Enumeration on Graphs and Hypergraphs: Algorithms and Complexity. Enumération dans les graphes et hypergraphes : algorithmes et complexité

Transcription

1 Enumeration on Graphs and Hypergraphs: Algorithms and Complexity Enumération dans les graphes et hypergraphes : algorithmes et complexité Acronym : GraphEn Project leader: Dieter Kratsch, Université de Lorraine Partners: LITA, Université de Lorraine LIMOS, Université de Clermont-Ferrand LaBRI, Université de Bordeaux Duration: 48 months Requested grant: EXECUTIVE SUMMARY AND TABLE OF PARTICIPANTS CONTEXT, POSITION AND OBJECTIVES OF THE PROPOSAL State of the art Objectives, originality and novelty of the project SCIENTIFIC AND TECHNICAL PROGRAMME, PROJECT ORGANISATION Scientific programme and project structure Description by task Task 1 Organisation, synthesis and knowledge dissemination Task 2 Output-sensitive enumeration Task 3 Input-sensitive enumeration Task 4 Parameterized enumeration Task 5 Tractable classes of graphs and hypergraphs Task 6 Complexity theory and hardness results Schedule Consortium description Scientific justification of requested ressources DISSEMINATION AND EXPLOITATION OF RESULTS, INTELLECTUAL PROPERTY REFERENCES /29

2 1. EXECUTIVE SUMMARY AND TABLE OF PARTICIPANTS The P vs. NP question is arguably the most important open question in Theoretical Computer Science these days. Under the widely believed assumption that the complexity classes P and NP are not equal, there are problems that cannot be solved efficiently with the help of computers. Thus it is important to identify such problems and to find other ways of dealing with them, different from the traditional means of polynomial-time algorithms. Unfortunately, many problems of great theoretical importance and also many problems that arise from real applications turn out to be intractable in the general case. While optimisation is ubiquitous in Computer Science and a lot of research has been done on algorithms and complexity on optimisation problems, surprisingly little attention has been given to enumeration. A solution of the enumeration version of a problem typically provides an immediate solution for the optimisation version of the problem. This seems to suggest that enumeration is much harder than optimisation, which, among others, directed the search for tractability and efficient algorithms to optimisation problems. New insights from the recent research on the exact complexity of hard problems indicate that the relation between enumeration and optimisation is more subtle and worth a fundamental study from theoretical point of view. Listing, generating or enumerating objects of specified type and properties has important applications in various domains of Computer Science as e.g. data mining, machine learning and artificial intelligence, as well as in other sciences, in particular in biology, and also many applications in real life. This is one of the motivations of our interest in enumeration. The scientific goals of the project are of theoretical nature and oriented towards better understanding of the complexity of enumeration and the study of algorithmic techniques to solve enumeration problems. Thereby we shall concentrate on problems for graphs and hypergraphs and study three different approaches to the algorithmics of enumeration. For many enumeration problems the number of generated objects is exponential in the input size, e.g. the number of vertices of the input graph or hypergraph. This is one of the motivations of the so-called outputsensitive approach in which the running time of an enumeration algorithm depends on the size of input and the size of the output. This approach has a long tradition and is the classical one in enumeration algorithms. Recently research on exact exponential-time algorithms triggered the study of enumeration algorithms in which the running time depends on the input length only. It is of great use in establishing (exponential) bounds on the number of objects as well as in the study of exact complexity of hard problems. Another approach to solve enumeration problems on graphs is the use of algorithms parameterized by some width parameter like tree-width or clique-width. Our goal is to establish meta-theorems producing algorithms for enumeration problems based on logic, automata and the structure of graphs. This includes the implementation in the experimental system Autograph, which allows to verify the usability of algorithmic meta-theorems. Besides the common goals of such a project bringing together various groups of French researchers interested in algorithmic enumeration in graphs, like cumulating knowledge on particular enumeration problems and scientific production, our principal motivation is to establish an enumeration community in France and is also to help making enumeration an important research subject world-wide in algorithmics. A workshop at the Lorentz Center in Leiden co-organised by the leader of this project has the aim to bring researchers of different research direction in enumeration together. Writing a book on algorithmic enumeration will be another cornerstone in giving algorithmic enumeration the place it merits. 2/29

3 Table of participants Organisation Last name First name Position Involvem ent (months) Partner LaBRI, Université Bordeaux Université Bordeaux Université Bordeaux Université Bordeaux Université Bordeaux Université Bordeaux Université Bordeaux Université de Clermont-Ferrand Université de Clermont-Ferrand Université de Clermont-Ferrand Université Claude Bernard Lyon 1 Université de Clermont-Ferrand Université de Clermont-Ferrand Université de Clermont-Ferrand Université de Lorraine Université de Lorraine Université d Orléans Université d Orléans Université d Orléans Université de Lorraine COURCELLE Bruno Professor (pr. émérite) DORBEC Paul Associate professor Role 28 Co-coordinator Task 4: algorithmic meta-theorems 24 Local coordinator, LaBRI Coordinator Task 5: tractable classes of graphs and hyper-graphs DURAND Irène Associate 24 Co-coordinator Task 4: algorithmic professor meta-theorems GAVOILLE Cyril Professor 12 Coordinator Task 6: complexity and to be recruited, funded by this project to be recruited, funded by this project Research engineer Master students Partner LIMOS, Université de Clermont-Ferrand BEAUDOU Laurent Associate 12 professor KANTÉ Mamadou Associate Moustapha professor LIMOUZY Vincent Associate professor MARY Arnaud Associate professor hardness results 18 algorithmic meta-theorems 10 algorithmic meta-theorems 28 Local coordinator, LIMOS Coordinator Task 2: output-sensitive enumeration NOURINE Lhouari Professor 12 to be recruited PhD 28 output-sensitive enumeration to be recruited, funded by this project Post-doc 24 output-sensitive enumeration Partner LITA, Université de Lorraine 24 GÉLY Alain Associate professor KRATSCH Dieter Professor 32 Coordinator of the project and of LITA. Co-coordinator Task 1: organisation, synthesis and knowledge dissemination and Task 3: inputsensitive enumeration LIEDLOFF Mathieu Associate professor MONTEALE 12 Pedro PhD 12 GRE-BARBA TODINCA Ioan Professor 28 Co-coordinator Task 3: input-sensitive enumeration to be recruited, PhD 36 input-sensitive enumeration funded by this project 3/29

