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Nicolas Bousquet

Publications and source records attributed to Nicolas Bousquet.

At least 109 records · Page 6Linked to original sources
arXiv2015

Welfare and Rationality Guarantees for the Simultaneous Multiple-Round Ascending Auction

The simultaneous multiple-round auction (SMRA) and the combinatorial clock auction (CCA) are the two primary mechanisms used to sell bandwidth. Under truthful bidding, the SMRA is known to output a Walrasian equilibrium that maximizes social welfare provided the bidder valuation functions satisfy the gross substitutes property. Recently, it was shown that the combinatorial clock auction (CCA) provides good welfare guarantees for general classes of valuation functions. This motivates the question of whether similar welfare guarantees hold for the SMRA in the case of general valuation functions. We show the answer is no. But we prove that good welfare guarantees still arise if the degree of complementarities in the bidder valuations are bounded. In particular, if bidder valuations functions are $α$-near-submodular then, under truthful bidding, the SMRA has a welfare ratio (the worst case ratio between the social welfare of the optimal allocation and the auction allocation) of at most $(1+α)$. The special case of submodular valuations, namely $α=1$, and produces individually rational solutions. However, for $α>1$, this is a bicriteria guarantee, to obtain good welfare under truthful bidding requires relaxing individual rationality. Finally, we examine what strategies are required to ensure individual rationality in the SMRA with general valuation functions. First, we provide a weak characterization, namely \emph{secure bidding}, for individual rationality. We then show that if the bidders use a profit-maximizing secure bidding strategy the welfare ratio is at most $1+α$. Consequently, by bidding securely, it is possible to obtain the same welfare guarantees as truthful bidding without the loss of individual rationality.

cs.GT↗
arXiv2015

An adaptive kriging method for solving nonlinear inverse statistical problems

In various industrial contexts, estimating the distribution of unobserved random vectors Xi from some noisy indirect observations H(Xi) + Ui is required. If the relation between Xi and the quantity H(Xi), measured with the error Ui, is implemented by a CPU-consuming computer model H, a major practical difficulty is to perform the statistical inference with a relatively small number of runs of H. Following Fu et al. (2014), a Bayesian statistical framework is considered to make use of possible prior knowledge on the parameters of the distribution of the Xi, which is assumed Gaussian. Moreover, a Markov Chain Monte Carlo (MCMC) algorithm is carried out to estimate their posterior distribution by replacing H by a kriging metamodel build from a limited number of simulated experiments. Two heuristics, involving two different criteria to be optimized, are proposed to sequentially design these computer experiments in the limits of a given computational budget. The first criterion is a Weighted Integrated Mean Square Error (WIMSE). The second one, called Expected Conditional Divergence (ECD), developed in the spirit of the Stepwise Uncertainty Reduction (SUR) criterion, is based on the discrepancy between two consecutive approximations of the target posterior distribution. Several numerical comparisons conducted over a toy example then a motivating real case-study show that such adaptive designs can significantly outperform the classical choice of a maximin Latin Hypercube Design (LHD) of experiments. Dealing with a major concern in hydraulic engineering, a particular emphasis is placed upon the prior elicitation of the case-study, highlighting the overall feasibility of the methodology. Faster convergences and manageability considerations lead to recommend the use of the ECD criterion in practical applications.

stat.ME↗
arXiv2015

On the Economic Efficiency of the Combinatorial Clock Auction

Since the 1990s spectrum auctions have been implemented world-wide. This has provided for a practical examination of an assortment of auction mechanisms and, amongst these, two simultaneous ascending price auctions have proved to be extremely successful. These are the simultaneous multiround ascending auction (SMRA) and the combinatorial clock auction (CCA). It has long been known that, for certain classes of valuation functions, the SMRA provides good theoretical guarantees on social welfare. However, no such guarantees were known for the CCA. In this paper, we show that CCA does provide strong guarantees on social welfare provided the price increment and stopping rule are well-chosen. This is very surprising in that the choice of price increment has been used primarily to adjust auction duration and the stopping rule has attracted little attention. The main result is a polylogarithmic approximation guarantee for social welfare when the maximum number of items demanded $\mathcal{C}$ by a bidder is fixed. Specifically, we show that either the revenue of the CCA is at least an $Ω\Big(\frac{1}{\mathcal{C}^{2}\log n\log^2m}\Big)$-fraction of the optimal welfare or the welfare of the CCA is at least an $Ω\Big(\frac{1}{\log n}\Big)$-fraction of the optimal welfare, where $n$ is the number of bidders and $m$ is the number of items. As a corollary, the welfare ratio -- the worst case ratio between the social welfare of the optimum allocation and the social welfare of the CCA allocation -- is at most $O(\mathcal{C}^2 \cdot \log n \cdot \log^2 m)$. We emphasize that this latter result requires no assumption on bidders valuation functions. Finally, we prove that such a dependence on $\mathcal{C}$ is necessary. In particular, we show that the welfare ratio of the CCA is at least $Ω\Big(\mathcal{C} \cdot \frac{\log m}{\log \log m}\Big)$.

