Thomas FerniqueHSE, Moscow, Russia, Olga Mikhailovna SizovaSemenov Institute of Chemical Physics, Moscow, Russia
We introduce an elementary transformation called flips on tilings by squares and triangles and conjecture that it connects any two tilings of the same region of the Euclidean plane.
{"title":"Square-Triangle Tilings: Lift & Flip to Sample?","authors":"Thomas FerniqueHSE, Moscow, Russia, Olga Mikhailovna SizovaSemenov Institute of Chemical Physics, Moscow, Russia","doi":"arxiv-2406.16402","DOIUrl":"https://doi.org/arxiv-2406.16402","url":null,"abstract":"We introduce an elementary transformation called flips on tilings by squares\u0000and triangles and conjecture that it connects any two tilings of the same\u0000region of the Euclidean plane.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"210 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lapo CioniUniversity of Pisa, Italy, Luca FerrariUniversity of Firenze, Italy, Rebecca SmithSUNY Brockport
We consider sorting procedures for permutations making use of pop stacks with a bypass operation, and explore the combinatorial properties of the associated algorithms.
我们考虑了利用带旁路操作的 pop 栈对排列进行排序的过程，并探索了相关算法的组合特性。
{"title":"Pop Stacks with a Bypass","authors":"Lapo CioniUniversity of Pisa, Italy, Luca FerrariUniversity of Firenze, Italy, Rebecca SmithSUNY Brockport","doi":"arxiv-2406.16399","DOIUrl":"https://doi.org/arxiv-2406.16399","url":null,"abstract":"We consider sorting procedures for permutations making use of pop stacks with\u0000a bypass operation, and explore the combinatorial properties of the associated\u0000algorithms.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In graph theory, the minimum directed feedback vertex set (FVS) problem consists in identifying the smallest subsets of vertices in a directed graph whose deletion renders the directed graph acyclic. Although being known as NP-hard since 1972, this problem can be solved in a reasonable time on small instances, or on instances having special combinatorial structure. In this paper we investigate graph reductions preserving all or some minimum FVS and focus on their properties, especially the Church-Rosser property, also called confluence. The Church-Rosser property implies the irrelevance of reduction order, leading to a unique directed graph. The study seeks the largest subset of reductions with the Church-Rosser property and explores the adaptability of reductions to meet this criterion. Addressing these questions is crucial, as it may impact algorithmic implications, allowing for parallelization and speeding up sequential algorithms.
{"title":"On the Confluence of Directed Graph Reductions Preserving Feedback Vertex Set Minimality","authors":"Moussa Abdenbi, Alexandre Blondin Massé, Alain Goupil, Odile Marcotte","doi":"arxiv-2406.16390","DOIUrl":"https://doi.org/arxiv-2406.16390","url":null,"abstract":"In graph theory, the minimum directed feedback vertex set (FVS) problem\u0000consists in identifying the smallest subsets of vertices in a directed graph\u0000whose deletion renders the directed graph acyclic. Although being known as\u0000NP-hard since 1972, this problem can be solved in a reasonable time on small\u0000instances, or on instances having special combinatorial structure. In this\u0000paper we investigate graph reductions preserving all or some minimum FVS and\u0000focus on their properties, especially the Church-Rosser property, also called\u0000confluence. The Church-Rosser property implies the irrelevance of reduction\u0000order, leading to a unique directed graph. The study seeks the largest subset\u0000of reductions with the Church-Rosser property and explores the adaptability of\u0000reductions to meet this criterion. Addressing these questions is crucial, as it\u0000may impact algorithmic implications, allowing for parallelization and speeding\u0000up sequential algorithms.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Structural identifiability is an important property of parametric ODE models. When conducting an experiment and inferring the parameter value from the time-series data, we want to know if the value is globally, locally, or non-identifiable. Global identifiability of the parameter indicates that there exists only one possible solution to the inference problem, local identifiability suggests that there could be several (but finitely many) possibilities, while non-identifiability implies that there are infinitely many possibilities for the value. Having this information is useful since, one would, for example, only perform inferences for the parameters which are identifiable. Given the current significance and widespread research conducted in this area, we decided to create a database of linear compartment models and their identifiability results. This facilitates the process of checking theorems and conjectures and drawing conclusions on identifiability. By only storing models up to symmetries and isomorphisms, we optimize memory efficiency and reduce query time. We conclude by applying our database to real problems. We tested a conjecture about deleting one leak of the model states in the paper 'Linear compartmental models: Input-output equations and operations that preserve identifiability' by E. Gross et al., and managed to produce a counterexample. We also compute some interesting statistics related to the identifiability of linear compartment model parameters.
