arXiv subjects

Stéphane Vialette

Publications and source records attributed to Stéphane Vialette.

18 recordsLinked to original sources

arXiv2026

Resolving Envy by Adding Goods with Bounded Supply: A Type-Count Dichotomy and Two-Agent Hardness

We study envy elimination by adding goods (EEAG) when the additional pool has bounded supply and no separate budget bound. We establish a sharp type-count dichotomy for binary additive valuations. With one additional item type, EEAG is polynomial-time solvable for any number of agents. More generally, our algorithm permits arbitrary nonnegative integer per-copy values. The envy constraints form a system of difference constraints, and Bellman--Ford returns the componentwise least feasible extension. In contrast, with exactly two additional item types, EEAG is \textsf{NP}-complete even when both types have positive finite supply and the approvers of one type form a subset of the approvers of the other. This closes the two-type case left open by Bentert et al. Separately, we prove weak \textsf{NP}-completeness even for two agents with identical additive valuations, one initially endowed good, and a growing number of unit-supply item types. Thus, bounded-supply hardness appears both with two item-types and many agents and with two agents and many item-types.

cs.GT↗

arXiv2025

Branch Prediction Analysis of Morris-Pratt and Knuth-Morris-Pratt Algorithms

We analyze the classical Morris-Pratt and Knuth-Morris-Pratt pattern matching algorithms through the lens of computer architecture, investigating the impact of incorporating a simple branch prediction mechanism into the model of computation. Assuming a fixed pattern and a random text, we derive precise estimates of the number of mispredictions these algorithms produce using local predictors. Our approach is based on automata theory and Markov chains, providing a foundation for the theoretical analysis of other text algorithms and more advanced branch prediction strategies.

cs.DS↗

arXiv2024

Generalizing Roberts' characterization of unit interval graphs

For any natural number $d$, a graph $G$ is a (disjoint) $d$-interval graph if it is the intersection graph of (disjoint) $d$-intervals, the union of $d$ (disjoint) intervals on the real line. Two important subclasses of $d$-interval graphs are unit and balanced $d$-interval graphs (where every interval has unit length or all the intervals associated to a same vertex have the same length, respectively). A celebrated result by Roberts gives a simple characterization of unit interval graphs being exactly claw-free interval graphs. Here, we study the generalization of this characterization for $d$-interval graphs. In particular, we prove that for any $d \geq 2$, if $G$ is a $K_{1,2d+1}$-free interval graph, then $G$ is a unit $d$-interval graph. However, somehow surprisingly, under the same assumptions, $G$ is not always a \emph{disjoint} unit $d$-interval graph. This implies that the class of disjoint unit $d$-interval graphs is strictly included in the class of unit $d$-interval graphs. Finally, we study the relationships between the classes obtained under disjoint and non-disjoint $d$-intervals in the balanced case and show that the classes of disjoint balanced 2-intervals and balanced 2-intervals coincide, but this is no longer true for $d>2$.

cs.DM↗

arXiv2024

Quasi-kernels in split graphs

In a digraph, a quasi-kernel is a subset of vertices that is independent and such that the shortest path from every vertex to this subset is of length at most two. The ``small quasi-kernel conjecture,'' proposed by Erdős and Székely in 1976, postulates that every sink-free digraph has a quasi-kernel whose size is within a fraction of the total number of vertices. The conjecture is even more precise with a $1/2$ ratio, but even with larger ratio, this property is known to hold only for few classes of graphs. The focus here is on small quasi-kernels in split graphs. This family of graphs has played a special role in the study of the conjecture since it was used to disprove a strengthening that postulated the existence of two disjoint quasi-kernels. The paper proves that every sink-free split digraph $D$ has a quasi-kernel of size at most $\frac{2}{3}|V(D)|$, and even of size at most two when the graph is an orientation of a complete split graph. It is also shown that computing a quasi-kernel of minimal size in a split digraph is W[2]-hard.

