Parameterized complexity of DPLL search procedures

We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k-clique requires n steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in [11] of understanding the Resolution complexity of this family of formulas

Parameterized bounded-depth Frege is not optimal

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [9]. There the authors concentrate on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system. The main result of the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [9]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's

Unified characterisations of resolution hardness measures

Various "hardness" measures have been studied for resolution, providing theoretical insight into the proof complexity of resolution and its fragments, as well as explanations for the hardness of instances in SAT solving. In this paper we aim at a unified view of a number of hardness measures, including different measures of width, space and size of resolution proofs. Our main contribution is a unified game-theoretic characterisation of these measures. As consequences we obtain new relations between the different hardness measures. In particular, we prove a generalised version of Atserias and Dalmau's result on the relation between resolution width and space from [5]

XANTHENE DYES SHELL FORMATION ONTO NANOSCALE KEPLERATE {MO132} SURFACE: NMR AND PHOTOPHYSICAL STUDIES

This work was supported by Russian Science Foundation: Project No.21-73-00311

Separate, measure and conquer: faster polynomial-space algorithms for Max 2-CSP and counting dominating sets

We show a method resulting in the improvement of several polynomial-space, exponential-time algorithms. The method capitalizes on the existence of small balanced separators for sparse graphs, which can be exploited for branching to disconnect an instance into independent components. For this algorithm design paradigm, the challenge to date has been to obtain improvements in worst-case analyses of algorithms, compared with algorithms that are analyzed with advanced methods, such as Measure and Conquer. Our contribution is the design of a general method to integrate the advantage from the separator-branching into Measure and Conquer, for an improved running time analysis. We illustrate the method with improved algorithms for Max (r,2) -CSP and #Dominating Set. For Max (r,2) -CSP instances with domain size r and m constraints, the running time improves from r m/6 to r m/7.5 for cubic instances and from r 0.19⋅m to r 0.18⋅m for general instances, omitting subexponential factors. For #Dominating Set instances with n vertices, the running time improves from 1.4143 n to 1.2458 n for cubic instances and from 1.5673 n to 1.5183 n for general instances. It is likely that other algorithms relying on local transformations can be improved using our method, which exploits a non-local property of graphs

Sparser Random 3SAT Refutation Algorithms and the Interpolation Problem:Extended Abstract

We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek [12], as a family of unsatisfiable propositional formulas for which refutations of small size in any propo-sitional proof system that possesses the feasible interpolation property imply an efficient deterministic refutation algorithm for random 3SAT with n variables and Ω(n1.4) clauses. Such small size refutations would improve the state of the art (with respect to the clause density) efficient refutation algorithm, which works only for Ω(n1.5) many clauses [13]. We demonstrate polynomial-size refutations of the 3XOR principle in resolution operating with disjunctions of quadratic equations with small integer coefficients, denoted R(quad); this is a weak extension of cutting planes with small coefficients. We show that R(quad) is weakly autom-atizable iff R(lin) is weakly automatizable, where R(lin) is similar to R(quad) but with linear instead of quadratic equations (introduced in [25]). This reduces the problem of refuting random 3CNF with n vari-ables and Ω(n1.4) clauses to the interpolation problem of R(quad) and to the weak automatizability of R(lin)

Encoding Redundancy for Satisfaction-Driven Clause Learning

Satisfaction-Driven Clause Learning (SDCL) is a recent SAT solving paradigm that aggressively trims the search space of possible truth assignments. To determine if the SAT solver is currently exploring a dispensable part of the search space, SDCL uses the so-called positive reduct of a formula: The positive reduct is an easily solvable propositional formula that is satisfiable if the current assignment of the solver can be safely pruned from the search space. In this paper, we present two novel variants of the positive reduct that allow for even more aggressive pruning. Using one of these variants allows SDCL to solve harder problems, in particular the well-known Tseitin formulas and mutilated chessboard problems. For the first time, we are able to generate and automatically check clausal proofs for large instances of these problems

