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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 Θ(μ(n)) for any reasonable function μ, 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/ )n o(1/ ) for any function f unless an unlikely collapse occurs in parameterized complexity theory. ✩ A preliminary version of this paper "Linear FPT reductions and computational lower bounds" was presented at

Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of well-known NP-hard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on depth-t circuits cannot be solved in time n o(k) poly(m), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t − 1)-st level W [t − 1] of the Whierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted sat, dominating set, hitting set, set cover, and feature set, cannot be solved in time n o(k) poly(m), where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W [1] of the W-hierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted q-sat (for any fixed q ≥ 2), clique, and independent set, cannot be solved in time n o(k) unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n k poly(m) or O(n k).

In this letter, we use transmission electron microscopy to study the microstructure feature of recently reported Pb-free piezoceramic Ba(Zr0.2Ti0.8)O3-x(Ba0.7Ca0.3)TiO3 across its piezoelectricity-optimal morphotropic phase boundary (MPB) by varying composition and temperature, respectively. The domain structure evolutions during such processes show that in MPB regime, the domains become miniaturized down to nanometer size with a domain hierarchy, which coincides with the d33-maximum region. Further convergent beam electron diffraction measurement shows that rhombohedral and tetragonal crystal symmetries coexist among the miniaturized domains. Strong piezoelectricity reported in such a system is due to easy polarization rotation between the coexisting nano-scale tetragonal and rhombohedral domains.

Plasmonic metal nanostructures have attracted considerable attention for solar energy harvesting due to their capability in photothermal conversion. However, the narrow resonant band of the conventional plasmonic nanoparticles greatly limits their application as only a small fraction of the solar energy can be utilized. Herein, a unique confined seeded growth strategy is developed to synthesize black silver nanostructures with broadband absorption in the visible and near-infrared spectrum. Through this novel strategy, assemblages of silver nanoparticles with widely distributed interparticle distances are generated in rod-shaped tubular spaces, leading to strong random plasmonic coupling and accordingly broadband absorption for significantly improved utilization of solar energy. With excellent efficiency in converting solar energy to heat, the resulting black Ag nanostructures can be made into thin films floating at the air/water interface for efficient generation of clean water steam through localized interfacial heating.

The design of bright NIR-II luminescent nanomaterials that enable efficient labelling of proteins without disturbing their physiological properties in vivo is challenging. We developed an efficient strategy to synthesizebright NIR-II gold nanoclusters (AuN Cs) protected by biocompatible cyclodextrin (CD). Leveraging the ultrasmall sizeo fA uN Cs (< 2nm) and strong macrocycle-based host-guest chemistry, the as-synthesized CD-AuN Cs can readily label proteins/ antibodies.M oreover,t he labelled proteins/antibodies enable highly efficient in vivo trackingd uring blood circulation, without disturbing their biodistribution and tumor targeting ability,thus leading to asensitive tumor-targeted imaging. CD-Au NCs are stable in the harsh biological environment and show good biocompatibility and high renal clearance efficiency.T herefore,t he NIR-II biolabels developed in this study provideapromising platform to monitor the physiological behavior of biomolecules in living organisms.

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Abstract. Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable if and only if it is fixed-parameter tractable. Practically, applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance (i.e., a kernel) has led to very efficient algorithms and now goes hand-in-hand with the design of practical algorithms for solving N P-hard problems. Well-known examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by 2k, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by 335k. In this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems. In terms of our lower bound results, we show, for example, that unless P = N P, planar vertex cover does not have a problem kernel of size smaller than 4k/3, and planar independent set and planar dominating set do not have kernels of size smaller than 2k. In terms of our upper bound results, we further reduce the upper bound on the kernel size for the planar dominating set problem to 67k, improving significantly the 335k previous upper bound given by Alber, Fellows, and Niedermeier [J. ACM, 51 (2004), pp. 363-384]. This latter result is obtained by introducing a new set of reduction and coloring rules, which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel. The paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem.Key words. parameterized algorithm, planar graph, dominating set, vertex cover, independent set, kernel AMS subject classifications. 05C85, 68Q17 DOI. 10.1137/050646354 1. Introduction. Many practical algorithms for N P-hard problems start by applying data reduction subroutines to the input instances of the problem. The hope is that after the data reduction phase the instance of the problem has shrunk to a moderate size. This makes the applicability of a second phase, such as a branchand-bound phase, to the resulting instance more feasible. Weihe showed in [41] how a practical preprocessing algorithm for a variation of the dominating set problem, called the red/blue dominating set problem, resulted in breaking up input instances of the problem into much smaller instances. Langston, and Shanbhag [2], in their implementation of algorithms for the vertex cover problem,