Accurate simulation of complex biological processes is an essential component of developing and validating new technologies and inference approaches. As an effort to help contain the COVID-19 pandemic, large numbers of SARS-CoV-2 genomes have been sequenced from most regions in the world. More than 5.5 million viral sequences are publicly available as of November 2021. Many studies estimate viral genealogies from these sequences, as these can provide valuable information about the spread of the pandemic across time and space. Additionally such data are a rich source of information about molecular evolutionary processes including natural selection, for example allowing the identification of new variants with transmissibility and immunity evasion advantages. To our knowledge, there is no framework that is both efficient and flexible enough to simulate the pandemic to approximate world-scale scenarios and generate viral genealogies of millions of samples. Here, we introduce a new fast simulator VGsim which addresses the problem of simulation genealogies under epidemiological models. The simulation process is split into two phases. During the forward run the algorithm generates a chain of population-level events reflecting the dynamics of the pandemic using an hierarchical version of the Gillespie algorithm. During the backward run a coalescent-like approach generates a tree genealogy of samples conditioning on the population-level events chain generated during the forward run. Our software can model complex population structure, epistasis and immunity escape.

Òåîðåìû ñóùåñòâîâàíèÿ è äîñòàòî÷íîñòè, ñâÿçàííûå ñ ëîêàëüíûìè ïðåîáðàçîâàíèÿìè ãðàôîâ äëÿ çàäà÷è îÀííîòàöèÿ.  äàííîé ðàáîòå ââîäèòñÿ íåêîòîðûé êëàññ çàìåí ïîäãðàôîâ â ãðàôàõ, ïðè÷åì çàìåíû èç ýòîãî êëàññà ñîõðàíÿþò k -ðàñêðàøèâàåìîñòü. Êàaeäîå òàêîå ëîêàëüíîå ïðåîáðà-çîâàíèå ãðàôîâ îïðåäåëÿåòñÿ íåêîòîðûì øàáëîíîì íàáîðîì ðàçáèåíèé ìíîaeåñòâà íà åãî ïîäìíîaeåñòâà. Ïîêàçûâàåòñÿ, ÷òî çàìåíÿþùèé ïîäãðàô ñóùåñòâóåò äëÿ ëþáîãî øàáëîíà, à òàêaeå ïðèâîäèòñÿ îöåíêà íà êîëè÷åñòâî åãî âåðøèí â çàâèñèìîñòè îò ðàçìåðà øàáëîíà. Äàí-íûé ðåçóëüòàò ÿâëÿåòñÿ îñíîâíûì â ðàáîòå, äëÿ åãî ïîëó÷åíèÿ áûëè èñïîëüçîâàíû ìåòîäû òåîðèè ãðàôîâ è êîìáèíàòîðíîãî àíàëèçà. Ðàññìàòðèâàåìûé â ðàáîòå êëàññ ïðåîáðàçîâàíèé ìîaeåò áûòü ïîëåçåí ïðè ïîñòðîåíèè ïîëèíîìèàëüíûõ ñâåäåíèé äëÿ çàäà÷è î k -ðàñêðàñêå.  ÷àñòíîñòè, âìåñòå ñ îñíîâíûì ðåçóëüòàòîì ðàáîòû, îí ìîaeåò áûòü èñïîëüçîâàí ïðè ðåäóêöèè äàííûõ äëÿ ðåøåíèÿ çàäà÷è î k -ðàñêðàñêå.Êëþ÷åâûå ñëîâà: çàäà÷à î k -ðàñêðàñêå, ëîêàëüíîå ïðåîáðàçîâàíèå, çàäà÷à ðåàëèçàöèè, ôóíêöèÿ Øåííîíà.

Î ñëîaeíîñòè ïîñòðîåíèÿ 3-ðàñêðàñêè ïëàíàðíûõ ãðàôîâ ñ êîðîòêèìè ãðàíÿìè c ⃝ Ä. Â. Ñèðîòêèí 1 Àííîòàöèÿ. Çàäà÷à î âåðøèííîé 3-ðàñêðàñêå äëÿ çàäàííîãî ãðàôà ñîñòîèò â òîì, ÷òîáû ïðîâåðèòü, ìîaeíî ëè ìíîaeåñòâî åãî âåðøèí ðàçáèòü íà òðè ïîäìíîaeåñòâà ïîïàðíî íåñìåaeíûõ âåðøèí. Èçâåñòíî, ÷òî ýòà çàäà÷à ÿâëÿåòñÿ NP-ïîëíîé â êëàññå ïëàíàðíûõ ãðàôîâ è ÷òî îíà ñòàíîâèòñÿ ïîëèíîìèàëüíî ðàçðåøèìîé äëÿ ïëîñêèõ òðèàíãóëÿöèé ïëàíàðíûõ ãðàôîâ, ó êîòîðûõ âñå ãðàíè (âêëþ÷àÿ è âíåøíþþ) ÿâëÿþòñÿ òðåóãîëüíèêàìè. Èçâåñòíî òàêaeå, ÷òî îíà ÿâëÿåòñÿ NP-ïîëíîé â êëàññå ïëàíàðíûõ ãðàôîâ ñî ñòåïåíÿìè âñåõ âåðøèí íå áîëåå ÷åì 4, íî ñòàíîâèòñÿ ðàçðåøèìîé çà ëèíåéíîå âðåìÿ â êëàññå ãðàôîâ ñ ìàêñèìèëüíîé ñòåïåíüþ âåðøèí íå áîëåå ÷åì 3. Ïîýòîìó èíòåðåñåí âîïðîñ î ïîèñêå ïîðîãà íà çíà÷åíèÿ äëèí ãðàíåé è ìàêñèìàëüíîé ñòåïåíè âåðøèí ïëàíàðíûõ ãðàôîâ, ïðè ïåðåõîäå ÷åðåç êîòîðûé äëÿ çàäà÷è î âåðøèííîé 3-ðàñêðàñêå ïîëèíîìèàëüíàÿ ðàçðåøèìîñòü ìåíÿåòñÿ íà NP-ïîëíîòó.  äàííîé ðàáîòå äàåòñÿ îòâåò íà ýòîò âîïðîñ è äîêàçûâàåòñÿ NP-ïîëíîòà çàäà÷è î âåðøèííîé 3-ðàñêðàñêå â êëàññå ïëàíàðíûõ ãðàôîâ, ãðàíÿìè êîòîðûõ ÿâëÿþòñÿ òîëüêî òðåóãîëüíèêè è ÷åòûðåõóãîëüíèêè, ñ ìàêñèìàëüíîé ñòåïåíüþ âåðøèí íå áîëåå ÷åì 5.