Families with infants: a general approach to solve hard partition problems

We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NP-hard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of so-called families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NP-hard problems as well as to simplify the proofs of several known results. For the chromatic number problem we present an algorithm with $O^*((2-\varepsilon(d))^n)$ time and exponential space for graphs of average degree $d$. This improves the algorithm by Bj\"{o}rklund et al. [Theory Comput. Syst. 2010] that works for graphs of bounded maximum (as opposed to average) degree and closes an open problem stated by Cygan and Pilipczuk [ICALP 2013]. For the traveling salesman problem we give an algorithm working in $O^*((2-\varepsilon(d))^n)$ time and polynomial space for graphs of average degree $d$. The previously known results of this kind is a polyspace algorithm by Bj\"{o}rklund et al. [ICALP 2008] for graphs of bounded maximum degree and an exponential space algorithm for bounded average degree by Cygan and Pilipczuk [ICALP 2013]. For counting perfect matching in graphs of average degree~$d$ we present an algorithm with running time $O^*((2-\varepsilon(d))^{n/2})$ and polynomial space. Recent algorithms of this kind due to Cygan, Pilipczuk [ICALP 2013] and Izumi, Wadayama [FOCS 2012] (for bipartite graphs only) use exponential space.Comment: 18 pages, a revised version of this paper is available at http://arxiv.org/abs/1410.220

D-harmonic distributions and global hypoellipticity on nilmanifolds

Let M = Γ\N be a compact nilmanifold. A system of differential operators D1, …, Dk on M is globally hypoelliptic (GH) if when D1f = g1,…, Dkf=gk with f ∈ D’ (M), g1, …, gk ∈ C∞ (M) then f ∈ C∞ (M). Let X1, …, Xk be real vector fields on M induced by the Lie algebra N of N. We study the relationships between (GH) of the system X1, …, Xk on M, (GH) of the operator D = X12 + … + Xk2, the constancy of D-harmonic distributions on M, and related algebraic conditions on X1, …, Xk ∈ N. © 1991 by Pacific Journal of Mathematics

{On Subexponential Running Times for Approximating Directed Steiner Tree and Related Problems}

This paper concerns proving almost tight (super-polynomial) running times, for achieving desired approximation ratios for various problems. To illustrate, the question we study, let us consider the Set-Cover problem with n elements and m sets. Now we specify our goal to approximate Set-Cover to a factor of (1-d)ln n, for a given parameter 0= 2^{n^{c d}}, for some constant 0= exp((1+o(1)){log^{d-c}n}), for any c>0, unless the ETH is false. Our result follows by analyzing the work of Halperin and Krauthgamer [STOC, 2003]. The same lower and upper bounds hold for CST

Formation des prix des légumes à la Réunion. Analyse de la variabilité des prix et des différentiels entre producteurs et consommateurs

Les productions légumières, potentiellement très rémunératrices, sont attractives pour les petits agriculteurs qui cherchent à valoriser davantage leur foncier, toutefois l'ampleur des fluctuations de leur prix entraîne un risque économique important. Outre l'instabilité des prix à court terme, les producteurs sont également préoccupés par l'évolution de leurs rapports commerciaux avec la distribution qui est dans une phase de concentration accélérée. Cette étude apporte un éclairage sur les conditions de formation des prix en abordant deux aspects. D'une part les fluctuations des prix au producteur, caractérisées en distinguant leurs composantes de tendance, saisonnalité et aléa représentatif du risque. D'autre part les performances de la distribution, évaluées à partir d'une analyse des différentiels de prix entre stades de gros et de détail. Les quinze principaux légumes ont ainsi été classés selon l'importance de la variabilité de leur prix et la contribution de chaque composante. Il ressort de cette décomposition des prix que les aléas constituent le principal facteur de variabilité. Les produits connaissant les aléas les plus forts sont la petite tomate, le chou-fleur, le piment et le poivron. La saisonnalité n'apporte une contribution majeure à la variabilité des prix que dans les cas de la pomme de terre et du haricot demi-sec. La composante de tendance est nulle sur la période 1993-1998 pour la majorité des produits. Les différentiels de prix entre stades de gros à St Pierre et de détail à St Denis sont contrastés selon le circuit, marché forain ou grande distribution. Le marché forain présente une meilleure compétitivité en terme de prix et accroît encore son avantage durant la période 1994-1998. La tendance des différentiels est en effet à la baisse pour les marchés forains et stationnaire pour la grande distribution. L'évolution de la structure des prix dans la filière légumes n'est donc pas défavorable aux producteurs, contrairement au cas d'autres filières agricoles. (Résumé d'auteur

Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

Given a graph $G$ and an integer $k$, the Feedback Vertex Set (FVS) problem asks if there is a vertex set $T$ of size at most $k$ that hits all cycles in the graph. The fixed-parameter tractability status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC '08) showed that it is FPT by giving a $4^{k}k!n^{O(1)}$ time algorithm. In the subset versions of this problems, we are given an additional subset $S$ of vertices (resp., edges) and we want to hit all cycles passing through a vertex of $S$ (resp. an edge of $S$). Recently, the Subset Feedback Vertex Set in undirected graphs was shown to be FPT by Cygan et al. (ICALP '11) and independently by Kakimura et al. (SODA '12). We generalize the result of Chen et al. (STOC '08) by showing that Subset Feedback Vertex Set in directed graphs can be solved in time $2^{O(k^3)}n^{O(1)}$. By our result, we complete the picture for feedback vertex set problems and their subset versions in undirected and directed graphs. Besides proving the fixed-parameter tractability of Directed Subset Feedback Vertex Set, we reformulate the random sampling of important separators technique in an abstract way that can be used for a general family of transversal problems. Moreover, we modify the probability distribution used in the technique to achieve better running time; in particular, this gives an improvement from $2^{2^{O(k)}}$ to $2^{O(k^2)}$ in the parameter dependence of the Directed Multiway Cut algorithm of Chitnis et al. (SODA '12).Comment: To appear in ACM Transactions on Algorithms. A preliminary version appeared in ICALP '12. We would like to thank Marcin Pilipczuk for pointing out a missing case in the conference version which has been considered in this version. Also, we give an single exponential FPT algorithm improving on the double exponential algorithm from the conference versio

Fast algorithms for min independent dominating set

We first devise a branching algorithm that computes a minimum independent dominating set on any graph with running time O*(2^0.424n) and polynomial space. This improves the O*(2^0.441n) result by (S. Gaspers and M. Liedloff, A branch-and-reduce algorithm for finding a minimum independent dominating set in graphs, Proc. WG'06). We then show that, for every r>3, it is possible to compute an r-((r-1)/r)log_2(r)-approximate solution for min independent dominating set within time O*(2^(nlog_2(r)/r))

Parameterized lower bound and NP-completeness of some HH-free Edge Deletion problems

For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. We prove that $H$-free Edge Deletion is NP-complete if $H$ is a graph with at least two edges and $H$ has a component with maximum number of vertices which is a tree or a regular graph. Furthermore, we obtain that these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time $2^{o(k)}\cdot |G|^{O(1)}$, unless Exponential Time Hypothesis fails.Comment: 15 pages, COCOA 15 accepted pape

Connecting Terminals and 2-Disjoint Connected Subgraphs

Given a graph $G=(V,E)$ and a set of terminal vertices $T$ we say that a superset $S$ of $T$ is $T$-connecting if $S$ induces a connected graph, and $S$ is minimal if no strict subset of $S$ is $T$-connecting. In this paper we prove that there are at most ${|V \setminus T| \choose |T|-2} \cdot 3^{\frac{|V \setminus T|}{3}}$ minimal $T$-connecting sets when $|T| \leq n/3$ and that these can be enumerated within a polynomial factor of this bound. This generalizes the algorithm for enumerating all induced paths between a pair of vertices, corresponding to the case $|T|=2$. We apply our enumeration algorithm to solve the {\sc 2-Disjoint Connected Subgraphs} problem in time $O^*(1.7804^n)$, improving on the recent $O^*(1.933^n)$ algorithm of Cygan et al. 2012 LATIN paper.Comment: 13 pages, 1 figur