Planar Disjoint Paths, Treewidth, and Kernels

In the Planar Disjoint Paths problem, one is given an undirected planar graph with a set of $k$ vertex pairs $(s_i,t_i)$ and the task is to find $k$ pairwise vertex-disjoint paths such that the $i$-th path connects $s_i$ to $t_i$. We study the problem through the lens of kernelization, aiming at efficiently reducing the input size in terms of a parameter. We show that Planar Disjoint Paths does not admit a polynomial kernel when parameterized by $k$ unless coNP $\subseteq$ NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e}, Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel unless the WK-hierarchy collapses. Our reduction carries over to the setting of edge-disjoint paths, where the kernelization status remained open even in general graphs. On the positive side, we present a polynomial kernel for Planar Disjoint Paths parameterized by $k + tw$, where $tw$ denotes the treewidth of the input graph. As a consequence of both our results, we rule out the possibility of a polynomial-time (Turing) treewidth reduction to $tw= k^{O(1)}$ under the same assumptions. To the best of our knowledge, this is the first hardness result of this kind. Finally, combining our kernel with the known techniques [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver, SICOMP'94] yields an alternative (and arguably simpler) proof that Planar Disjoint Paths can be solved in time $2^{O(k^2)}\cdot n^{O(1)}$, matching the result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].Comment: To appear at FOCS'23, 82 pages, 30 figure

Impact of the COVID-19 pandemic on the differentiation of selected macroeconomic variables characterizing the EU economies over a short period

The 2020+ pandemic was not only the most serious global health crisis since the Spanish flu of 1918 but also one of the most economically expensive pandemics on a global scale. The scale and dynamics of the spread of the COVID-19 pandemic caused by a coronavirus entailed serious disturbances in social and economic life. Chapter 2 aims to analyze and assess the impact of the COVID-19 pandemic on the differentiation of selected macroeconomic aggregates in the European Union economies. The study examined 27 European countries that were members of the European Union at the time of the announcement of the pandemic by the World Health Organization. The following macro-indicators were analyzed and assessed: gross domestic product (GDP) per capita, the unemployment rate, gross fixed capital formation (investments) per capita, exports and imports per capita. The research project included both a long (2006-2020) and a short (Q2 of 2020 vs. Q2 of 2019) period. The depth of decreases in macroeconomic aggregates, including GDP (and consequently GDP per capita) in the second quarter of 2020 was unprecedented in the post-war history of the member states of the former European Communities and the current European Union. The sharpest decrease was observed in the countries that in 2006-2019 belonged to the group of economies with high and highest values of GDP per capita

Multilingualism in Greater Poland court records (1386-1448): tagging discourse boundaries and code-switching

No abstract available

Fiscal interventions in 2020

Chapter 3 aims to make a comparative analysis of fiscal responses of the European Union member states and of the United Kingdom to the outbreak of the COVID-19 pandemic. The survey covered measures implemented in 28 European economies between January and November 2020. This study discusses assistance packages offered in all countries, and fiscal and monetary responses in all member states of the European Union. The countries analyzed are divided into the eurozone and other economies. The lockdown, with its social and economic restrictions imposed in the surveyed countries, resulted in an economic downturn and a loss of 4.5% of GDP in total. Unemployment rose in those countries by 24% on average. Assistance packages were offered to stimulate economies and preserve employment, e.g. subsidies to wages, moratoriums (grace periods) on payment of taxes and repayment of loans, loans and guarantees granted to businesses, purchase of treasury and corporate securities by central banks. Those government responses were financed by public debt that rose in the countries surveyed from 80% to 91% of GDP on average. The Next Generation EU fund was approved at the EU level (EUR 750 billion), and was aimed to support the 2021-2027 financial framework (EUR 1.8 trillion in total). It was observed that governmental assistance was aimed at improved innovation, competitiveness and energy transition