4 2. CONTEXT, POSITION AND OBJECTIVES OF THE PROPOSAL The P vs. NP question is arguably the most important open question in Theoretical Computer Science these days. Under the widely believed assumption that the complexity classes P and NP are not equal, there are problems which cannot be solved efficiently with the help of computers. Thus it is important to identify such problems and to find other ways of dealing with them than via the traditional means of polynomial-time algorithms. Unfortunately, many problems of great theoretical importance and also many problems that arise from real applications turn out to be intractable in the general case. These problems are typically in one of the following forms. An optimisation problem asks for a best solution (defined under some criteria) from a set of feasible solutions. Typically, the set of feasible solutions is huge, which makes the optimisation problem intractable. A counting problem asks for the number of feasible solutions, and an enumeration problem asks to list all feasible solutions. A decision problem, on the other hand, simply asks whether or not the answer to a given question is yes. For an example, the various versions of the famous Dominating Set problem can be defined for a given graph G as: find a minimal dominating set of smallest size (optimisation), find the number of minimal dominating sets (counting), list all minimal dominating sets (enumeration), and is there a minimal dominating set of size at most k? (decision). While optimisation problems are ubiquitous in Computer Science and decision problems are well-studied in particular in Algorithms and Complexity Theory, surprisingly little attention has been given to counting and enumeration. One possible explanation for this fact is the natural relation between the complexities of decision, optimisation, counting and enumeration versions of a problem. As can be seen from the above, a solution for the enumeration version of a problem typically provides an immediate solution for the optimisation, decision and counting versions of the problem, and clearly enumeration is at least as hard as the other versions. There seems to be a feeling saying that enumeration is much harder than optimisation, which directed the search for tractability and efficient algorithms of many researchers to decision and optimisation problems. However new insights from the study of the exact complexity of hard problems show that the relation between enumeration and optimisation is more subtle and worth a fundamental study. In this project we want to focus on algorithms and complexity of enumeration problems for graphs and hypergraphs. For many enumeration problems the number of generated objects is exponential in the input size. This motivated the output-sensitive approach in which the running time of an enumeration algorithm depends on the size of the input and the size of the output. This approach has a long tradition and is the classical one in enumeration algorithms. Recently research on exact exponential-time algorithms triggered a new approach to the design of enumeration algorithms. We call this the input-sensitive approach, which indeed is the one using classical worst-case running time analysis, i.e., the running time only depends on the length of the input. Another approach to solve enumeration problems on graphs is the use of algorithms parameterized by some parameter, among them in particular width parameters like tree-width, clique-width, rank-width, etc. Our principal goal in parameterized enumeration is to establish FPT delay results, where the delay between two enumerated objects is polynomial in the input size but may be exponential in the parameter. An important part of the last approach consists in studying a powerful and general FPT delay result (metatheorem) and showing that such meta-theorems are not necessarily impracticable, using a fly-automaton as a major tool in the implementations. The overall goals of all three approaches are similar and include in particular the construction and analysis of algorithms to enumerate certain objects in graphs and hypergraphs. The approaches are quite related, all study algorithms to solve particular enumeration problems on graphs and hypergraphs and heavily rely on combinatorics. They mainly differ in the way the running time of their algorithms is measured and which algorithms are considered as good or best ones. All approaches will contribute to a better understanding and an increasing knowledge on enumeration algorithms. 4/29

5 2.1. STATE OF THE ART Trivial enumeration, sometimes also called exhaustive search, is a simple algorithm design technique. Trivial enumeration of all possible solutions is at the heart of exact complexity. Kurt Gödel asked the following question already in 1956 in a famous letter to John von Neumann [Sip92]: It would be interesting to know,... how strongly in general the number of steps in finite combinatorial problems can be reduced with respect to simple exhaustive search. In modern wording this means: Is it possible that every problem (in NP) can be solved faster than by trivial exhaustive search, i.e., trivial enumeration of all solutions? Nowadays we are far from having any answer to this challenging question, see e.g. [Wil10]. In fact we cannot exclude that for many (decision and optimisation) problems the essentially best algorithm is one solving a corresponding enumeration problem (in input-sensitive approach). This is one of our principal