Polynomial formulations as a barrier for reduction-based hardness proofs

The Strong Exponential Time Hypothesis (SETH) asserts that for every $\varepsilon>0$ there exists $k$ such that $k$-SAT requires time $(2-\varepsilon)^n$. The field of fine-grained complexity has leveraged SETH to prove quite tight conditional lower bounds for dozens of problems in various domains and complexity classes, including Edit Distance, Graph Diameter, Hitting Set, Independent Set, and Orthogonal Vectors. Yet, it has been repeatedly asked in the literature whether SETH-hardness results can be proven for other fundamental problems such as Hamiltonian Path, Independent Set, Chromatic Number, MAX-$k$-SAT, and Set Cover. In this paper, we show that fine-grained reductions implying even $\lambda^n$-hardness of these problems from SETH for any $\lambda>1$, would imply new circuit lower bounds: super-linear lower bounds for Boolean series-parallel circuits or polynomial lower bounds for arithmetic circuits (each of which is a four-decade open question). We also extend this barrier result to the class of parameterized problems. Namely, for every $\lambda>1$ we conditionally rule out fine-grained reductions implying SETH-based lower bounds of $\lambda^k$ for a number of problems parameterized by the solution size $k$. Our main technical tool is a new concept called polynomial formulations. In particular, we show that many problems can be represented by relatively succinct low-degree polynomials, and that any problem with such a representation cannot be proven SETH-hard (without proving new circuit lower bounds)

Computations with polynomial evaluation oracle: ruling out superlinear SETH-based lower bounds

The field of fine-grained complexity aims at proving conditional lower bounds on the time complexity of computational problems. One of the most popular assumptions, Strong Exponential Time Hypothesis (SETH), implies that SAT cannot be solved in $2^{(1-\epsilon)n}$ time. In recent years, it has been proved that known algorithms for many problems are optimal under SETH. Despite the wide applicability of SETH, for many problems, there are no known SETH-based lower bounds, so the quest for new reductions continues. Two barriers for proving SETH-based lower bounds are known. Carmosino et al. (ITCS 2016) introduced the Nondeterministic Strong Exponential Time Hypothesis (NSETH) stating that TAUT cannot be solved in time $2^{(1-\epsilon)n}$ even if one allows nondeterminism. They used this hypothesis to show that some natural fine-grained reductions would be difficult to obtain: proving that, say, 3-SUM requires time $n^{1.5+\epsilon}$ under SETH, breaks NSETH and this, in turn, implies strong circuit lower bounds. Recently, Belova et al. (SODA 2023) introduced the so-called polynomial formulations to show that for many NP-hard problems, proving any explicit exponential lower bound under SETH also implies strong circuit lower bounds. We prove that for a range of problems from P, including $k$-SUM and triangle detection, proving superlinear lower bounds under SETH is challenging as it implies new circuit lower bounds. To this end, we show that these problems can be solved in nearly linear time with oracle calls to evaluating a polynomial of constant degree. Then, we introduce a strengthening of SETH stating that solving SAT in time $2^{(1-\varepsilon)n}$ is difficult even if one has constant degree polynomial evaluation oracle calls. This hypothesis is stronger and less believable than SETH, but refuting it is still challenging: we show that this implies circuit lower bounds

Structure and properties of (AlB

A systematic search for energetically lowest lying structures of neutral (AlB2)n and (MgB2)n clusters with n = 1, …, 10 is performed using density functional theory within a multistep hierarchical algorithm specially adapted for the global optimization of relatively large structures. For obtained clusters, different physical properties (energetic, electrostatic, electronic, and thermodynamic) are determined. The variation of these properties with increasing cluster size is discussed in detail. The bulk values of binding energy, specific zero point energy, ionization potential, electron affinity, collision diameter and formation enthalpy for aluminum and magnesium diborides have been obtained by means of physically sound extrapolation of the calculated data to the particles of infinite size. The temperature-dependent thermodynamic functions of (AlB2)n and (MgB2)n clusters, such as enthalpy, entropy, specific heat capacity, and reduced Gibbs energy, are evaluated with allowance for vibrational anharmonicity and for the existence of excited electronic states. The appropriate data are fitted to seven-parameter NASA (Chemkin) polynomials. The approximations of the reduced Gibbs energy applicable for extrapolation towards large clusters and even small nanoparticles are also elaborated