math.CO↗

arXiv2023

Recognizing unit multiple intervals is hard

Multiple interval graphs are a well-known generalization of interval graphs introduced in the 1970s to deal with situations arising naturally in scheduling and allocation. A $d$-interval is the union of $d$ intervals on the real line, and a graph is a $d$-interval graph if it is the intersection graph of $d$-intervals. In particular, it is a unit $d$-interval graph if it admits a $d$-interval representation where every interval has unit length. Whereas it has been known for a long time that recognizing 2-interval graphs and other related classes such as 2-track interval graphs is NP-complete, the complexity of recognizing unit 2-interval graphs remains open. Here, we settle this question by proving that the recognition of unit 2-interval graphs is also NP-complete. Our proof technique uses a completely different approach from the other hardness results of recognizing related classes. Furthermore, we extend the result for unit $d$-interval graphs for any $d\geq 2$, which does not follow directly in graph recognition problems --as an example, it took almost 20 years to close the gap between $d=2$ and $d> 2$ for the recognition of $d$-track interval graphs. Our result has several implications, including that recognizing $(x, \dots, x)$ $d$-interval graphs and depth $r$ unit 2-interval graphs is NP-complete for every $x\geq 11$ and every $r\geq 4$.

cs.CC↗

arXiv2021

Algorithmic aspects of quasi-kernels

In a digraph, a quasi-kernel is a subset of vertices that is independent and such that every vertex can reach some vertex in that set via a directed path of length at most two. Whereas Chvátal and Lovász proved in 1974 that every digraph has a quasi-kernel, very little is known so far about the complexity of finding small quasi-kernels. In 1976 Erdős and Székely conjectured that every sink-free digraph $D = (V, A)$ has a quasi-kernel of size at most $|V|/2$. Obviously, if $D$ has two disjoint quasi-kernels then it has a quasi-kernel of size at most $|V|/2$, and in 2001 Gutin, Koh, Tay and Yeo conjectured that every sink-free digraph has two disjoint quasi-kernels. Yet, they constructed in 2004 a counterexample, thereby disproving this stronger conjecture. We shall show that, not only sink-free digraphs occasionally fail to contain two disjoint quasi-kernels, but it is computationally hard to distinguish those that do from those that do not. We also prove that the problem of computing a small quasi-kernel is polynomial time solvable for orientations of trees but is computationally hard in most other cases (and in particular for restricted acyclic digraphs).

cs.DM↗

arXiv2018

Algorithmic and algebraic aspects of unshuffling permutations

A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy allows to obtain easy proofs for algebraic and combinatorial properties of our shuffle product. We besides exhibit a bijection between square $(213,231)$-avoiding permutations and square binary words. On the other hand, by using a pattern avoidance criterion on directed perfect matchings, we prove that recognizing square permutations is {\bf NP}-complete.

cs.DS↗

arXiv2018

The Clever Shopper Problem

We investigate a variant of the so-called "Internet Shopping Problem" introduced by Blazewicz et al. (2010), where a customer wants to buy a list of products at the lowest possible total cost from shops which offer discounts when purchases exceed a certain threshold. Although the problem is NP-hard, we provide exact algorithms for several cases, e.g. when each shop sells only two items, and an FPT algorithm for the number of items, or for the number of shops when all prices are equal. We complement each result with hardness proofs in order to draw a tight boundary between tractable and intractable cases. Finally, we give an approximation algorithm and hardness results for the problem of maximising the sum of discounts.

cs.DS↗

arXiv2017

Sorting With Forbidden Intermediates

A wide range of applications, most notably in comparative genomics, involve the computation of a shortest sorting sequence of operations for a given permutation, where the set of allowed operations is fixed beforehand. Such sequences are useful for instance when reconstructing potential scenarios of evolution between species, or when trying to assess their similarity. We revisit those problems by adding a new constraint on the sequences to be computed: they must \emph{avoid} a given set of \emph{forbidden intermediates}, which correspond to species that cannot exist because the mutations that would be involved in their creation are lethal. We initiate this study by focusing on the case where the only mutations that can occur are exchanges of any two elements in the permutations, and give a polynomial time algorithm for solving that problem when the permutation to sort is an involution.

