2,227 research outputs found

    Algorithms and complexity for approximately counting hypergraph colourings and related problems

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    The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Δ, this problem falls into the regime of Lovász local lemma when Δ ≲ qᵏ. In prior, however, fast approximate counting algorithms exist when Δ ≲ qᵏ/³, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows. • When q, k ≥ 4 are evens and Δ ≥ 5·qᵏ/², approximating the number of hypergraph colourings is NP-hard. • When the input hypergraph is linear and Δ ≲ qᵏ/², a fast approximate counting algorithm does exist

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Fair Correlation Clustering in Forests

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    The study of algorithmic fairness received growing attention recently. This stems from the awareness that bias in the input data for machine learning systems may result in discriminatory outputs. For clustering tasks, one of the most central notions of fairness is the formalization by Chierichetti, Kumar, Lattanzi, and Vassilvitskii [NeurIPS 2017]. A clustering is said to be fair, if each cluster has the same distribution of manifestations of a sensitive attribute as the whole input set. This is motivated by various applications where the objects to be clustered have sensitive attributes that should not be over- or underrepresented. Most research on this version of fair clustering has focused on centriod-based objectives. In contrast, we discuss the applicability of this fairness notion to Correlation Clustering. The existing literature on the resulting Fair Correlation Clustering problem either presents approximation algorithms with poor approximation guarantees or severely limits the possible distributions of the sensitive attribute (often only two manifestations with a 1:1 ratio are considered). Our goal is to understand if there is hope for better results in between these two extremes. To this end, we consider restricted graph classes which allow us to characterize the distributions of sensitive attributes for which this form of fairness is tractable from a complexity point of view. While existing work on Fair Correlation Clustering gives approximation algorithms, we focus on exact solutions and investigate whether there are efficiently solvable instances. The unfair version of Correlation Clustering is trivial on forests, but adding fairness creates a surprisingly rich picture of complexities. We give an overview of the distributions and types of forests where Fair Correlation Clustering turns from tractable to intractable. As the most surprising insight, we consider the fact that the cause of the hardness of Fair Correlation Clustering is not the strictness of the fairness condition. We lift most of our results to also hold for the relaxed version of the fairness condition. Instead, the source of hardness seems to be the distribution of the sensitive attribute. On the positive side, we identify some reasonable distributions that are indeed tractable. While this tractability is only shown for forests, it may open an avenue to design reasonable approximations for larger graph classes