motivations to concentrate our interest on algorithms and complexity of enumeration. Input-sensitive enumeration. Enumeration algorithms of exponential running time have been given small emphasis so far and the number of publications is very small as well. Nevertheless, within the recently established domain of exact exponential algorithms the study of enumeration problems in input-sensitive approach has found increasing interest. In a natural way such enumeration algorithms also provide combinatorial upper bounds on the maximum number of the objects to be enumerated, e.g. in a graph on n vertices. Amazingly this allows establishing combinatorial results via branching algorithms that had not been achieved within classical combinatorics before. Contrary to the classical output-sensitive enumeration, the input-sensitive one measures the running time of algorithms depending only on the length of the input. The interest in this approach is highly motivated by the success of exact exponential algorithms in the last decade. Enumeration is a natural part of research in exact exponential algorithms. Lawler's well-known coloring algorithm from 1976 is a dynamic programming algorithm over all maximal independent sets of a graph [Law76]. This requires an algorithm enumerating all maximal independent sets as well as an upper bound for the number of maximal independent sets in an n-vertex graph. On one hand, Moon and Moser had shown in 1965 that the maximum number of maximal independent sets in an n-vertex graph is exactly 3 n/3 [MoMo65]. On the other hand, Tsukiyama et al. provided a polynomial delay algorithm to enumerate all maximal independent sets of a graph [TIAS77]. Put together, the two results provide an O*(3 n/3 ) time algorihm enumerating all maximal independent sets (the O* notation ignores polynomial factors). Nowadays a simple branching algorithm provides both the enumeration within the same running time, and the combinatorial bound on the number of enumerated objects. This is a nice example for the interplay of output-sensitive enumeration, input-sensitive enumeration and combinatorial upper bounds. Most of the work on input-sensitive enumeration has been done in the last ten years. The typical question is the following. Given a property of graphs, what is the maximum number (n) of vertex subsets satisfying property in a graph on n vertices?. Besides the combinatorial question, one wishes to obtain an algorithm enumerating all vertex subsets with property in time O*( (n)). In fact, typically the combinatorial bound is achieved via an enumeration algorithm. For most important properties (with the notable exception of maximal independent sets), the best upper bounds for (n) are far from the best lower bounds: objects lower bound upper bound bibliography minimal dominating sets n n [FGPS08] minimal feedback vertex sets n n [FGPR08] minimal separators n n [GaMa15,FoVi12] potential maximal cliques n n [GaMa15,FTV15] The maximum number of minimal dominating sets in an n-vertex graph is a long-standing open question. The only answer available before the groundbreaking paper of Fomin et al. [FGPS08] and the use of exact exponential algorithms was the trivial one of 2 n. Several of those intriguing open problems are mentioned in 5/29

6 the book Exact exponential algorithms of Fomin and Kratsch 1 [FoKr10]. It was believed that in the case of (large) gaps between upper and lower bound, it is much more likely that the lower bound provides the correct value. Recent insights indicate that more efforts in the construction of lower bounds, possibly via computer, are needed, and they may indeed lead to improvements over the currently best known lower bounds (often constructed by hand ). Summarizing, progress on the currently known upper and lower bounds requires new tools and techniques. Improved upper bounds may be achieved by improved methods to analyse branching algorithms. The analysis of branching algorithms is a fundamental problem in the area of exact exponential algorithms and any progress would have important consequences throughout the whole area. The nowadays standard method for the analysis of these algorithms is the Measure & Conquer approach due to Fomin, Grandoni and Kratsch [FGK09]. However it is believed that all the current methods to analyse branching algorithms, including Measure & Conquer, do not suffice to obtain the worst-case running time of such algorithms, in the sense that the running time provided by the methods is over-estimated. Indeed, for some important branching algorithms even lower bounds of their running time have been studied, and even those weak lower bounds (for the complexity of a fixed algorithm and not for the complexity of the problem) do typically not match the upper bounds obtained by the analysis of the algorithm. Improved upper bounds on the number of enumerated objects and faster enumeration algorithms are not only of combinatorial interest. They would often imply faster algorithms to solve NP-hard problems exactly. For example, Fomin, Villanger and Todinca [FTV15] show that an improved estimation of the number of potential maximal cliques (currently O*( n )), together with a faster enumeration algorithm, would directly imply better bounds for challenging problems like computing tree-width, minimum fill-in, minimum feedback vertex set and also many other optimisation problems about induced subgraphs of bounded treewidth. Output-sensitive enumeration. The algorithmic study of enumeration problems has long been concentrating on what we shall call the output-sensitive approach. Since the number of objects to be enumerated is often exponential, any analysis of the running time in the so-called input-sensitive approach normally provides only exponential algorithms. Therefore it makes sense to measure the running time in dependence on the size of the input and the size of the output (also called total time). This allows enumeration algorithms of polynomial (total) running time, even when the number of generated objects is exponential. Output-sensitive enumeration has attracted several well-known researchers in the past [ChNi85, JPY88, LLK80, ReTa75, Tar73, TIAS77]. These results concentrate on listing objects in time polynomial in the number of listed objects and the size of the input. Algorithms achieving such time complexity are called output-polynomial