cs.DS↗

arXiv2016

Unshuffling Permutations

A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy allows to obtain easy proofs for algebraic and combinatorial properties of our shuffle product. We besides exhibit a bijection between square $(213,231)$-avoiding permutations and square binary words. On the other hand, by using a pattern avoidance criterion on oriented perfect matchings, we prove that recognizing square permutations is $\mathbf{NP}$-complete.

cs.DS↗

arXiv2015

Pattern matching in $(213,231)$-avoiding permutations

Given permutations $σ\in S_k$ and $π\in S_n$ with $k<n$, the \emph{pattern matching} problem is to decide whether $π$ matches $σ$ as an order-isomorphic subsequence. We give a linear-time algorithm in case both $π$ and $σ$ avoid the two size-$3$ permutations $213$ and $231$. For the special case where only $σ$ avoids $213$ and $231$, we present a $O(max(kn^2,n^2\log(\log(n)))$ time algorithm. We extend our research to bivincular patterns that avoid $213$ and $231$ and present a $O(kn^4)$ time algorithm. Finally we look at the related problem of the longest subsequence which avoids $213$ and $231$.

cs.DS↗

arXiv2015

Locating a Tree in a Phylogenetic Network in Quadratic Time

A fundamental problem in the study of phylogenetic networks is to determine whether or not a given phylogenetic network contains a given phylogenetic tree. We develop a quadratic-time algorithm for this problem for binary nearly-stable phylogenetic networks. We also show that the number of reticulations in a reticulation visible or nearly stable phylogenetic network is bounded from above by a function linear in the number of taxa.

cs.DS↗

arXiv2013

Flexible RNA design under structure and sequence constraints using formal languages

The problem of RNA secondary structure design (also called inverse folding) is the following: given a target secondary structure, one aims to create a sequence that folds into, or is compatible with, a given structure. In several practical applications in biology, additional constraints must be taken into account, such as the presence/absence of regulatory motifs, either at a specific location or anywhere in the sequence. In this study, we investigate the design of RNA sequences from their targeted secondary structure, given these additional sequence constraints. To this purpose, we develop a general framework based on concepts of language theory, namely context-free grammars and finite automata. We efficiently combine a comprehensive set of constraints into a unifying context-free grammar of moderate size. From there, we use generic generic algorithms to perform a (weighted) random generation, or an exhaustive enumeration, of candidate sequences. The resulting method, whose complexity scales linearly with the length of the RNA, was implemented as a standalone program. The resulting software was embedded into a publicly available dedicated web server. The applicability demonstrated of the method on a concrete case study dedicated to Exon Splicing Enhancers, in which our approach was successfully used in the design of \emph{in vitro} experiments.

q-bio.QM↗

arXiv2013

Single and multiple consecutive permutation motif search

Let $t$ be a permutation (that shall play the role of the {\em text}) on $[n]$ and a pattern $p$ be a sequence of $m$ distinct integer(s) of $[n]$, $m\leq n$. The pattern $p$ occurs in $t$ in position $i$ if and only if $p_1... p_m$ is order-isomorphic to $t_i... t_{i+m-1}$, that is, for all $1 \leq k< \ell \leq m$, $p_k>p_\ell$ if and only if $t_{i+k-1}>t_{i+\ell-1}$. Searching for a pattern $p$ in a text $t$ consists in identifying all occurrences of $p$ in $t$. We first present a forward automaton which allows us to search for $p$ in $t$ in $O(m^2\log \log m +n)$ time. We then introduce a Morris-Pratt automaton representation of the forward automaton which allows us to reduce this complexity to $O(m\log \log m +n)$ at the price of an additional amortized constant term by integer of the text. Both automata occupy $O(m)$ space. We then extend the problem to search for a set of patterns and exhibit a specific Aho-Corasick like algorithm. Next we present a sub-linear average case search algorithm running in $O(\frac{m\log m}{\log\log m}+\frac{n\log m}{m\log\log m})$ time, that we eventually prove to be optimal on average.