cs.GT↗
arXiv2015

Coalition Games on Interaction Graphs: A Horticultural Perspective

We examine cooperative games where the viability of a coalition is determined by whether or not its members have the ability to communicate amongst themselves independently of non-members. This necessary condition for viability was proposed by Myerson (1977) and is modeled via an interaction graph $G=(V,E)$; a coalition $S\subseteq V$ is then viable if and only if the induced graph $G[S]$ is connected. The non-emptiness of the core of a coalition game can be tested by a well-known covering LP. Moreover, the integrality gap of its dual packing LP defines exactly the multiplicative least-core and the relative cost of stability of the coalition game. This gap is upper bounded by the packing-covering ratio which, for graphical coalition games, is known to be at most the treewidth of the interaction graph plus one (Meir et al. 2013). We examine the packing-covering ratio and integrality gaps of graphical coalition games in more detail. We introduce the thicket parameter of a graph, and prove it precisely measures the packing-covering ratio. It also approximately measures the primal and dual integrality gaps. The thicket number provides an upper bound of both integrality gaps. Moreover we show that for any interaction graph, the primal integrality gap is, in the worst case, linear in terms of the thicket number while the dual integrality gap is polynomial in terms of it. At the heart of our results, is a graph theoretic minmax theorem showing the thicket number is equal to the minimum width of a vine decomposition of the coalition graph (a vine decomposition is a generalization of a tree decomposition). We also explain how the thicket number relates to the VC-dimension of the set system produced by the game.

cs.GT↗
arXiv2014

VC-dimension and Erdős-Pósa property

Let $G=(V,E)$ be a graph. A $k$-neighborhood in $G$ is a set of vertices consisting of all the vertices at distance at most $k$ from some vertex of $G$. The hypergraph on vertex set $V$ which edge set consists of all the $k$-neighborhoods of $G$ for all $k$ is the neighborhood hypergraph of $G$. Our goal in this paper is to investigate the complexity of a graph in terms of its neighborhoods. Precisely, we define the distance VC-dimension of a graph $G$ as the maximum taken over all induced subgraphs $G'$ of $G$ of the VC-dimension of the neighborhood hypergraph of $G'$. For a class of graphs, having bounded distance VC-dimension both generalizes minor closed classes and graphs with bounded clique-width. Our motivation is a result of Chepoi, Estellon and Vaxès asserting that every planar graph of diameter $2\ell$ can be covered by a bounded number of balls of radius $\ell$. In fact, they obtained the existence of a function $f$ such that every set $\cal F$ of balls of radius $\ell$ in a planar graph admits a hitting set of size $f(ν)$ where $ν$ is the maximum number of pairwise disjoint elements of $\cal F$. Our goal is to generalize the proof of Chepoi, Estellon and Vaxès with the unique assumption of bounded distance VC-dimension of neighborhoods. In other words, the set of balls of fixed radius in a graph with bounded distance VC-dimension has the Erdős-Pósa property.

math.CO↗
arXiv2014

Fast Recoloring of Sparse Graphs

In this paper, we show that for every graph of maximum average degree bounded away from $d$, any $(d+1)$-coloring can be transformed into any other one within a polynomial number of vertex recolorings so that, at each step, the current coloring is proper. In particular, it implies that we can transform any $8$-coloring of a planar graph into any other $8$-coloring with a polynomial number of recolorings. These results give some evidence on a conjecture of Cereceda, van den Heuvel and Johnson which asserts that any $(d+2)$ coloring of a $d$-degenerate graph can be transformed into any other one using a polynomial number of recolorings. We also show that any $(2d+2)$-coloring of a $d$-degenerate graph can be transformed into any other one using a linear number of recolorings.

math.CO↗
arXiv2014

On the practical interest of discrete Inverse Polya and Weibull-1 models in industrial reliability studies