{"title":"Database for identifiability properties of linear compartmental models","authors":"Natali Gogishvili","doi":"arxiv-2406.16132","DOIUrl":"https://doi.org/arxiv-2406.16132","url":null,"abstract":"Structural identifiability is an important property of parametric ODE models.\u0000When conducting an experiment and inferring the parameter value from the\u0000time-series data, we want to know if the value is globally, locally, or\u0000non-identifiable. Global identifiability of the parameter indicates that there\u0000exists only one possible solution to the inference problem, local\u0000identifiability suggests that there could be several (but finitely many)\u0000possibilities, while non-identifiability implies that there are infinitely many\u0000possibilities for the value. Having this information is useful since, one\u0000would, for example, only perform inferences for the parameters which are\u0000identifiable. Given the current significance and widespread research conducted\u0000in this area, we decided to create a database of linear compartment models and\u0000their identifiability results. This facilitates the process of checking\u0000theorems and conjectures and drawing conclusions on identifiability. By only\u0000storing models up to symmetries and isomorphisms, we optimize memory efficiency\u0000and reduce query time. We conclude by applying our database to real problems.\u0000We tested a conjecture about deleting one leak of the model states in the paper\u0000'Linear compartmental models: Input-output equations and operations that\u0000preserve identifiability' by E. Gross et al., and managed to produce a\u0000counterexample. We also compute some interesting statistics related to the\u0000identifiability of linear compartment model parameters.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of efficiently coloring $3$-colorable graphs with few colors has received much attention on both the algorithmic and inapproximability fronts. We consider exponential time approximations, in which given a parameter $r$, we aim to develop an $r$-approximation algorithm with the best possible runtime, providing a tradeoff between runtime and approximation ratio. In this vein, an algorithm to $O(n^varepsilon)$-color a 3-colorable graphs in time $2^{Theta(n^{1-2varepsilon}log(n))}$ is given in (Atserias and Dalmau, SODA 2022.) We build on tools developed in (Bansal et al., Algorithmic, 2019) to obtain an algorithm to color $3$-colorable graphs with $O(r)$ colors in $expleft(tilde{O}left(frac {nlog^{11/2}r} {r^3}right)right)$ time, asymptotically improving upon the bound given by Atserias and Dalmau.
{"title":"Exponential Time Approximation for Coloring 3-Colorable Graphs","authors":"Venkatesan Guruswami, Rhea Jain","doi":"arxiv-2406.15563","DOIUrl":"https://doi.org/arxiv-2406.15563","url":null,"abstract":"The problem of efficiently coloring $3$-colorable graphs with few colors has\u0000received much attention on both the algorithmic and inapproximability fronts.\u0000We consider exponential time approximations, in which given a parameter $r$, we\u0000aim to develop an $r$-approximation algorithm with the best possible runtime,\u0000providing a tradeoff between runtime and approximation ratio. In this vein, an\u0000algorithm to $O(n^varepsilon)$-color a 3-colorable graphs in time\u0000$2^{Theta(n^{1-2varepsilon}log(n))}$ is given in (Atserias and Dalmau, SODA\u00002022.) We build on tools developed in (Bansal et al., Algorithmic, 2019) to obtain\u0000an algorithm to color $3$-colorable graphs with $O(r)$ colors in\u0000$expleft(tilde{O}left(frac {nlog^{11/2}r} {r^3}right)right)$ time,\u0000asymptotically improving upon the bound given by Atserias and Dalmau.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
David L. Fairbairn, George B. Mertzios, Norbert Peyerimhoff
The $k$-CombDMR problem is that of determining whether an $n times n$ distance matrix can be realised by $n$ vertices in some undirected graph with $n + k$ vertices. This problem has a simple solution in the case $k=0$. In this paper we show that this problem is polynomial time solvable for $k=1$ and $k=2$. Moreover, we provide algorithms to construct such graph realisations by solving appropriate 2-SAT instances. In the case where $k geq 3$, this problem is NP-complete. We show this by a reduction of the $k$-colourability problem to the $k$-CombDMR problem. Finally, we discuss the simpler polynomial time solvable problem of tree realisability for a given distance matrix.