algorithms. Particularly interesting output-polynomial algorithms are incremental polynomial algorithms and algorithms of polynomial delay. The latter are typically considered as the best enumeration algorithms (in the output-sensitive approach). From the complexity point of view it is worth mentioning that there are techniques to prove hardness results for output-sensitive enumeration of certain problems [KBEG08, LLK80], i.e., there is no output-polynomial enumeration algorithm for such a problem unless P=NP. There are also techniques for proving that a given problem does not admit an incremental polynomial or a polynomial delay algorithm (see for instance [Str10]). One of the most classical and widely studied enumeration problems is that of listing all minimal transversals of a hypergraph, i.e., all minimal hitting sets of its set of hyperedges. This problem, also called hypergraph dualization, is not only of great theoretical interest but it has also a number of applications in many areas, among them data mining and artificial intelligence [KRRT99, EiMa02, AgSr94, DoLi05, AMS+96, GKMT97, StUe05]. Whether or not all minimal transversals of a hypergraph can be listed in outputpolynomial time has been identified as a fundamental challenge in a long list of seminal papers. The quest for an output-polynomial algorithm to enumerate all minimal transversals of a hypergraph is illustrated by a 1 Underlined names correspond to members of this project. 6/29

7 large number of articles on the topic [GKMT97, EGM03, Elb02, KaSt05, BGKM02, FrKh96, JPY88, KBEG07, ElRa10], including output-polynomial algorithms for particular classes of hypergraphs. The problem remains unresolved despite continuous attempts since the 1980's and the fastest currently known algorithm for hypergraph dualization has quasi-polynomial (total) running time [FrKh96]. A polynomialspace, output-polynomial algorithm for generating all minimal transversals of a bounded intersection hypergraph is given in [KBEG07] which generalizes previous results in [EGM03]. Many enumeration algorithms (output-polynomial or not) for the dualization of general hypergraphs as well as for particular classes of hypergraphs have been proposed in the last decade [GKMT97, EGM03, EGM08, KaSt05, BGKM02, FrKh96, MuUn14]. Recently, Kanté, Limouzy, Mary and Nourine [KLMN14] have shown that there is an output-polynomial time algorithm to enumerate all minimal dominating sets of a given graph if and only if there is an outputpolynomial time algorithm to enumerate all minimal transversals of a given hypergraph. We say that both enumeration problems are equivalent (actually, they are also equivalent w.r.t. polynomial delay algorithms). We recall that a subset of vertices of a graph G is a dominating set of G if every vertex is either in the set or has a neighbor in the set. Such a set is minimal if no proper subset of it is a dominating set. Dominating sets form one of the best studied notions in graph theory and graph algorithms. The number of papers on domination in graphs is in the thousands, and several well known surveys and books are dedicated to the topic (see, e.g., [HaHe98]). The newly proved equivalence of hypergraph dualization and enumerating all minimal dominating sets of a graph allows new ways to attack the long-standing open problem of hypergraph dualization. This triggered the research on the enumeration of all minimal dominating sets. Many tractable graph classes, i.e., those for which enumeration of all minimal dominating sets can be done in output-polynomial time, have been found recently [KLMN12, KLMN13, GHKV13]. Finally this also triggered the interest in enumeration of other types of objects in graphs. Nevertheless for various objects in graphs (with corresponding NP-hard optimisation problems) little is known about enumeration either outputsensitive or input-sensitive or both. To mention a few examples: enumerating minimal connected dominating sets, maximal k-colorable induced subgraphs and minimal Steiner trees. Parameterized enumeration. Parameterized computation and complexity is a very successful approach to solve hard (graph) problems and to study their complexity. Technically the problem is equipped with a parameter k and running time is measured in input length and parameter. The holy grail are fixed-parameter tractable (FPT) algorithms and polynomial kernels. We shall mainly be interested in FPT algorithms, having running time f(k) p(n), where f is a computable function 2 and p is a polynomial in n. Naturally parameterized algorithms, and in particular FPT algorithms may be studied for enumeration problems as well; they might be seen as parameterized input-sensitive. Fernau [Fer02] was the first to study such algorithms. Some but not much work has been done in this direction. Output-sensitive algorithms for parameterized enumeration problems have a running time that is measured in input length, output length and parameter. The most interesting class of such algorithms are those having fixed-parameter tractable delay, i.e. delay f(k) p(n), also called delay-fpt [CMM+13]. Very little is known about combining FPT and enumeration except for the important case when k is a width parameter. A powerful generic tool for the design of FPT algorithms are the so-called algorithmic meta-theorems, initiated by the seminal paper of Courcelle [Cou90], stating that every monadic second-order formula can be checked in linear time in graph classes of bounded tree-width. Algorithmic meta-theorems are of the following general form: For each class C of graphs of type T, for each problem P specified in logical language L, one can build an efficient algorithm for solving P over the graphs in C. Typically the class of graphs T depends on a width parameter, and the running time of the algorithms is FPT in this parameter. In the literature, the most famous meta-theorems are the following. 1. Monadic Second-Order (MSO for short) decision problems (model-checking) for graphs of bounded tree-width or clique-width [DoFe99, FlGr06, CoEn12, CoDu12]; they give FPT algorithms, exponential in the width of the decomposition, but linear in the size of the input graph. 2 Any computable function f in theory, but hopefully, a reasonable one for having usable algorithms. 7/29