    Jake pozicione igre

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    In this thesis, we study 2-player combinatorial games on graphs. We devote a lot of attention to strong positional games, where both players have the same goal. First, we consider the so-called fixed graph strong Avoider-Avoider game in which two players called Red and Blue alternately claim edges of the complete graph Kn, and the player who first completes a copy of a fixed graph F loses the game. If neither of the players claimed a copy of F in his graph and all the elements of the board are claimed, the game is declared a draw. Even though these games have been studied for decades, there are very few known results. We make a step forward by proving that Blue has a winning strategy it two different games of this kind. Furthermore, we introduce strong CAvoiderCAvoider F games where the claimed edges of each player must form a connected graph throughout the game. This is a natural extension of the strong Avoider-Avoider games, with a connectedness constraint. We prove that Blue can win in three standard CAvoider-CAvoider F games. Next, we study strong Maker-Maker F games, where now, the player who first occupies a copy of F is the winner. It is well-known that the outcome of these games when both players play optimally can be either the first player's win or a draw. We are interested in finding the achievement number a(F) of a strong Maker-Maker F game, that is, the smallest n for which Red has a winning strategy. We can find the exact value a(F) for several graphs F, including paths, cycles, perfect matchings, and a subclass of trees on n vertices. We also give the upper and lower bounds for the achievement number of stars and trees. Finally, we introduce generalized saturation games as a natural extension of two different types of combinatorial games, saturation games and Constructor-Blocker games. In the generalized saturation game, two graphs H and F are given in advance. Two players called Max and Mini alternately claim unclaimed edges of the complete graph Kn and together gradually building the game graph G, the graph that consists of all edges claimed by both players. The graph G must never contain a copy of F, and the game ends when there are no more moves, i.e. when G is a saturated F-free graph. We are interested in the score of this game, that is, the number of copies of the graph H in G at the end of the game. Max wants to maximize this score, whereas Mini tries to minimize it. The game is played under the assumption that both players play optimally. We study several generalized saturation games for natural choices of F and H, in an effort to locate the score of the game as precisely as possible.У овој тези проучавамо комбинаторне игре на графовима које играју 2 играча. Посебну пажњу посвећујемо јаким позиционим играма, у којима оба играча имају исти циљ. Прво, посматрамо такозвану јаку Авојдер-Авојдер игру са задатим фиксним графом у којој два играча, Црвени и Плави наизменично селектују гране комплетног графа Kn, а играч који први селектује копију фиксног графа F губи игру. Ако ниједан од играча не садржи копију од F у свом графу и сви елементи табле су селектовани, игра се проглашава нерешеном. Иако су ове игре проучаване деценијама, врло је мало познатих резултата. Ми смо направили корак напред доказавши да Плави има победничку стратегију у две различите игре ове врсте. Такође, уводимо јаке ЦАвојдер-ЦАвојдер F игре у којима граф сваког играча мора остати повезан током игре. Ово је природно проширење јаких Авојдер-Авојдер игара, са ограничењем повезаности. Доказујемо да Плави може да победи у три стандардне ЦАвојдер-ЦАвојдер F игре. Затим проучавамо јаке Мејкер-Мејкер F игре, у којима је играч који први селектује копију од F победник. Познато је да исход ових игара уколико оба играча играју оптимално може бити или победа првог играча или нерешено. Циљ нам је да пронађемо ачивмент број а(F) јаке Мејкер-Мејкер F игре, односно најмање n за које Црвени има победничку стратегију. Дајемо тачну вредност a(F) за неколико графова F, укључујући путеве, циклусе, савршене мечинге и поткласу стабала са n чворова. Такође, дајемо горње и доње ограничење ачивмент броја за звезде и стабла. Коначно, уводимо уопштене игре сатурације као природно проширење две различите врсте комбинаторних игара, игара сатурације и Конструктор-Блокер игара. У уопштеној игри сатурације унапред су дата два графа H и F. Два играча по имену Макс и Мини наизменично селектују слободне гране комплетног графа Kn и заједно постепено граде граф игре G, који се састоји од свих грана које су селектовала оба играча. Граф G не сме да садржи копију од F, а игра се завршава када више нема потеза, односно када је G сатуриран граф који не садржи F. Занима нас резултат ове игре, односно, број копија графа H у G на крају игре. Макс жели да максимизира овај резултат, док Мини покушава да га минимизира. Игра се под претпоставком да оба играча играју оптимално. Проучавамо неколико уопштених игара сатурације за природне изборе F и H, у настојању да што прецизније одредимо резултат игре.U ovoj tezi proučavamo kombinatorne igre na grafovima koje igraju 2 igrača. Posebnu pažnju posvećujemo jakim pozicionim igrama, u kojima oba igrača imaju isti cilj. Prvo, posmatramo takozvanu jaku Avojder-Avojder igru sa zadatim fiksnim grafom u kojoj dva igrača, Crveni i Plavi naizmenično selektuju grane kompletnog grafa Kn, a igrač koji prvi selektuje kopiju fiksnog grafa F gubi igru. Ako nijedan od igrača ne sadrži kopiju od F u svom grafu i svi elementi table su selektovani, igra se proglašava nerešenom. Iako su ove igre proučavane decenijama, vrlo je malo poznatih rezultata. Mi smo napravili korak napred dokazavši da Plavi ima pobedničku strategiju u dve različite igre ove vrste. Takođe, uvodimo jake CAvojder-CAvojder F igre u kojima graf svakog igrača mora ostati povezan tokom igre. Ovo je prirodno proširenje jakih Avojder-Avojder igara, sa ograničenjem povezanosti. Dokazujemo da Plavi može da pobedi u tri standardne CAvojder-CAvojder F igre. Zatim proučavamo jake Mejker-Mejker F igre, u kojima je igrač koji prvi selektuje kopiju od F pobednik. Poznato je da ishod ovih igara ukoliko oba igrača igraju optimalno može biti ili pobeda prvog igrača ili nerešeno. Cilj nam je da pronađemo ačivment broj a(F) jake Mejker-Mejker F igre, odnosno najmanje n za koje Crveni ima pobedničku strategiju. Dajemo tačnu vrednost a(F) za nekoliko grafova F, uključujući puteve, cikluse, savršene mečinge i potklasu stabala sa n čvorova. Takođe, dajemo gornje i donje ograničenje ačivment broja za zvezde i stabla. Konačno, uvodimo uopštene igre saturacije kao prirodno proširenje dve različite vrste kombinatornih igara, igara saturacije i Konstruktor-Bloker igara. U uopštenoj igri saturacije unapred su data dva grafa H i F. Dva igrača po imenu Maks i Mini naizmenično selektuju slobodne grane kompletnog grafa Kn i zajedno postepeno grade graf igre G, koji se sastoji od svih grana koje su selektovala oba igrača. Graf G ne sme da sadrži kopiju od F, a igra se završava kada više nema poteza, odnosno kada je G saturiran graf koji ne sadrži F. Zanima nas rezultat ove igre, odnosno, broj kopija grafa H u G na kraju igre. Maks želi da maksimizira ovaj rezultat, dok Mini pokušava da ga minimizira. Igra se pod pretpostavkom da oba igrača igraju optimalno. Proučavamo nekoliko uopštenih igara saturacije za prirodne izbore F i H, u nastojanju da što preciznije odredimo rezultat igre