{"title":"NP-Completeness of the Combinatorial Distance Matrix Realisation Problem","authors":"David L. Fairbairn, George B. Mertzios, Norbert Peyerimhoff","doi":"arxiv-2406.14729","DOIUrl":"https://doi.org/arxiv-2406.14729","url":null,"abstract":"The $k$-CombDMR problem is that of determining whether an $n times n$\u0000distance matrix can be realised by $n$ vertices in some undirected graph with\u0000$n + k$ vertices. This problem has a simple solution in the case $k=0$. In this\u0000paper we show that this problem is polynomial time solvable for $k=1$ and\u0000$k=2$. Moreover, we provide algorithms to construct such graph realisations by\u0000solving appropriate 2-SAT instances. In the case where $k geq 3$, this problem\u0000is NP-complete. We show this by a reduction of the $k$-colourability problem to\u0000the $k$-CombDMR problem. Finally, we discuss the simpler polynomial time\u0000solvable problem of tree realisability for a given distance matrix.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The assignment game, introduced by Shapley and Shubik (1971), is a classic model for two-sided matching markets between buyers and sellers. In the original assignment game, it is assumed that payments lead to transferable utility and that buyers have unit-demand valuations for the items being sold. Two important and mostly independent lines of work have studied more general settings with imperfectly transferable utility and gross substitutes valuations. Multiple efficient algorithms have been proposed for computing a competitive equilibrium, the standard solution concept in assignment games, in these two settings. Our main result is an efficient algorithm for computing competitive equilibria in a setting with both imperfectly transferable utility and gross substitutes valuations. Our algorithm combines augmenting path techniques from maximum matching and algorithms for matroid intersection. We also show that, in a mild generalization of our model, computing a competitive equilibrium is NP-hard.
{"title":"An Algorithm for the Assignment Game Beyond Additive Valuations","authors":"Eric Balkanski, Christopher En, Yuri Faenza","doi":"arxiv-2406.13620","DOIUrl":"https://doi.org/arxiv-2406.13620","url":null,"abstract":"The assignment game, introduced by Shapley and Shubik (1971), is a classic\u0000model for two-sided matching markets between buyers and sellers. In the\u0000original assignment game, it is assumed that payments lead to transferable\u0000utility and that buyers have unit-demand valuations for the items being sold.\u0000Two important and mostly independent lines of work have studied more general\u0000settings with imperfectly transferable utility and gross substitutes\u0000valuations. Multiple efficient algorithms have been proposed for computing a\u0000competitive equilibrium, the standard solution concept in assignment games, in\u0000these two settings. Our main result is an efficient algorithm for computing\u0000competitive equilibria in a setting with both imperfectly transferable utility\u0000and gross substitutes valuations. Our algorithm combines augmenting path\u0000techniques from maximum matching and algorithms for matroid intersection. We\u0000also show that, in a mild generalization of our model, computing a competitive\u0000equilibrium is NP-hard.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The conference GASCom brings together researchers in combinatorics, algorithms, probabilities, and more generally mathematical computer science, around the theme of random and exhaustive generation of combinatorial structures, mostly considered from a theoretical point of view. In connection with this main theme, the conference is also interested in contributions in enumerative or analytic combinatorics, and interactions with other areas of mathematics, computer science, physics or biology. The conference is both interested in methods for random or exhaustive generation and in original results on combinatorial or algorithmic questions, whose solution has been made possible by an approach involving random or exhaustive generation. The present edition of the conference includes a specific bunch of talks dedicated to polyominoes and tilings. A (not exhaustive) list of topics of the conference is: random and exhaustive generation of combinatorial objects; tilings and polyominoes; bijective, enumerative, algebraic and analytic combinatorics; algorithmic aspects: analysis of algorithms, probabilistic algorithms; interactions: bio-informatics, combinatorics on words, number theory.
