arXiv subjects

Lefteris Kirousis

Publications and source records attributed to Lefteris Kirousis.

15 recordsLinked to original sources

arXiv2026

An improvement on the bound for the acyclic chromatic index

The acyclic chromatic index (or acyclic edge-chromatic number) of a graph is the least number of colors needed to properly color its edges so that none of its cycles has only two colors. We show that for a graph of max degree $\Delta$, the acyclic chromatic index is at most $3.142(\Delta-1)+1$, improving on the (best to date) bound of Fialho et al. (2020). Our improvement is made possible by considering unordered (non-plane) trees, instead of ordered (plane) ones, as witness structures for the Lov\'{a}sz Local Lemma, a key combinatorial tool often used in related works. The counting of these witness structures entails methods of Analytic Combinatorics.

math.CO↗

arXiv2024

Improved bounds for acyclic coloring parameters

The {\em acyclic chromatic number} of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. The {\em acyclic chromatic index} is the analogous graph parameter for edge colorings. We first show that the acyclic chromatic index is at most $2Δ-1$, where $Δ$ is the maximum degree of the graph. We then show that for all $ε>0$ and for $Δ$ large enough (depending on $ε$), the acyclic chromatic number of the graph is at most $\lceil(4^{-1/3} +ε) Δ^{4/3} \rceil +Δ+ 1$. Both results improve long chains of previous successive advances. Both are algorithmic, in the sense that the colorings are generated by randomized algorithms. Previous randomized algorithms assume the availability of enough colors to guarantee properness deterministically and use additional colors in dealing with the bichromatic cycles in a randomized fashion. In contrast, our algorithm initially generates colorings that are not necessarily proper; it only aims at avoiding cycles where all pairs of edges, or vertices, that are one edge, or vertex, apart in a traversal of the cycle are homochromatic (of the same color). When this goal is reached, the algorithm checks for properness and if necessary it repeats until properness is attained. Thus savings in the number of colors is attained.

math.CO↗

arXiv2023

Perception, performance, and detectability of conversational artificial intelligence across 32 university courses

The emergence of large language models has led to the development of powerful tools such as ChatGPT that can produce text indistinguishable from human-generated work. With the increasing accessibility of such technology, students across the globe may utilize it to help with their school work -- a possibility that has sparked discussions on the integrity of student evaluations in the age of artificial intelligence (AI). To date, it is unclear how such tools perform compared to students on university-level courses. Further, students' perspectives regarding the use of such tools, and educators' perspectives on treating their use as plagiarism, remain unknown. Here, we compare the performance of ChatGPT against students on 32 university-level courses. We also assess the degree to which its use can be detected by two classifiers designed specifically for this purpose. Additionally, we conduct a survey across five countries, as well as a more in-depth survey at the authors' institution, to discern students' and educators' perceptions of ChatGPT's use. We find that ChatGPT's performance is comparable, if not superior, to that of students in many courses. Moreover, current AI-text classifiers cannot reliably detect ChatGPT's use in school work, due to their propensity to classify human-written answers as AI-generated, as well as the ease with which AI-generated text can be edited to evade detection. Finally, we find an emerging consensus among students to use the tool, and among educators to treat this as plagiarism. Our findings offer insights that could guide policy discussions addressing the integration of AI into educational frameworks.

cs.CY↗

arXiv2022

An Improved Bound of Acyclic Vertex-Coloring

The acyclic chromatic number of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. We show that for all $α>2^{-1/3}$ there exists an integer $Δ_α$ such that if the maximum degree $Δ$ of a graph is at least $Δ_α$, then the acyclic chromatic number of the graph is at most $\lceilαΔ^{4/3} \rceil +Δ+ 1$. The previous best bound, due to Gonçalves et al (2020), was $(3/2) Δ^{4/3} + O(Δ)$.

math.CO↗

arXiv2022

The Acyclic Chromatic Index is Less than the Double of the Max Degree

The acyclic chromatic index of a graph $G$ is the least number of colors needed to properly color its edges so that none of its cycles is bichromatic. In this work, we show that $2Δ-1$ colors are sufficient to produce such a coloring, where $Δ$ is the maximum degree of the graph. In contrast with most extant randomized algorithmic approaches to the chromatic index, where the algorithms presuppose enough colors to guarantee properness deterministically and use randomness only to deal with the bichromatic cycles, our randomized, Moser-type algorithm produces a not necessarily proper random coloring, in a structured way, trying to avoid cycles whose edges of the same parity are homochromatic, and only when this goal is reached it checks for properness. It repeats until properness is attained.