    On the existence of (r,g,χ)(r,g,\chi)-cages

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    In this paper, we work with simple and finite graphs. We study a generalization of the \emph{Cage Problem}, which has been widely studied since cages were introduced by Tutte \cite{T47} in 1947 and after Erd\" os and Sachs \cite{ES63} proved their existence in 1963. An \emph{(r,g)(r,g)-graph} is an rr-regular graph in which the shortest cycle has length equal to gg; that is, it is an rr-regular graph with girth gg. An \emph{(r,g)(r,g)-cage} is an (r,g)(r,g)-graph with the smallest possible number of vertices among all (r,g)(r,g)-graphs; the order of an (r,g)(r,g)-cage is denoted by n(r,g)n(r,g). The Cage Problem consists of finding (r,g)(r,g)-cages; it is well-known that (r,g)(r,g)-cages have been determined only for very limited sets of parameter pairs (r,g)(r, g). There exists a simple lower bound for n(r,g)n(r,g), given by Moore and denoted by n0(r,g)n_0(r,g). The cages that attain this bound are called \emph{Moore cages}.Comment: 18 page

    Map enumeration on surfaces: from bijective techniques to asymptotic counting

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    L'objectiu principal del treball és l'enumeració de mapes a superfícies orientables amb l'ús majoritari de tècniques bijectives. Deduirem la fórmula exacta per mapes planars amb n arestes amb 3 mètodes diferents. Un d'ells explica en detall la relació que hi ha entre mapes i arbres planars, a més d'oferir una base per l'enumeració de mapes en gèneres majors. En aquestes superfícies, enumerarem asimptòticament els mapes a partir d'una intel·ligent bijecció entre mapes i mapes amb una sola cara, anomenats g-trees.El objetivo principal del trabajo es la enumeración de mapas en superficies orientables con el uso mayoritario de técnicas biyectivas. Deduciremos la fórmula exacta para mapas planos con n aristas con tres métodos diferentes. Uno de ellos explica en detalle la relación que hay entre mapas y árboles planos, a más de dar una base para la enumeración de mapas en superficies de género mayor. En dichas superficies, enumeraremos asintóticamente los mapas a partir de una biyección entre mapas y mapas con una sola cara, llamados g-trees.The main objective of this work is the enumeration of maps on orientable surfaces with the use of mostly bijective techniques. We are going to deduce the exact formula for planar maps with n edges by three different methods. One of them explains in detail the relation that planar maps and trees have, in addition to giving the foundations for the enumeration of maps in higher genus surfaces. In said surfaces, we are going to asymptotically enumerate maps via a clever bijection between maps and maps with one face, which are called g-trees

    Topological methods in zero-sum Ramsey theory

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    A cornerstone result of Erd\H os, Ginzburg, and Ziv (EGZ) states that any sequence of 2n12n-1 elements in Z/n\mathbb{Z}/n contains a zero-sum subsequence of length nn. While algebraic techniques have predominated in deriving many deep generalizations of this theorem over the past sixty years, here we introduce topological approaches to zero-sum problems which have proven fruitful in other combinatorial contexts. Our main result (1) is a topological criterion for determining when any Z/n\mathbb{Z}/n-coloring of an nn-uniform hypergraph contains a zero-sum hyperedge. In addition to applications for Kneser hypergraphs, for complete hypergraphs our methods recover Olson's generalization of the EGZ theorem for arbitrary finite groups. Furthermore, we (2) give a fractional generalization of the EGZ theorem with applications to balanced set families and (3) provide a constrained EGZ theorem which imposes combinatorial restrictions on zero-sum sequences in the original result.Comment: 18 page

    University of Windsor Graduate Calendar 2023 Spring

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    https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1027/thumbnail.jp

    Extremal problems for a matching and any other graph

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    For a family of graphs \F, a graph is called \F-free if it does not contain any member of \F as a subgraph. The generalized Tur\'an number \ex(n,K_r,\F) is the maximum number of KrK_r in an nn-vertex \F-free graph and \ex(n,K_2,\F)=\ex(n,\F), i.e., the classical Tur\'an number. Let Ms+1M_{s+1} be a matching on s+1s+1 edges and FF be any graph. In this paper, we determine \ex(n,K_r, \{M_{s+1},F\}) apart from a constant additive term and also give a condition when the error constant term can be determined. In particular, we give the exact value of \ex(n,\{M_{s+1},F\}) for FF being any non-bipartite graph or some bipartite graphs. Furthermore, we determine \ex(n,K_r,\{M_{s+1},F\}) when FF is color critical with χ(F)max{r+1,4}\chi(F)\ge \max\{r+1,4\}. These extend the results in [2,11,18]
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