Parameterized (in)approximability of subset problems

We discuss approximability and inapproximability in FPT-time for a large class of subset problems where a feasible solution $S$ is a subset of the input data and the value of $S$ is $|S|$. The class handled encompasses many well-known graph, set, or satisfiability problems such as Dominating Set, Vertex Cover, Set Cover, Independent Set, Feedback Vertex Set, etc. In a first time, we introduce the notion of intersective approximability that generalizes the one of safe approximability and show strong parameterized inapproximability results for many of the subset problems handled. Then, we study approximability of these problems with respect to the dual parameter $n-k$ where $n$ is the size of the instance and $k$ the standard parameter. More precisely, we show that under such a parameterization, many of these problems, while W[$\cdot$]-hard, admit parameterized approximation schemata.Comment: 7 page

Complexity of Grundy coloring and its variants

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a given graph has Grundy number at least $k$. We also study the variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper) and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that GRUNDY COLORING can be solved in time $O^*(2.443^n)$ and WEAK GRUNDY COLORING in time $O^*(2.716^n)$ on graphs of order $n$. While GRUNDY COLORING and WEAK GRUNDY COLORING are known to be solvable in time $O^*(2^{O(wk)})$ for graphs of treewidth $w$ (where $k$ is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time $O^*(2^{o(w\log w)})$. We also describe an $O^*(2^{2^{O(k)}})$ algorithm for WEAK GRUNDY COLORING, which is therefore \fpt for the parameter $k$. Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for GRUNDY COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with the two other problems, we show that CONNECTED GRUNDY COLORING is \np-complete already for $k=7$ colors.Comment: 24 pages, 7 figures. This version contains some new results and improvements. A short paper based on version v2 appeared in COCOON'1

Designing RNA secondary structures is hard

An RNA sequence is a word over an alphabet on four elements {A, C, G, U} called bases. RNA sequences fold into secondary structures where some bases match one another while others remain unpaired. Pseudoknot-free secondary structures can be represented as well-parenthesized expressions with additional dots, where pairs of matching parentheses symbolize paired bases and dots, unpaired bases. The two fundamental problems in RNA algorithmic are to predict how sequences fold within some model of energy and to design sequences of bases which will fold into targeted secondary structures. Predicting how a given RNA sequence folds into a pseudoknot-free secondary structure is known to be solvable in cubic time since the eighties and in truly subcubic time by a recent result of Bringmann et al. (FOCS 2016), whereas Lyngsø has shown it is NP-complete if pseudoknots are allowed (ICALP 2004). As a stark contrast, it is unknown whether or not designing a given RNA secondary structure is a tractable task; this has been raised as a challenging open question by Anne Condon (ICALP 2003). Because of its crucial importance in a number of fields such as pharmaceutical research and biochemistry, there are dozens of heuristics and software libraries dedicated to RNA secondary structure design. It is therefore rather surprising that the computational complexity of this central problem in bioinformatics has been unsettled for decades. In this paper we show that, in the simplest model of energy which is the Watson-Crick model the design of secondary structures is NP-complete if one adds natural constraints of the form: index i of the sequence has to be labeled by base b. This negative result suggests that the same lower bound holds for more realistic models of energy. It is noteworthy that the additional constraints are by no means artificial: they are provided by all the RNA design pieces of software and they do correspond to the actual practice (see for example the instances of the EteRNA project). Our reduction from a variant of 3-Sat has as main ingredients: arches of parentheses of different widths, a linear order interleaving variables and clauses, and an intended rematching strategy which increases the number of pairs iff the three literals of a same clause are not satisfied. The correctness of the construction is also quite intricate; it relies on the polynomial algorithm for the design of saturated structures – secondary structures without dots – by Haleš et al. (Algorithmica 2016), counting arguments, and a concise case analysis

A multi-step solution algorithm for Maxwell boundary integral equations applied to low-frequency electromagnetic testing of conductive objects

International audienceWe consider the solution, using boundary elements (BE), of the surface integral equation system arising in electromagnetic testing of conducting bodies, with emphasis on situations such that $o(1) \leq \sqrt{\omega\varepsilon_{0}/\sigma} \leq O(1)$, $L \sqrt{\omega\sigma\mu_{0}} =O(1)$ which includes in particular the case of eddy current testing) and assuming $L\omega\sqrt{\varepsilon_0 \mu_{0}}\leq 2\pi$, i.e. low-frequency conditions ($L$: diameter of conducting body). Earlier approaches for dielectric objects at low frequencies are not applicable in the present context. After showing that a simple normalization of the BE system significantly improves its conditioning, we propose a multi-step solution method based on block SOR iterations, which facilitates the use of direct solvers and converges within a few iterations for the considered range of physical parameters. This novel, albeit simple, treatment allows to perform eddy current-type analyses using standard Maxwell SIE formulations, avoiding the adverse consequences of ill-conditioning for low frequencies and high conductivities. Its performance and limitations are studied on three numerical examples involfing low frequencies and high conductivities

Parameterized Exact and Approximation Algorithms for Maximum kk-Set Cover and Related Satisfiability Problems

Given a family of subsets $\mathcal S$ over a set of elements~$X$ and two integers~$p$ and~$k$, Max k-Set Cover consists of finding a subfamily~$\mathcal T \subseteq \mathcal S$ of cardinality at most~$k$, covering at least~$p$ elements of~$X$. This problem is W[2]-hard when parameterized by~$k$, and FPT when parameterized by $p$. We investigate the parameterized approximability of the problem with respect to parameters~$k$ and~$p$. Then, we show that Max Sat-k, a satisfiability problem generalizing Max k-Set Cover, is also FPT with respect to parameter~$p$.Comment: Accepted in RAIRO - Theoretical Informatics and Application

Generalized feedback vertex set problems on bounded-treewidth graphs: chordality is the key to single-exponential parameterised algorithms

It has long been known that Feedback Vertex Set can be solved in time 2^O(w log w)n^O(1) on graphs of treewidth w, but it was only recently that this running time was improved to 2^O(w)n^O(1), that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class of graphs P, Bounded P-Block Vertex Deletion asks, given a graph G on n vertices and positive integers k and d, whether G contains a set S of at most k vertices such that each block of G-S has at most d vertices and is in P. Assuming that P is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of d: - if P consists only of chordal graphs, then the problem can be solved in time 2^O(wd^2) n^{O}(1), - if P contains a graph with an induced cycle of length ell>= 4, then the problem is not solvable in time 2^{o(w log w)} n^O(1)} even for fixed d=ell, unless the ETH fails. We also study a similar problem, called Bounded P-Component Vertex Deletion, where the target graphs have connected components of small size instead of having blocks of small size, and present analogous results

Parameterized Hardness of Art Gallery Problems

International audienceGiven a simple polygon P on n vertices, two points x, y in P are said to be visible to each other if the line 2 segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum set S such that every point in P is visible from a point in S. The Vertex Guard Art Gallery problem asks for such a set S subset of the vertices of P. A point in the set S is referred to as a guard. For both variants, we rule out any f (k)n o(k /log k) algorithm, where k := |S | is the number of guards, for any computable function f , unless the Exponential Time Hypothesis fails. These lower bounds almost match the n O (k) algorithms that exist for both problems