math.CO↗

arXiv2021

On the Computational Complexity of Non-dictatorial Aggregation

We investigate when non-dictatorial aggregation is possible from an algorithmic perspective, where non-dictatorial aggregation means that the votes cast by the members of a society can be aggregated in such a way that there is no single member of the society that always dictates the collective outcome. We consider the setting in which the members of a society take a position on a fixed collection of issues, where for each issue several different alternatives are possible, but the combination of choices must belong to a given set X of allowable voting patterns. Such a set X is called a possibility domain if there is an aggregator that is non-dictatorial, operates separately on each issue, and returns values among those cast by the society on each issue. We design a polynomial-time algorithm that decides, given a set X of voting patterns, whether or not X is a possibility domain. Furthermore, if X is a possibility domain, then the algorithm constructs in polynomial time a non-dictatorial aggregator for X. Furthermore, we show that the question of whether a Boolean domain X is a possibility domain is in NLOGSPACE. We also design a polynomial-time algorithm that decides whether X is a uniform possibility domain, that is, whether X admits an aggregator that is non-dictatorial even when restricted to any two positions for each issue. As in the case of possibility domains, the algorithm also constructs in polynomial time a uniform non-dictatorial aggregator, if one exists. Then, we turn our attention to the case where X is given implicitly, either as the set of assignments satisfying a propositional formula, or as a set of consistent evaluations of a sequence of propositional formulas. In both cases, we provide bounds to the complexity of deciding if X is a (uniform) possibility domain.

math.CO↗

arXiv2020

Directed Lovász Local Lemma and Shearer's Lemma

Moser and Tardos (2010) gave an algorithmic proof of the lopsided Lovász local lemma (LLL) in the variable framework, where each of the undesirable events is assumed to depend on a subset of a collection of independent random variables. For the proof, they define a notion of a lopsided dependency between the events suitable for this framework. In this work, we strengthen this notion, defining a novel directed notion of dependency and prove LLL for the corresponding graph. We show that this graph can be strictly sparser (thus the sufficient condition for LLL weaker) compared with graphs that correspond to other extant lopsided versions of dependency. Thus, in a sense, we address the problem "find other simple local conditions for the constraints (in the variable framework) that advantageously translate to some abstract lopsided condition" posed by Szegedy (2013). We also give an example where our notion of dependency graph gives better results than the classical Shearer lemma. Finally, we prove Shearer's lemma for the dependency graph we define. For the proofs, we perform a direct probabilistic analysis that yields an exponentially small upper bound for the probability of the algorithm that searches for the desired assignment to the variables not to return a correct answer within $n$ steps. In contrast, the method of proof that became known as the entropic method, gives an estimate of only the expectation of the number of steps until the algorithm returns a correct answer, unless the probabilities are tinkered with.

math.CO↗

arXiv2019

Algorithmically Efficient Syntactic Characterization of Possibility Domains

In the field of Judgment Aggrgation, a domain, that is a subset of a Cartesian power of $\{0,1\}$, is considered to reflect abstract rationality restrictions on vectors of two-valued judgments on a number of issues. We are interested in the ways we can aggregate the positions of a set of individuals, whose positions over each issue form vectors of the domain, by means of unanimous (idempotent) functions, whose output is again an element of the domain. Such functions are called non-dictatorial, when their output is not simply the positions of a single individual. Here, we consider domains admitting various kinds of non-dictatorial aggregators, which reflect various properties of majority aggregation: (locally) non-dictatorial, generalized dictatorships, anonymous, monotone, StrongDem and systematic. We show that interesting and, in some sense, democratic voting schemes are always provided by domains that can be described by propositional formulas of specific syntactic types we define. Furthermore, we show that we can efficiently recognize such formulas and that, given a domain, we can both efficiently check if it is described by such a formula and, in case it is, construct it. Our results fall in the realm of classical results concerning the syntactic characterization of domains with specific closure properties, like domains closed under logical AND which are the models of Horn formulas. The techniques we use to obtain our results draw from judgment aggregation as well as propositional logic and universal algebra.

cs.CC↗

arXiv2018

An interactive version of the Lovász local lemma

Assume we are given (finitely many) mutually independent variables and (finitely many) "undesirable" events, each depending on a subset of the variables of at most $k$ elements, called the scope of the event. Assume that the probability of a variable belonging to the scope of an occurring event is bounded by $q$. We prove that if $ekq \leq 1$ then there exists at least one assignment to the variables for which none of the events occurs. Examples are given where the criterion $ekq \leq 1$ is applicable, whereas that of the classical version of the Lovász local lemma is not. The proof of the result is through an interactive, private-coin implementation of the algorithm by Moser. The original implementation, which yields the classical result, finds efficiently, but probabilistically, an assignment to the events that avoids all undesirable events. Interestingly, the interactive implementation given in this work does not constitute an efficient, even if probabilistic, algorithm to find an assignment as desired under the weaker assumption $ekq \leq 1$. We can only conclude that under the hypothesis that $ekq \leq 1$, the interactive protocol will produce an assignment as desired within $n$ rounds, with probability high with respect to $n$; however, the provers' choices remain non-deterministic. Plausibly finding such an assignment is inherently hard, as the situation is reminiscent, in a probabilistic framework, of problems complete for syntactic subclasses of TFNP.