cs.DS↗

arXiv2012

On the combinatorics of suffix arrays

We prove several combinatorial properties of suffix arrays, including a characterization of suffix arrays through a bijection with a certain well-defined class of permutations. Our approach is based on the characterization of Burrows-Wheeler arrays given in [1], that we apply by reducing suffix sorting to cyclic shift sorting through the use of an additional sentinel symbol. We show that the characterization of suffix arrays for a special case of binary alphabet given in [2] easily follows from our characterization. Based on our results, we also provide simple proofs for the enumeration results for suffix arrays, obtained in [3]. Our approach to characterizing suffix arrays is the first that exploits their relationship with Burrows-Wheeler permutations.

cs.DS↗

arXiv2008

Comparing RNA structures using a full set of biologically relevant edit operations is intractable

Arc-annotated sequences are useful for representing structural information of RNAs and have been extensively used for comparing RNA structures in both terms of sequence and structural similarities. Among the many paradigms referring to arc-annotated sequences and RNA structures comparison (see \cite{IGMA_BliDenDul08} for more details), the most important one is the general edit distance. The problem of computing an edit distance between two non-crossing arc-annotated sequences was introduced in \cite{Evans99}. The introduced model uses edit operations that involve either single letters or pairs of letters (never considered separately) and is solvable in polynomial-time \cite{ZhangShasha:1989}. To account for other possible RNA structural evolutionary events, new edit operations, allowing to consider either silmutaneously or separately letters of a pair were introduced in \cite{jiangli}; unfortunately at the cost of computational tractability. It has been proved that comparing two RNA secondary structures using a full set of biologically relevant edit operations is {\sf\bf NP}-complete. Nevertheless, in \cite{DBLP:conf/spire/GuignonCH05}, the authors have used a strong combinatorial restriction in order to compare two RNA stem-loops with a full set of biologically relevant edit operations; which have allowed them to design a polynomial-time and space algorithm for comparing general secondary RNA structures. In this paper we will prove theoretically that comparing two RNA structures using a full set of biologically relevant edit operations cannot be done without strong combinatorial restrictions.

cs.DS↗

arXiv2008

On the Approximability of Comparing Genomes with Duplicates

A central problem in comparative genomics consists in computing a (dis-)similarity measure between two genomes, e.g. in order to construct a phylogeny. All the existing measures are defined on genomes without duplicates. However, we know that genes can be duplicated within the same genome. One possible approach to overcome this difficulty is to establish a one-to-one correspondence (i.e. a matching) between genes of both genomes, where the correspondence is chosen in order to optimize the studied measure. In this paper, we are interested in three measures (number of breakpoints, number of common intervals and number of conserved intervals) and three models of matching (exemplar, intermediate and maximum matching models). We prove that, for each model and each measure M, computing a matching between two genomes that optimizes M is APX-hard. We also study the complexity of the following problem: is there an exemplarization (resp. an intermediate/maximum matching) that induces no breakpoint? We prove the problem to be NP-Complete in the exemplar model for a new class of instances, and we show that the problem is in P in the maximum matching model. We also focus on a fourth measure: the number of adjacencies, for which we give several approximation algorithms in the maximum matching model, in the case where genomes contain the same number of duplications of each gene.

q-bio.QM↗

arXiv2007

On restrictions of balanced 2-interval graphs

The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatic problems. Some of those applications imply restrictions on the 2-interval graphs, and justify the introduction of a hierarchy of subclasses of 2-interval graphs that generalize line graphs: balanced 2-interval graphs, unit 2-interval graphs, and (x,x)-interval graphs. We provide instances that show that all the inclusions are strict. We extend the NP-completeness proof of recognizing 2-interval graphs to the recognition of balanced 2-interval graphs. Finally we give hints on the complexity of unit 2-interval graphs recognition, by studying relationships with other graph classes: proper circular-arc, quasi-line graphs, K_{1,5}-free graphs, ...

cs.DM↗