Theoretical Study of the Reaction of Ethane with Oxygen Molecules in the Ground Triplet and Singlet Delta States

Quantum chemical calculations are carried out to study the reaction of ethane with molecular oxygen in the ground triplet and singlet delta states. Transition states, intermediates, and possible products of the reaction on the triplet and singlet potential energy surfaces are identified on the basis of the coupled-cluster method. The basis set dependence of coupled-cluster energy values is estimated by the second-order perturbation theory. The values of energy barriers are also refined by using the compound CBS-Q and G3 techniques. It was found that the C₂H₆ + O₂(X³Σ_g^<i>–</i>) reaction leads to the formation of C₂H₅ and HO₂ products, whereas the C₂H₆ + O₂(a¹Δ_g) process produces C₂H₄ and H₂O₂ molecules. The appropriate rate constants of these reaction paths are estimated on the basis of variational and nonvariational transition-state theories assuming tunneling and possible nonadiabatic transitions in the temperature range 500–4000 K. The calculations showed that the rate constant of the C₂H₆ + O₂(a¹Δ_g) reaction path is much greater than that of the C₂H₆ + O₂(X³Σ_g^–) one. At the same time, the singlet and triplet potential surface intersection is detected that leads to the appearance of the nonadiabatic quenching channel O₂(a¹Δ_g) + C₂H₆ → O₂(X ³Σ_g^–) + C₂H₆. The rate constant of this process is estimated with the use of the Landau–Zener model. It is demonstrated that, in the case of the existence of thermal equilibrium in the distribution of molecules over the electronic states, at low temperatures (<i>T</i> < 1200 K) the main products of the reaction of C₂H₆ with O₂ are C₂H₄ and H₂O₂, rather than C₂H₅ and HO₂. At higher temperature (<i>T</i> > 1200 K) the situation is inverted

Theoretical study of physical and thermodynamic properties of Al

Geometrical structures and physical properties, such as collision diameter, rotational constants, characteristic vibrational temperatures, dipole moment, static isotropic polarizability, enthalpy of formation of various forms of AlnNm clusters with n = 0,...,5, m = 0,...,5, are analyzed with the usage of density functional theory. Different isomeric forms of these clusters with the isomerization energy up to 5 eV have been identified by using the original multistep heuristic algorithm that was based on semiempirical calculations, ab initio and density functional theory approaches and comprises the elements of genetic algorithms. Temperature dependencies of enthalpy, entropy and specific heat capacity have been calculated both for the individual isomers and for the Boltzmann ensemble of each class of clusters taking into account the anharmonicity of cluster vibrations and the contribution of excited electronic states of clusters. Novel criterion of the stability of isomeric forms, based on the maximal vibrational energy of the modes of cluster, has been proposed. The potentialities of the application of small AlnNm clusters as the components of energetic materials are also considered

Quantum chemical study of small B

Different isomeric forms of BnCm clusters with n = 0, ..., 5, m = 0, ..., 5 with the isomerization energy up to 5 eV have been identified by using the multi-step heuristic algorithm based on semiempirical, ab initio and density functional theory calculations. Physical properties, such as rotational constants and characteristic vibrational temperatures, collision diameter, enthalpy of formation, cohesive energy, dipole moment, static isotropic polarizability and magnetic moment of different isomeric forms have been obtained with the usage of density functional theory. It has been revealed that the electric properties of clusters depend on their structure. It was found that the isomers with linear structure contribute mostly to the average polarizability of the ensemble of the isomeric forms of given class of clusters. Temperature-dependent thermodynamic properties of clusters including specific heat capacity and entropy were calculated taking into account the contribution of excited electronic states and possible isomeric forms in the anharmonic oscillator approximation for vibrational degrees of freedom. It was shown that the effect of structural isomers on the thermodynamic properties of the Boltzmann ensemble of clusters can be significant