cs.DM↗

arXiv2018

Acyclic Edge Coloring through the Lovász Local Lemma

We give a probabilistic analysis of a Moser-type algorithm for the Lovász Local Lemma (LLL), adjusted to search for acyclic edge colorings of a graph. We thus improve the best known upper bound to acyclic chromatic index, also obtained by analyzing a similar algorithm, but through the entropic method (basically counting argument). Specifically we show that a graph with maximum degree $Δ$ has an acyclic proper edge coloring with at most $\lceil 3.74(Δ-1)\rceil+1 $ colors, whereas, previously, the best bound was $4(Δ-1)$. The main contribution of this work is that it comprises a probabilistic analysis of a Moser-type algorithm applied to events pertaining to dependent variables.

cs.DM↗

arXiv2016

Aggregation of Votes with Multiple Positions on Each Issue

We consider the problem of aggregating votes cast by a society on a fixed set of issues, where each member of the society may vote for one of several positions on each issue, but the combination of votes on the various issues is restricted to a set of feasible voting patterns. We require the aggregation to be supportive, i.e. for every issue $j$ the corresponding component $f_j$ of every aggregator on every issue should satisfy $f_j(x_1, ,\ldots, x_n) \in \{x_1, ,\ldots, x_n\}$. We prove that, in such a set-up, non-dictatorial aggregation of votes in a society of some size is possible if and only if either non-dictatorial aggregation is possible in a society of only two members or a ternary aggregator exists that either on every issue $j$ is a majority operation, i.e. the corresponding component satisfies $f_j(x,x,y) = f_j(x,y,x) = f_j(y,x,x) =x, \forall x,y$, or on every issue is a minority operation, i.e. the corresponding component satisfies $f_j(x,x,y) = f_j(x,y,x) = f_j(y,x,x) =y, \forall x,y.$ We then introduce a notion of uniformly non-dictatorial aggregator, which is defined to be an aggregator that on every issue, and when restricted to an arbitrary two-element subset of the votes for that issue, differs from all projection functions. We first give a characterization of sets of feasible voting patterns that admit a uniformly non-dictatorial aggregator. Then making use of Bulatov's dichotomy theorem for conservative constraint satisfaction problems, we connect social choice theory with combinatorial complexity by proving that if a set of feasible voting patterns $X$ has a uniformly non-dictatorial aggregator of some arity then the multi-sorted conservative constraint satisfaction problem on $X$, in the sense introduced by Bulatov and Jeavons, with each issue representing a sort, is tractable; otherwise it is NP-complete.

math.CO↗

arXiv2016

The problème des ménages revisited

We present an alternative proof to the Touchard-Kaplansky formula for the problème des ménages, which, we believe, is simpler than the extant ones and is in the spirit of the elegant original proof by Kaplansky (1943). About the latter proof, Bogart and Doyle (1986) argued that despite its cleverness, suffered from opting to give precedence to one of the genders for the couples involved (Bogart and Doyle supplied an elegant proof that avoided such gender-dependent bias).

math.CO↗

arXiv2015

An alternative proof for the constructive Asymmetric Lovász Local Lemma

We provide an alternative constructive proof of the Asymmetric Lovász Local Lemma. Our proof uses the classic algorithmic framework of Moser and the analysis introduced by Giotis, Kirousis, Psaromiligkos, and Thilikos in "On the algorithmic Lovász Local Lemma and acyclic edge coloring", combined with the work of Bender and Richmond on the multivariable Lagrange Inversion formula.

cs.DM↗

arXiv2014

Optimizing the Social Cost of Congestion Games by Imposing Variable Delays

We describe a new coordination mechanism for non-atomic congestion games that leads to a (selfish) social cost which is arbitrarily close to the non-selfish optimal. This mechanism does not incur any additional extra cost, like tolls, which are usually differentiated from the social cost as expressed in terms of delays only.

cs.GT↗

arXiv2013

On the Stability of Generalized Second Price Auctions with Budgets

The Generalized Second Price (GSP) auction used typically to model sponsored search auctions does not include the notion of budget constraints, which is present in practice. Motivated by this, we introduce the different variants of GSP auctions that take budgets into account in natural ways. We examine their stability by focusing on the existence of Nash equilibria and envy-free assignments. We highlight the differences between these mechanisms and find that only some of them exhibit both notions of stability. This shows the importance of carefully picking the right mechanism to ensure stable outcomes in the presence of budgets

cs.GT↗