8 2. First-order (FO for short) decision problems (model-checking) for graphs of locally bounded treewidth [Fri04] or more generally those that locally exclude a minor [DGK07], graphs of bounded expansion [DKT10] and nowhere dense graphs [GKS14]. 3. MSO query evaluation (includes enumerating or counting solutions) [FFG02]. 4. Enumeration problems (with linear FPT delay between two answers) for MSO queries on graphs of bounded tree-width or clique-width [Bag02, Cou09]), and FO queries on subclasses of graphs of locally bounded tree-width or clique-width [Fri04, CGK11]. 5. Labelling problems for MSO queries (or optimisation of MSO queries such as distance queries) in graphs of bounded tree-width or clique-width [CoVa03], or for FO queries in some subclasses of graphs of locally bounded tree-width or clique-width [CGK11]. While this list is not exhaustive, it is representative of the kinds of meta-theorems encountered in the literature. A key tool used in almost all cases is the construction of a finite automaton on terms representing graphs of bounded tree-width or clique-width, checking that the graph defined by the considered term satisfies a fixed given MSO property. The difficulty met when trying to implement these meta-theorems is that the considered automata are huge and cannot be constructed in the usual sense, with a transition table. Hence is often considered that these meta-theorems are only of theoretical interest because they are intractable due to the size of the automata. Some solutions (as e.g. one based on game theory by Kneis et al. [KLR11]) have been proposed in the literature to cope with the impracticability, but none of them is satisfactory. Recently, Courcelle and Durand introduced the notion of fly-automaton, that computes its necessary transitions on the fly [CoDu12, CoDu15] and that seems to be a remedy for the impracticability. This notion is flexible in several respects: these automata can be combined so as to reflect the structure of a logical formula describing the problem. Those for some basic graph properties can be defined directly, without using logic. They allow infinite sets of states (for example a tuple of counters) so that they can check properties that are not MSO expressible. They have been implemented and tested on large graphs (see [CoDu12]). Fly-automata can be easily adapted to optimisation, counting and enumeration problems as shown by Durand [Dur12]. However, a lot remains to do concerning their theory, applications and implementation OBJECTIVES, ORIGINALITY AND NOVELTY OF THE PROJECT Input-sensitive enumeration. Our major goal is to make significant contributions to the design and analysis of algorithms for input-sensitive enumeration problems and the study of their complexity. Finding better tools to analyse branching algorithms is a major issue in the research on exact exponential algorithms and one of our ambitious objectives. Branching algorithms are the main approach for inputsensitive enumeration. However analysing branching algorithms such as to obtain the worst-case running time is a tough task. Measure & Conquer was a great progress in this direction and might be the best possible when using linear recurrences in the time analysis, but generally it does not produce tight bounds. The question is whether one can analyse branching algorithms without using linear recurrences. Another direction is the study of non-standard and often surprising measures, which may lead (see e.g. [GaMa15]) to better and simpler analysis. We also plan to improve upon the gap between lower and upper bounds for well-studied combinatorial questions like the maximum number of minimal dominating sets, minimal feedback vertex sets and minimal separators of an n-vertex graph. This can be done by improving upon the time analysis, improved enumeration algorithms, and also by improving the construction of lower bounds, including automatic construction. Finally we shall try in some of the most interesting cases, like enumerating all minimal dominating sets of a graph, to prove that the best known lower bound is optimal, for example by an inductive proof. We shall also study the enumeration of all maximal vertex subsets S of an undirected graph G such that G[S], the subgraph induced by S, does not have an induced subgraph isomorphic to a fixed graph H. This is also equivalent to a certain enumeration problem on hypergraphs. 8/29

9 The complexity and hardness in input-sensitive enumeration is even less studied than combinatorial bounds. There are two directions here. First we would like to know good candidates for optimisation/decision problems that cannot be solved faster than by trivial enumeration. Clearly this cannot be answered without a suitable hypothesis. The three classical ones have the following consequences (see [LMS11] for definitions of ETH and SETH): (1) The problem SAT on n variables cannot be solved in polynomial time (the P NP hypothesis). (2) The problem 3-SAT on n variables cannot be solved in subexponential time (Exponential Time Hypothesis, ETH). (3) The problem SAT on n variables cannot be solved in time O*((2-ε) n ), for any ε>0 (Strong Exponential Time Hypothesis, SETH). The last one says that trivial enumeration is (essentially) the best algorithm to solve SAT. It was shown by Cygan et al. that, assuming SETH, this implies related hardness properties for other problems [CDLM12]. We intend to show similar results for enumeration problems either by using similar reductions or by using variants of ETH and SETH for enumeration. Second, we intend to compare the time complexities of optimisation, counting and enumeration of benchmark problems, by comparing the bases of their exponential running times. There are problems for which optimisation currently is much faster than enumeration. E.g., enumerating all minimal dominating sets can be done in time O( n ) [FGPS08] while the best known algorithm to compute a minimum dominating set runs in time O( n ) [Iwa12]. For other problems the best known exact exponential-time complexities of optimisation and enumeration are the same, or almost the same, as e.g. for the problem SAT. We intend to study this type of complexity questions for a collection of well-studied graph problems. Currently, most algorithms in input-sensitive enumeration concern the enumeration of vertex subsets with particular properties in undirected graphs. We plan to study other types of problems, like coloring and permutation problems, as