Theoretical Study of the Reactions of Ethanol with Aluminum and Aluminum Oxide

Quantum chemical calculations with the use of B2PLYP method were carried out to study the reactions of Al and AlO with the C₂H₅OH molecule. The values of energy barriers were estimated by means of extrapolation to the basis set limit. Examination of the potential energy surface revealed the energetically favorable reaction pathways. It has been found that for the Al + C₂H₅OH reaction, the OH-abstraction process leading to the formation of AlOH and C₂H₅ prevails. During investigation of the AlO + C₂H₅OH reaction it has been found that resulting products of this reaction were AlOH and C₂H₅O in different isomeric forms: hydroxyethyl and ethoxyl radicals. Appropriate rate constants for revealed channels have been estimated by using a canonical variational theory and capture model. The Arrhenius approximations for these processes have been proposed for the temperature range <i>T</i> = 400–4000 K

Theoretical evaluation of diffusion coefficients of (Al

The binary diffusion coefficients of two low lying isomers of (Al2O3)n, n = 1...4, clusters in different bath gases, that most frequently met in the nature and in the technical applications: H2, N2, O2, CO, H2O as well as their self-diffusion coefficients have been calculated on the basis of kinetic theory and dipole reduced formalism. The parameters of interaction potential have been determined taking into account the contributions of a dispersion, dipole-dipole and dipole-induced dipole interactions between alumina clusters and bath molecules. The dipole moments, polarizabilities and collision diameters of clusters have been obtained by using quantum chemical calculations of cluster structure. The approximations for temperature dependencies of diffusion coefficients for two low-lying isomers of each considered alumina clusters are reported. It is demonstrated that an account for the contributions of the second for each type of clusters does not affect substantially the value of net diffusion coefficient. The diffusion coefficients of the isomers of small (Al2O3)n clusters can differ notably in the case when their dipole moments are distinct and they interact with strongly dipole molecules

Physical and Thermodynamic Properties of Al_<i>n</i>C_<i>m</i> Clusters: Quantum-Chemical Study

Geometrical structures and physical properties, such as rotational constants and characteristic vibrational temperatures, collision diameter, enthalpy of formation, dipole moment, static isotropic polarizability, and magnetic moment of different forms of Al_<i>n</i>C_<i>m</i> clusters with <i>n</i> = 0–5, <i>m</i> = 0–5, have been studied with the usage of density functional theory. Different forms of clusters with the electronic energy up to 5 eV have been identified by using the original multistep heuristic algorithm based on semiempirical calculations and density functional theory. Temperature dependencies of thermodynamic properties such as enthalpy, entropy, and specific heat capacity were calculated for both the individual isomers and the Boltzmann ensembles of each class of clusters

Theoretical Study of the Reactions of Methane and Ethane with Electronically Excited N₂(A³Σ_u⁺)

Comprehensive quantum chemical analysis with the usage of density functional theory and post-Hartree–Fock approaches were carried out to study the processes in the N₂(A³Σ_u⁺) + CH₄ and N₂(A³Σ_u⁺) + C₂H₆ systems. The energetically favorable reaction pathways have been revealed on the basis of the examination of potential energy surfaces. It has been shown that the reactions N₂(A³Σ_u⁺) + CH₄ and N₂(A³Σ_u⁺) + C₂H₆ occur with very small or even zero activation barriers and, primarily, lead to the formation of N₂H + CH₃ and N₂H + C₂H₅ products, respectively. Further, the interaction of these species can give rise the ground state N₂(X¹Σ_g⁺) and CH₄ (or C₂H₆) products, i.e., quenching of N₂(A³Σ_u⁺) by CH₄ and C₂H₆ molecules is the complex two-step process. The possibility of dissociative quenching in the course of the interaction of N₂(A³Σ_u⁺) with CH₄ and C₂H₆ molecules has been analyzed on the basis of RRKM theory. It has been revealed that, for the reaction of N₂(A³Σ_u⁺) with CH₄, the dissociative quenching channel could occur with rather high probability, whereas in the N₂(A³Σ_u⁺) + C₂H₆ reacting system, an analogous process was little probable. Appropriate rate constants for revealed reaction channels have been estimated by using a canonical variational theory and capture approximation. The estimations showed that the rate constant of the N₂(A³Σ_u⁺) + C₂H₆ reaction path is considerably greater than that for the N₂(A³Σ_u⁺) + CH₄ one