Engineers often cope with the problem of assessing the lifetime of industrial components, under the basis of observed industrial feedback data. Usually, lifetime is modelled as a continuous random variable, for instance exponentially or Weibull distributed. However, in some cases, the features of the piece of equipment under investigation rather suggest the use of discrete probabilistic models. This happens for an equipment which only operates on cycles or on demand. In these cases, the lifetime is rather measured in number of cycles or number of demands before failure, therefore, in theory, discrete models should be more appropriate. This article aims at bringing some light to the practical interest for the reliability engineer in using two discrete models among the most popular: the Inverse Polya distribution (IPD), based on a Polya urn scheme, and the so-called Weibull-1 (W1) model. It is showed that, for different reasons, the practical use of both models should be restricted to specific industrial situations. In particular, when nothing is a priori known over the nature of ageing and/or data are heavily right-censored, they can remain of limited interest with respect to more flexible continuous lifetime models such as the usual Weibull distribution. Nonetheless, the intuitive meaning given to the IPD distribution favors its use by engineers in low (decelerated) ageing situations.

stat.AP↗
arXiv2014

The Erdős-Hajnal Conjecture for Long Holes and Anti-holes

Erdős and Hajnal conjectured that, for every graph $H$, there exists a constant $c_H$ such that every graph $G$ on $n$ vertices which does not contain any induced copy of $H$ has a clique or a stable set of size $n^{c_H}$. We prove that for every $k$, there exists $c_k>0$ such that every graph $G$ on $n$ vertices not inducing a cycle of length at least $k$ nor its complement contains a clique or a stable set of size $n^{c_k}$.

cs.DM↗
arXiv2014

A Near-Optimal Mechanism for Impartial Selection

We examine strategy-proof elections to select a winner amongst a set of agents, each of whom cares only about winning. This impartial selection problem was introduced independently by Holzman and Moulin and Alon et al. Fisher and Klimm showed that the permutation mechanism is impartial and $1/2$-optimal, that is, it selects an agent who gains, in expectation, at least half the number of votes of most popular agent. Furthermore, they showed the mechanism is $7/12$-optimal if agents cannot abstain in the election. We show that a better guarantee is possible, provided the most popular agent receives at least a large enough, but constant, number of votes. Specifically, we prove that, for any $ε>0$, there is a constant $N_ε$ (independent of the number $n$ of voters) such that, if the maximum number of votes of the most popular agent is at least $N_ε$ then the permutation mechanism is $(\frac{3}{4}-ε)$-optimal. This result is tight. Furthermore, in our main result, we prove that near-optimal impartial mechanisms exist. In particular, there is an impartial mechanism that is $(1-ε)$-optimal, for any $ε>0$, provided that the maximum number of votes of the most popular agent is at least a constant $M_ε$.

cs.GT↗
arXiv2014

Clique versus Independent Set

Yannakakis' Clique versus Independent Set problem (CL-IS) in communication complexity asks for the minimum number of cuts separating cliques from stable sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial CS-separator, i.e. of size $O(n^{\log n})$, and addresses the problem of finding a polynomial CS-separator. This question is still open even for perfect graphs. We show that a polynomial CS-separator almost surely exists for random graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a clique and a stable set) then there exists a constant $c_H$ for which we find a $O(n^{c_H})$ CS-separator on the class of H-free graphs. This generalizes a result of Yannakakis on comparability graphs. We also provide a $O(n^{c_k})$ CS-separator on the class of graphs without induced path of length k and its complement. Observe that on one side, $c_H$ is of order $O(|H| \log |H|)$ resulting from Vapnik-Chervonenkis dimension, and on the other side, $c_k$ is exponential. One of the main reason why Yannakakis' CL-IS problem is fascinating is that it admits equivalent formulations. Our main result in this respect is to show that a polynomial CS-separator is equivalent to the polynomial Alon-Saks-Seymour Conjecture, asserting that if a graph has an edge-partition into k complete bipartite graphs, then its chromatic number is polynomially bounded in terms of k. We also show that the classical approach to the stubborn problem (arising in CSP) which consists in covering the set of all solutions by $O(n^{\log n})$ instances of 2-SAT is again equivalent to the existence of a polynomial CS-separator.

cs.DM↗
arXiv2014

Reconfiguring Independent Sets in Cographs

Two independent sets of a graph are adjacent if they differ on exactly one vertex (i.e. we can transform one into the other by adding or deleting a vertex). Let $k$ be an integer. We consider the reconfiguration graph $TAR_k(G)$ on the set of independent sets of size at least $k$ in a graph $G$, with the above notion of adjacency. Here we provide a cubic-time algorithm to decide whether $TAR_k(G)$ is connected when $G$ is a cograph, thus solving an open question of~[Bonsma 2014]. As a by-product, we also describe a linear-time algorithm which decides whether two elements of $TAR_k(G)$ are in the same connected component.