well as problems on directed graphs. We intend to find new techniques for the design of input-sensitive enumeration algorithms. In particular, we want to find out whether (and which) algorithmic techniques of output-sensitive enumeration can also be applied for input-sensitive enumeration. Output-sensitive enumeration. Our major goal is to make significant contributions to the construction and analysis of output-polynomial enumeration algorithms for graphs and hypergraphs. The main problem we are planning to attack is the hypergraph dualization problem (also called transversal problem), which asks for an output-sensitive algorithm enumerating all minimal transversals of an input hypergraph. This question is open for more than 50 years and has created many publications and seminal papers. Until now, the best algorithm known is the one by Fredman and Khachiyan, which is quasi-polynomial [FrKh96]. However, for several important hypergraph classes it was shown that an output-polynomial algorithm exists, e.g., k- conformal hypergraphs [KBEG07], k-degenerate hypergraphs [EGM03], bounded-intersection edge hypergraphs [KBEG07], circuits, bases, hyperplanes, flats of a matroid [KBE+05], some geometric hypergraphs [ElRa10]; all such classes are called tractable. Contrary to the original research on hypergraph dualization including in particular the finding of tractable classes of hypergraphs, we shall concentrate on the enumeration of all minimal dominating sets in graphs. It is the most ambitious goal of our project to find out whether hypergraph dualization can be solved in output-polynomial time. To be honest, there is clearly a risk of not being able to answer this principal open question. Nevertheless the new approach, i.e., the quest for an output-polynomial enumeration algorithm for all minimal dominating sets of a graph is very promising and worth to be exploited even if it would lead only to a collection of tractable graph classes. Another attack is trying to prove that there is no output-polynomial algorithm enumerating all minimal dominating sets of a graph unless P=NP, which would also settle the question concerning hypergraph dualization. The currently known reductions for such proofs seem hard to apply to that problem; but of course the nonexistence of an output-polynomial algorithms needs to be studied seriously. The lack of success of so many well-reputated researchers over such a long period of time also might be explained by a lack of knowledge on the structure of transversals in hypergraphs. Using graphs instead of hypergraphs, one can hope to better understand the structure of the objects to be enumerated, e.g. minimal dominating sets. Indeed, structural graph theory has proven its importance in algorithmic graph theory and some members of the project have proven their expertise in studying graph structures and their use in order to obtain new algorithmic results (as e.g. Courcelle, Gavoille and Kratsch). We intend to use graph structural properties in order to understand the structure of minimal dominating sets and therefore obtain algorithms for new (hyper)graph classes and also get an output-polynomial algorithm that summarizes many of the known tractable cases in (hyper)graphs. Moreover, Kanté et al. [KLMN12, KLMN14] have given some evidence 9/29

10 that by using graph structure one may obtain simple and new output-polynomial algorithms for enumerating all minimal dominating sets in certain graph classes. The most interesting results in this direction are for line graphs [GHKV13] co-authored by Kratsch, and for chordal graphs (unpublished work by Kanté, Limouzy, Mary, Nourine and Uno). If we fail to settle the principal question whether there is an output-polynomial algorithm to enumerate all minimal dominating sets of a graph, then we would like to understand at least why it admits a quasipolynomial time algorithm and whether there exists a complexity class for which it is complete. Such a task is not easy since we need to compare it with other problems that admit quasi-polynomial time algorithms like the ones in the LOGNP or NP[log2] class (see [PaYa96] for some examples). However, we believe that understanding the structure of minimal dominating sets will be extremely helpful in this task. Another goal of our research on output-sensitive enumeration is to develop new techniques for the construction and analysis of output-polynomial algorithms. Let us first mention some known techniques. Incremental enumeration allows to obtain upper bounds for the output-sensitive approach in enumeration [Elb02]. The backtrack technique is to enumerate objects that satisfy a (given) property and then the objects that do not satisfy this property. For example to enumerate all minimal transversals, one considers the subtransversal problem, which asks whether there exists a minimal transversal that contains a given set of vertices. A polynomial time algorithm for the subtransversal problem would imply an output-polynomial algorithm to enumerate all minimal transversals. Finally let us mention the reverse search technique, proposed by Avis and Fukuda [AvFu96], which is one of the most powerful techniques in enumeration algorithms. Typically the enumeration will be done by traversing a transition graph, often in a depth-firstsearch or a breadth-first search manner. Based on this method, a so-called flipping method to enumerate all minimal dominating sets of a graph has been developed [GHKV13] which enables output-polynomial algorithms for various graph classes. Finally we shall study output-sensitive algorithms for enumerating other objects in graphs. Many natural graph problems have not been studied with respect to output-sensitive enumeration. Doing this will among others enable a better understanding of enumeration in graph algorithms. Interesting questions that we intend to study are output-sensitive enumeration of minimal connected dominating sets of a graph, minimal dominating cliques, minimal total dominating sets, square roots of a graph, k-colored induced subgraphs of a graph (k 2), minimal Steiner trees, longest paths etc. Particularly interesting are problems where connectedness is part of the definition, which are inherently different from enumeration of minimal transversals in a hypergraphs and minimal dominating sets in graphs. We intend to define and study notions of