (P-FPT). Applying this technique to clique-width, an important graph parameter, remained to be done. In this article, we study several graph-theoretic problems for which hardness results exist such as cycle problems, distance problems, and maximum matching. We give hardness results and P-FPT algorithms, using clique-width and some of its upper bounds as parameters. We believe that our most important result is an algorithm in O(k4 ⋅ n + m)-time for computing a maximum matching, where k is either the modular-width of the graph or the P4-sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs. Our algorithms are based on preprocessing methods using modular decomposition and split decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width. This paper explains when and how communication and computational lower bounds for algorithms for an optimization problem translate to lower bounds on the worst-case quality of equilibria in games derived from the problem. We give three families of lower bounds on the quality of equilibria, each motivated by a different set of problems: congestion, scheduling, and distributed welfare games, welfare-maximization in combinatorial auctions with "black-box" bidder valuations, and welfare-maximization in combinatorial auctions with succinctly described valuations. The most straightforward use of our lower bound framework is to harness an existing computational or communication lower bound to derive a lower bound on the worst-case price of anarchy (POA) in a class of games. This is a new approach to POA lower bounds, which relies on reductions in lieu of explicit constructions. More generally, the POA lower bounds implied by our framework apply to all classes of games that share the same underlying optimization problem, independent of the details of players' utility functions. For this reason, our lower bounds are particularly significant for problems of game design--ranging from the design of simple combinatorial auctions to the computation of tolls for routing networks--where the goal is to design a game that has only near-optimal equilibria. For example, our results imply that the simultaneous first-price auction format is optimal among all "simple combinatorial auctions" in several settings. In this paper we study general @math l p regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of the first- and second-order stationary points and hence also of local minimizers of the @math l p minimization problems. We extend some existing iterative reweighted @math l 1 ( @math IRL 1 ) and @math l 2 ( @math IRL 2 ) minimization methods to solve these problems and propose new variants for them in which each subproblem has a closed-form solution. Also, we provide a unified convergence analysis for these methods. In addition, we propose a novel Lipschitz continuous @math ∈ -approximation to @math ? x ? p p . Using this result, we develop new @math IRL 1 methods for the @math l p minimization problems and show that any accumulation point of the sequence generated by these methods is a first-order stationary point, provided that the approximation parameter @math ∈ is below a computable threshold value. This is a remarkable result since all existing iterative reweighted minimization methods require that @math ∈ be dynamically updated and approach zero. Our computational results demonstrate that the new @math IRL 1 method and the new variants generally outperform the existing @math IRL 1 methods (Chen and Zhou in 2012; Foucart and Lai in Appl Comput Harmon Anal 26:395---407, 2009). We investigate the existence of approximation algorithms for maximization of submodular functions, that run in a fixed parameter tractable (FPT) time. Given a non-decreasing submodular set function v:2 X →R v : 2 X → R the goal is to select a subset S of K elements from X such that v(S) v ( S ) is maximized. We identify three properties of set functions, referred to as p -separability properties, and we argue that many real-life problems can be expressed as maximization of submodular, p -separable functions, with low values of the parameter p . We present FPT approximation schemes for the minimization and maximization variants of the problem, for several parameters that depend on characteristics of the optimized set function, such as p and K . In this paper we study the worst-case complexity of an inexact Augmented Lagrangian method for nonconvex inequality-constrained problems. Assuming that the penalty parameters are bounded, we prove a complexity bound of @math iterations for the referred algorithm generate an @math -approximate KKT point, for @math . When the penalty parameters are unbounded, we prove an iteration complexity bound of @math , where @math controls the rate of increase of the penalty parameters. For linearly constrained problems, these bounds yield to evaluation complexity bounds of @math and @math , respectively, when suitable @math -order methods ( @math ) are used to approximately solve the unconstrained subproblems at each iteration of our Augmented Lagrangian scheme. Combining the techniques of approximation algorithms and parameterized complexity has long been considered a promising research area, but relatively few results are currently known. In this paper we study the parameterized approximability of a number of problems which are known to be hard to solve exactly when parameterized by treewidth or clique-width. Our main contribution is to present a natural randomized rounding technique that extends well-known ideas and can be used for both of these widths. Applying this very generic technique we obtain approximation schemes for a number of problems, evading both polynomial-time inapproximability and parameterized intractability bounds. 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 . Papadimitriou introduced several classes of NP search problems based on combinatorial principles which guarantee the existence of solutions to the problems. Many interesting search problems not known to be solvable in polynomial time are contained in these classes, and a number of them are complete problems. We consider the question of the relative complexity of these search problem classes. We prove several separations which show that in a generic relativized world the search classes are distinct and there is a standard search problem in each of them that is not computationally equivalent to any decision problem. (Naturally, absolute separations would imply thatP?NP.) Our separation proofs have interesting combinatorial content and go to the heart of the combinatorial principles on which the classes are based. We derive one result via new lower bounds on the degrees of poly- nomials asserted to exist by Hilbert's nullstellensatz over finite fields. Recently, Forster [7] proved a new lower bound on probabilistic communication complexity in terms of the operator norm of the communication matrix. In this paper, we want to exploit the various relations between communication complexity of distributed Boolean functions, geometric questions related to half space representations of these functions, and the computational complexity of these functions in various restricted models of computation. In order to widen the range of applicability of Forster's bound, we start with the derivation of a generalized lower bound. We present a concrete family of distributed Boolean functions where the generalized bound leads to a linear lower bound on the probabilistic communication complexity (and thus to an exponential lower bound on the number of Euclidean dimensions needed for a successful half space representation), whereas the old bound fails. We move on to a geometric characterization of the well known communication complexity class C-PP in terms of half space representations achieving a large margin. Our characterization hints to a close connection between the bounded error model of probabilistic communication complexity and the area of large margin classification. In the final section of the paper, we describe how our techniques can be used to prove exponential lower bounds on the size of depth-2 threshold circuits (with still some technical restrictions). Similar results can be obtained for read-k-times randomized ordered binary decision diagram and related models.