Медико-демографическая ситуация в Амурской области как основа здоровьесбережения

Aim. To analyze medical and demographic situation and morbidity in the population of different age groups in the Russian Federation, the Far Eastern Federal District and the Amur Region in order to develop management decisions for the health protection of the population at the regional levels.Methods. Statistical and analytical analysis were conducted. The reports of the Ministry of Health of the Russian Federation and Federal State Statistics Services were used as the material.Results. The mortality rates in the population of the Amur Region are higher than in the Far Eastern Federal District and in the Russian Federation for all years of analysis (2016–2020) and have amounted to 16.2 in 2020; 13.9; 14.6 per 1000 population respectively. In the Amur Region, mortality from injuries, poisoning and some other consequences of external causes is 2 times higher than in the Russian Federation; from diseases of the digestive system 1.6 times higher, from diseases of the respiratory system 1.5 times higher. Mortality from COVID-19 in the Amur Region is 84.3‰00, which is lower than in the Russian Federation (98.8). There is a higher rate of general morbidity in the Amur Region (2020, 162484.1 per 100 thousand population) compared to the Russian Federation (156111.4), and to the Far Eastern Federal District (146365.3), respectively. A significantly high incidence of diseases of the digestive system (16871.3‰00), compared to the Russian Federation (10092.1), and the Far Eastern Federal District (11230.8) has been noted. The incidence of COVID-19 in the Amur Region have decreased from 84678.4 per 100 thousand population (2019) to 80294.8‰00 (2020), while 3141.3‰00 cases have been registered in 2020; a 2.6-fold increase in the incidence of pneumonia has been observed for the first time. At the same time, the primary incidence of circulatory system diseases has decreased by 12.1%, which is the result of lacking dispensary and preventive services associated with the pandemic. The frequency of COVID-19 in the Amur Region in 2020 is 3141.3‰00, in the Russian Federation – 3384.5, in the Far Eastern Federal District – 3394.9, respectively. Higher rates of morbidity (2020) are noted in children – 166656.9 per 100 thousand of the corresponding population compared to the adolescents – 147023.43‰00, which is significantly higher compared to the entire population – 80294.83‰00.Conclusion. The results of the study should be used by government representatives to develop managerial decisions regarding the health of the population, especially the future generation.Цель. Анализ медико-демографической ситуации и заболеваемости населения разных возрастных групп в Российской Федерации (РФ), Дальневосточном федеральном округе (ДФО) и Амурской области с целью разработки управленческих решений по здоровьесбережению населения на региональном уровне.Материалы и методы. Статистический, аналитический. Использованы материалы официальной государственной статистики Минздрава РФ и Росстата.Результаты. Показатели смертности населения в Амурской области были выше, чем в ДФО и РФ, за все годы анализа (2016–2020 гг.) и составили в 2020 г. 16,2, 13,9 и 14,6 на 1 000 населения соответственно. В Амурской области в два раза выше, чем в РФ, смертность от травм, отравлений и некоторых других последствий внешних причин; от болезней органов пищеварения выше – в 1,6 раза, от болезней органов дыхания больше – в 1,5 раза. Смертность от COVID-19 в Амурской области составила 84,3‰00, что ниже, чем в РФ (98,8‰00). В Амурской области (2020 г.) зарегистрирован более высокий, чем в РФ (156 111,4) и ДФО (146 365,3), показатель общей заболеваемости (162 484,1 на 100 тыс. населения). Отмечена значительно более высокая заболеваемость в классе болезней органов пищеварения (16 871,3‰00) – в сравнении с РФ (10 092,1) и ДФО (11 230,8). Впервые выявленная заболеваемость в Амурской области снизилась с 84 678,4‰00 на 100 тыс. населения (2019 г.) до 80 294,8‰00 (2020 г.), при этом в 2020 г. зарегистрировано 3 141,3‰00 случая COVID-19, отмечен рост в 2,6 раза впервые выявленной заболеваемости пневмонией. Первичная заболеваемость болезнями системы кровообращения снизилась на 12,1% в результате ослабления диспансерной и профилактической работы в связи с пандемией. Частота выявленного COVID-19 в Амурской области в 2020 г. составила 3 141,3‰00, в РФ – 3 384,5‰00, в ДФО – 3394,9‰00. Отмечены более высокие показатели первичной заболеваемости (2020 г.) детей – 166 656,9‰00 на 100 тыс. соответствующего населения, чем подростков – 147 023,43‰00, и значительно более высокие, чем всего населения, – 80 294,83‰00.Заключение. Результаты исследования следует использовать руководителям органов управления регионального и федерального уровней для разработки управленческих решений по здоровьесбережению населения, в первую очередь будущего поколения