cs.DM↗
arXiv2014

Recoloring graphs via tree decompositions

Let $k$ be an integer. Two vertex $k$-colorings of a graph are \emph{adjacent} if they differ on exactly one vertex. A graph is \emph{$k$-mixing} if any proper $k$-coloring can be transformed into any other through a sequence of adjacent proper $k$-colorings. Jerrum proved that any graph is $k$-mixing if $k$ is at least the maximum degree plus two. We first improve Jerrum's bound using the grundy number, which is the worst number of colors in a greedy coloring. Any graph is $(tw+2)$-mixing, where $tw$ is the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two $(tw+2)$-colorings is at most quadratic (which is optimal up to a constant factor), a problem left open in Bonamy et al. (2012). We also prove that given any two $(χ(G)+1)$-colorings of a cograph (resp. distance-hereditary graph) $G$, we can find a linear (resp. quadratic) sequence between them. In both cases, the bounds cannot be improved by more than a constant factor for a fixed $χ(G)$. The graph classes are also optimal in some sense: one of the smallest interesting superclass of distance-hereditary graphs corresponds to comparability graphs, for which no such property holds (even when relaxing the constraint on the length of the sequence). As for cographs, they are equivalently the graphs with no induced $P_4$, and there exist $P_5$-free graphs that admit no sequence between two of their $(χ(G)+1)$-colorings. All the proofs are constructivist and lead to polynomial-time recoloring algorithm

cs.DM↗
arXiv2014

Density modification based reliability sensitivity analysis

Sensitivity analysis of a numerical model, for instance simulating physical phenomena, is useful to quantify the influence of the inputs on the model responses. This paper proposes a new sensitivity index, based upon the modification of the probability density function (pdf) of the random inputs, when the quantity of interest is a failure probability (probability that a model output exceeds a given threshold). An input is considered influential if the input pdf modification leads to a broad change in the failure probability. These sensitivity indices can be computed using the sole set of simulations that has already been used to estimate the failure probability, thus limiting the number of calls to the numerical model. In the case of a Monte Carlo sample, asymptotical properties of the indices are derived. Based on Kullback-Leibler divergence, several types of input perturbations are introduced. The relevance of this new sensitivity analysis method is analysed through three case studies.

math.ST↗
arXiv2013

Brooks' theorem on powers of graphs

We prove that for $k\geq 3$, the bound given by Brooks' theorem on the chromatic number of $k$-th powers of graphs of maximum degree $Δ\geq 3$ can be lowered by 1, even in the case of online list coloring.

cs.DM↗
arXiv2013

Recoloring bounded treewidth graphs

Let $k$ be an integer. Two vertex $k$-colorings of a graph are \emph{adjacent} if they differ on exactly one vertex. A graph is \emph{$k$-mixing} if any proper $k$-coloring can be transformed into any other through a sequence of adjacent proper $k$-colorings. Any graph is $(tw+2)$-mixing, where $tw$ is the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two $(tw+2)$-colorings is at most quadratic, a problem left open in Bonamy et al. (2012). Jerrum proved that any graph is $k$-mixing if $k$ is at least the maximum degree plus two. We improve Jerrum's bound using the grundy number, which is the worst number of colors in a greedy coloring.

cs.DM↗
arXiv2012

Accelerated Monte Carlo estimation of failure probabilities in output of monotone computer codes

The problem of estimating the probability p=P(g(X<0) is considered when X represents a multivariate stochastic input of a monotone function g. First, a heuristic method to bound p is formally described, involving a specialized design of numerical experiments. Then a statistical estimation of p is considered based on a sequential stochastic exploration of the input space. A maximum likelihood estimator of p based on successive dependent Bernoulli data is defined and its theoretical convergence properties are studied. Under intuitive or mild conditions, the estimation is faster and more robust than the traditional Monte Carlo approach, therefore adapted to time-consuming computer codes g. The main result of the paper is related to the variance of the estimator. It appears as a new baseline measure of efficiency under monotone constraints, which could play a similar role to the usual Monte Carlo estimator variance in unconstrained frameworks. Furthermore the bias of the estimator is shown to be corrigible via bootstrap heuristics. The behavior of the method is illustrated by numerical tests led on a class of toy examples and a more realistic hydraulic case-study. Keywords : monotone function, deterministic computer codes, Monte Carlo acceleration, failure probability

math.ST↗
arXiv2012

Parameterized Domination in Circle Graphs

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction: - Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution. - Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs. - If T is a given tree, deciding whether a circle graph has a dominating set isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by |V(T)|. We prove that the FPT algorithm is subexponential.

cs.DS↗