connected dualization of hypergraphs which should generalize enumeration of connected objects in graphs. Parameterized complexity. On one hand, we are planning a fundamental study of parameterized inputsensitive and output-sensitive enumeration and the powers of those approaches comparing them with their classical counterparts. Searching the most interesting parameters, developing techniques to construct efficient algorithms of those types, as well as new hardness proof techniques are our main goals. Besides width parameters (see below) we are planning to study other input parameters and, if interesting, also output parameters. Only few publications adressing those approaches (on graphs) are known so far. Note that the standard parameter in parameterized algorithms, often called solution size, does not seem to provide interesting results in enumeration. Also worth to be mentioned, a desirable justification for using parameterization would be (like in classical parameterized complexity) a proof that no efficient nonparameterized algorithm of the corresponding type exists for the problem. On the other hand, a well-known meta-theorem provides a class of algorithms having FPT delay. To remedy the impracticability of meta-theorems, Courcelle and Durand have introduced the notion of fly-automata that helps in the understanding of this impracticability in concrete situations. The notion of fly-automata has also proven to be of interest for obtaining new algorithmic results beyond the ones based on MSO or FO logic (as considered by meta-theorems) since they allow infinite sets of states and infinite signature. We intend to develop the study of fly-automata in two steps. 10/29

11 In the first step, we plan to develop the pragmatics of fly-automata in several ways. First, since every term describing a graph has a linear order, we can take advantage of it in the writing of formulas. For example if we want to enumerate (that is, to list) the induced triangles, we can avoid repetitions due to the automorphisms of cliques by using the linear order on vertices coming from the input term. Second, several equivalent formulas do not yield the same automata. It is thus interesting to understand which ways of writing a formula will produce the best (smallest) automata. Third, we plan to push to the limit (for large, randomly generated graphs) the existing implementation, and understand ways to improve them. Fourth, we aim at understanding in which cases it is crucial to use a best possible term (implementing the exact treewidth or clique-width), and in which cases this is not that important. Such a goal can be combined with attempts in FPT theory and input-sensitive area looking for algorithms that are singly exponential in the width. This topic mixes theory and analysis of runs in the considered implementations. We cannot hope to get from meta-theorems fine complexity analysis as those described in Section 2.1. The interest is in the flexibility of using them. This part of the project includes implementations of meta-theorems in system Autograph, developed at LaBRI, which allows to check the usability of our algorithms. In the second step, we want to extend these tools to FO queries in graphs of locally bounded width on the basis of existing works ([Fri04, DKT10, Kan08]). Meta-theorems on graphs of locally bounded tree-width (or clique-width or those of bounded expansion) are based on coverings of the graphs by subgraphs of bounded tree-width (or clique-width or tree-depth) and the locality of FO formulas by Gaifman [Gai82]. However, the Gaifman locality theorem yields local formulas of huge size which are not usable in practice (this is the first difficulty to overcome). So, in order to get practical algorithms as those obtained by Courcelle and Durand with the use of fly-automata, one first needs to get a practical version of the Gaifman locality theorem or try another technique not based on the Gaifman locality theorem. Another challenge in graphs of locally bounded width is whether we can enumerate (or count) solutions of every FO query in graphs of locally bounded width. Attempts based on the techniques used for the model-checking have been tried without success [Fri04, CGK11], and we need to develop new equivalent definitions or structural decompositions of graphs of locally bounded width that are suitable for counting or enumeration of solutions of FO queries. This topic is intrinsically more difficult. We do not promise a working implementation, but we may expect a theoretical advance. Besides the common goals of such a project bringing together various groups of french researchers interested in algorithmic enumeration in graphs, like cumulating knowledge on particular enumeration problems and scientific production, our principal motivation is to establish an enumeration community in France and is also to help making enumeration an important research subject world-wide in algorithmics. A workshop at the Lorentz Center in Leiden in August co-organised by the leader of this project has the aim to bring researchers of different research direction in enumeration together. Like it has happened in other research directions we intend to provide strong support to creating an enumeration community. Writing a book on algorithmic enumeration will be another cornerstone in giving algorithmic enumeration the place it merits. 3. SCIENTIFIC AND TECHNICAL PROGRAMME, PROJECT ORGANISATION 3.1. SCIENTIFIC PROGRAMME AND PROJECT STRUCTURE The main tasks of the project correspond to the three research directions that have been already mentioned: output-sensitive enumeration, input-sensitive enumeration and parameterized enumeration. Moreover, we have identified two transversal tasks: tractable classes of graphs and hypergraphs, and complexity theory and hardness results. Each of these tasks has one or two leaders, which coordinate and animate the work. Besides these purely scientific tasks, we distinguish a task of organisation, synthesis and knowledge dissemination, coordinated by the project leader with the help of local leaders. Task Title/leaders Subtasks T1 Organisation, synthesis and knowledge T1.1 Project coordination dissemination T1.2 Knowledge acquisition and dissemination Leaders: D. Kratsch, P. Dorbec, M.M. Kanté T1.3 Organisation of meetings and workshops 11/29