{"title":"Proceedings of the 13th edition of the conference on Random Generation of Combinatorial Structures. Polyominoes and Tilings","authors":"Srečko Brlek, Luca Ferrari","doi":"arxiv-2406.14588","DOIUrl":"https://doi.org/arxiv-2406.14588","url":null,"abstract":"The conference GASCom brings together researchers in combinatorics,\u0000algorithms, probabilities, and more generally mathematical computer science,\u0000around the theme of random and exhaustive generation of combinatorial\u0000structures, mostly considered from a theoretical point of view. In connection\u0000with this main theme, the conference is also interested in contributions in\u0000enumerative or analytic combinatorics, and interactions with other areas of\u0000mathematics, computer science, physics or biology. The conference is both\u0000interested in methods for random or exhaustive generation and in original\u0000results on combinatorial or algorithmic questions, whose solution has been made\u0000possible by an approach involving random or exhaustive generation. The present\u0000edition of the conference includes a specific bunch of talks dedicated to\u0000polyominoes and tilings. A (not exhaustive) list of topics of the conference is: random and exhaustive\u0000generation of combinatorial objects; tilings and polyominoes; bijective,\u0000enumerative, algebraic and analytic combinatorics; algorithmic aspects:\u0000analysis of algorithms, probabilistic algorithms; interactions:\u0000bio-informatics, combinatorics on words, number theory.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141532102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the problem of allocating a set of indivisible goods to a set of agents with additive valuation functions, aiming to achieve approximate envy-freeness up to any good ($alpha$-EFX). The state-of-the-art results on the problem include that (exact) EFX allocations exist when (a) there are at most three agents, or (b) the agents' valuation functions can take at most two values, or (c) the agents' valuation functions can be represented via a graph. For $alpha$-EFX, it is known that a $0.618$-EFX allocation exists for any number of agents with additive valuation functions. In this paper, we show that $2/3$-EFX allocations exist when (a) there are at most emph{seven agents}, (b) the agents' valuation functions can take at most emph{three values}, or (c) the agents' valuation functions can be represented via a emph{multigraph}. Our results can be interpreted in two ways. First, by relaxing the notion of EFX to $2/3$-EFX, we obtain existence results for strict generalizations of the settings for which exact EFX allocations are known to exist. Secondly, by imposing restrictions on the setting, we manage to beat the barrier of $0.618$ and achieve an approximation guarantee of $2/3$. Therefore, our results push the emph{frontier} of existence and computation of approximate EFX allocations, and provide insights into the challenges of settling the existence of exact EFX allocations.
{"title":"Pushing the Frontier on Approximate EFX Allocations","authors":"Georgios Amanatidis, Aris Filos-Ratsikas, Alkmini Sgouritsa","doi":"arxiv-2406.12413","DOIUrl":"https://doi.org/arxiv-2406.12413","url":null,"abstract":"We study the problem of allocating a set of indivisible goods to a set of\u0000agents with additive valuation functions, aiming to achieve approximate\u0000envy-freeness up to any good ($alpha$-EFX). The state-of-the-art results on\u0000the problem include that (exact) EFX allocations exist when (a) there are at\u0000most three agents, or (b) the agents' valuation functions can take at most two\u0000values, or (c) the agents' valuation functions can be represented via a graph.\u0000For $alpha$-EFX, it is known that a $0.618$-EFX allocation exists for any\u0000number of agents with additive valuation functions. In this paper, we show that\u0000$2/3$-EFX allocations exist when (a) there are at most emph{seven agents}, (b)\u0000the agents' valuation functions can take at most emph{three values}, or (c)\u0000the agents' valuation functions can be represented via a emph{multigraph}. Our\u0000results can be interpreted in two ways. First, by relaxing the notion of EFX to\u0000$2/3$-EFX, we obtain existence results for strict generalizations of the\u0000settings for which exact EFX allocations are known to exist. Secondly, by\u0000imposing restrictions on the setting, we manage to beat the barrier of $0.618$\u0000and achieve an approximation guarantee of $2/3$. Therefore, our results push\u0000the emph{frontier} of existence and computation of approximate EFX\u0000allocations, and provide insights into the challenges of settling the existence\u0000of exact EFX allocations.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that any sequence of well-behaved (e.g. bounded and non-constant) real-valued functions of $n$ boolean variables ${f_n}$ admits a sequence of coordinates whose $L^1$ influence under the $p$-biased distribution, for any $pin(0,1)$, is $Omega(text{var}(f_n) frac{ln n}{n})$.
{"title":"On the maximal L1 influence of real-valued boolean functions","authors":"Andrew J. Young, Henry D. Pfister","doi":"arxiv-2406.10772","DOIUrl":"https://doi.org/arxiv-2406.10772","url":null,"abstract":"We show that any sequence of well-behaved (e.g. bounded and non-constant)\u0000real-valued functions of $n$ boolean variables ${f_n}$ admits a sequence of\u0000coordinates whose $L^1$ influence under the $p$-biased distribution, for any\u0000$pin(0,1)$, is $Omega(text{var}(f_n) frac{ln n}{n})$.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"346 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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