12 T2 T3 T4 T5 T6 Output-sensitive enumeration Leader: M.M. Kanté Input-sensitive enumeration Leaders: D. Kratsch, I. Todinca Parameterized enumeration Leaders: B. Courcelle, I. Durand Tractable classes of graphs and hypergraphs Leader: P. Dorbec Complexity theory and hardness results Leader: C. Gavoille T2.1 Minimal transversals and minimal dominating sets T2.2 Graphical objects T3.1 Algorithm design T3.2 Analysis techniques T3.3 Combinatorial bounds T3.4 Graphs of bounded tree-width and MSOL T4.1 Algorithmic meta-theorems T4.2 Output and input-sensitive parameterization T5.1 Structure of graphs and hypergraphs T5.2 Algorithm design T6.1 Complexity theory T6.1 Hardness of enumeration problems The relations between tasks are depicted below DESCRIPTION BY TASK TASK 1 ORGANISATION, SYNTHESIS AND KNOWLEDGE DISSEMINATION Leaders: D. Kratsch, P. Dorbec, M.M. Kanté The GraphEn project will be managed by the project leader Dieter Kratsch (LITA, Metz), with the support of the partners and mainly the local coordinators P. Dorbec (LaBRI, Bordeaux) and M.M. Kanté (LIMOS, Clermont-Ferrand). The goal is also to increase the scientific knowledge of all participants in the project's topics, and on the other hand to inform the community on our results. T1.1: Project coordination. The management of the project will be coordinated with the help of a wiki website, collecting scientific and practical information. A particular care will be given to the scientific production of the project (publications...) and to the project reports for the ANR. T1.2: Knowledge acquisition and dissemination. Each scientific leader will share his specific knowledge with the other participants. The kick-off meeting of the project will be mainly devoted to such tutorials, but we shall continue to give some overview talks at each meeting of the project. The 12/29

13 results obtained by our project will be presented to our members and to a broader audience. Besides the meetings organised by GraphEn, we will give talks in seminars, national meetings (e.g. Journées Graphes et Algorithmes and Complexité des Algorithmes meetings of the GDR Informatique Mathématique, Ecoles Jeunes Chrecheurs...) and will present our results in international conferences and journals. A survey document with state of the art, the most important achievements of the project and the bibliography will be maintained. T1.3: Organisation of meetings and workshops. At least twice a year, we will organise a workshop with two sections: a technical section reserved to the members of the project, and an open section for presenting the problems (state of the art) and the achieved results to a wider audience. In the second year of the project we will organise an international school on the topics of the project. During the fourth year of the project, we shall organise a workshop of international audience, e.g. in Dagstuhl, Leiden or CIRM (Marseille), as a successor of the one at Leiden in August TASK 2 OUTPUT-SENSITIVE ENUMERATION Leader: M.M. Kanté The aim of this task is to construct and analyse output-polynomial algorithms for enumeration problems on graphs and hypergraphs. Our main goal is to study the enumeration of all minimal transversals of a hypergraph (hypergraph dualization) which is one of the most ambitious tasks of this project. In particular, we are going to study the enumeration of all minimal dominating sets of a graph, for graphs in general and also for a collection of graph classes, as e.g. bipartite, unit disk, (multi) tolerance and nowhere dense graphs. A principal aim will be to achieve not only output-polynomial algorithms, but to also aim at even better algorithms as for example incremental polynomial algorithms and algorithms of polynomial delay. T2.1: Minimal transversals and minimal dominating sets. Constructing outpout-polynomial or even polynomial delay algorithms for the enumeration of all minimal dominating sets (or equivalently of all minimal transversals of hypergraphs) that use polynomial space is at the heart of this task. Several techniques have been used in the past to obtain outpout-polynomial and even polynomial delay algorithms for enumeration problems, as e.g., for enumerating spanning trees, maximal cliques and minimal feedback vertex sets. To mention one important technique, it appears that the reverse search strategy is often succesful; many known polynomial delay algorithms use it explicitely or implicitely. We intend to further develop this technique and adapt it to the problems we study. New techniques to design algorithms might be needed to establish output-polynomial algorithms for our major enumeration problems, as for example the recently developed flipping method. The major enumeration problems of this task will also be studied extensively for classes of graphs and classes of hypergraphs. Here the goal is to use structural properties of such a class to design best possible enumeration algorithms. To do this, like in optimisation algorithms, we have to find out which structure enables good algorithms. T2.2: Graphical objects. While some enumeration problems on graphs and hypergraphs are wellstudied and fundamental algorithmic and complexity-theoretic results are available, the research on enumeration for objects in graphs is in a nascent state. Despite the huge knowledge on graph algorithms for optimisation problems plenty of natural objects in graphs have never even been considered with respect to output-sensitive algorithms. Our goal is to construct and analyse enumeration algorithms for a large collection of objects in graphs. Particularly intriguing are objects for which connectedness is crucial, as e.g., minimal connected dominating sets, minimal Steiner trees, minimal connected vertex covers. Other objects in graphs to be studied w.r.t. output-sensitive enumeration are minimal dominating cliques, minimal (respectively maximal) induced subgraphs of property (e.g; k-colorable, tree-width at most k, rank-width at most k), square roots and matching cuts. Those problems will always be studied for general graphs and for well-chosen graph classes (restriced inputs). A principal goal of this task is to collect much more material and knowledge on enumeration in graphs than is available at the moment. This research will lead to a better understanding of outputsensitive enumerations, both its limits and the typical algorithms to achieve for certain problems and 13/29