No abstract

We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given ℓ terminals is separated from the others, (b) each of the given ℓ pairs of terminals is separated, (c) exactly ℓ vertices are cut away from the graph, (d) exactly ℓ connected vertices are cut away from the graph, (e) the graph is separated into at least ℓ components. We show that if both k and ℓ are parameters, then (a), (b) and (d) are fixed-parameter tractable, while (c) and (e) are W[1]-hard.

Relational joins are at the core of relational algebra, which in turn is the core of the standard database query language SQL. As their evaluation is expensive and very often dominated by the output size, it is an important task for database query optimisers to compute estimates on the size of joins and to find good execution plans for sequences of joins. We study these problems from a theoretical perspective, both in the worst-case model, and in an average-case model where the database is chosen according to a known probability distribution. In the former case, our first key observation is that the worst-case size of a query is characterised by the fractional edge cover number of its underlying hypergraph, a combinatorial parameter previously known to provide an upper bound. We complete the picture by proving a matching lower bound, and by showing that there exist queries for which the join-project plan suggested by the fractional edge cover approach may be substantially better than any join plan that does not use intermediate projections. On the other hand, we show that in the average-case model, every join-project plan can be turned into a plan containing no projections in such a way that the expected time to evaluate the plan increases only by a constant factor independent of the size of the database. Not surprisingly, the key combinatorial parameter in this context is the maximum density of the underlying hypergraph. We show how to make effective use of this parameter to eliminate the projections.

Many important combinatorial problems can be modeled as constraint satisfaction problems. Hence identifying polynomial-time solvable classes of constraint satisfaction problems has received a lot of attention. In this paper, we are interested in structural properties that can make the problem tractable. So far, the largest structural class that is known to be polynomial-time solvable is the class of bounded hypertree width instances introduced by Gottlob et al. [2002]. Here we identify a new class of polynomial-time solvable instances: those having bounded fractional edge cover number.Combining hypertree width and fractional edge cover number, we then introduce the notion of fractional hypertree width. We prove that constraint satisfaction problems with bounded fractional hypertree width can be solved in polynomial time (provided that a the tree decomposition is given in the input). Together with a recent approximation algorithm for finding such decompositions [Marx 2010a], it follows that bounded fractional hypertree width is now the most general known structural property that guarantees polynomialtime solvability.

Given an undirected graph G, a collection {(s 1 , t 1 ), . . . , (s k , t k )} of pairs of vertices, and an integer p, the Edge Multicut problem asks if there is a set S of at most p edges such that the removal of S disconnects every s i from the corresponding t i . Vertex Multicut is the analogous problem where S is a set of at most p vertices. Our main result is that both problems can be solved in time 2 O(p 3 ) · n O(1) , i.e., fixed-parameter tractable parameterized by the size p of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form f (p) · n O(1) exists for the directed version of the problem, as we show it to be W[1]-hard parameterized by the size of the cutset. Introduction. From the classical results of Ford and Fulkerson on minimum s − t cuts [20] to the more recent O(√ log n)-approximation algorithms for sparsest cut problems [44,1,18], the study of cut and separation problems has a deep and rich theory. One well-studied problem in this area is the Edge Multicut problem: given a graph G and pairs of vertices (s 1 , t 1 ), . . . , (s k , t k ), remove a minimum set of edges such that every s i is disconnected from its corresponding t i for every 1 ≤ i ≤ k. For k = 1, Edge Multicut is the classical s − t cut problem and can be solved in polynomial time. For k = 2, Edge Multicut remains polynomial-time solvable [46], but it becomes NP-hard for every fixed k ≥ 3 [15]. Edge Multicut can be approximated within a factor of O(log k) in polynomial time [22] (even in the weighted case, where the goal is to minimize the total weight of the removed edges). However, under the unique games conjecture of Khot [29], no constant factor approximation is possible [7]. One can analogously define the Vertex Multicut problem, where the task is to remove a minimum set of vertices. An easy reduction shows that the vertex version is more general than the edge version.Using brute force, one can decide in time n O(p) if a solution of size at most p exists. Our main result is a more efficient exact algorithm for small values of p (the O * notation hides factors that are polynomial in the input size).

Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type, and propose directions for future research.

We introduce graph motif parameters, a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs. Classical works by Lovász show that many interesting quantities have this form, including, for fixed graphs H, the number of H-copies (induced or not) in an input graph G, and the number of homomorphisms from H to G.Using the framework of graph motif parameters, we obtain faster algorithms for counting subgraph copies of fixed graphs H in host graphs G: For graphs H on k edges, we show how to count subgraph copies ofby a surprisingly simple algorithm. This improves upon previously known running times, such as O(n 0.91k+c ) time for k-edge matchings or O(n 0.46k+c ) time for k-cycles. Furthermore, we prove a general complexity dichotomy for evaluating graph motif parameters: Given a class C of such parameters, we consider the problem of evaluating f ∈ C on input graphs G, parameterized by the number of induced subgraphs that f depends upon. For every recursively enumerable class C, we prove the above problem to be either FPT or #W[1]-hard, with an explicit dichotomy criterion. This allows us to recover known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms in a uniform and simplified way, together with improved lower bounds.Finally, we extend graph motif parameters to colored subgraphs and prove a complexity trichotomy:

An important question in the study of constraint satisfaction problems (CSP) is understanding how the graph or hypergraph describing the incidence structure of the constraints influences the complexity of the problem. For binary CSP instances (i.e., where each constraint involves only two variables), the situation is well understood: the complexity of the problem essentially depends on the treewidth of the graph of the constraints [27,43]. However, this is not the correct answer if constraints with unbounded number of variables are allowed, and in particular, for CSP instances arising from query evaluation problems in database theory. Formally, if H is a class of hypergraphs, then let CSP(H) be CSP restricted to instances whose hypergraph is in H. Our goal is to characterize those classes of hypergraphs for which CSP(H) is polynomial-time solvable or fixed-parameter tractable, parameterized by the number of variables. Note that in the applications related to database query evaluation, we usually assume that the number of variables is much smaller than the size of the instance, thus parameterization by the number of variables is a meaningful question.The most general known property of H that makes CSP(H) polynomial-time solvable is bounded fractional hypertree width. Here we introduce a new hypergraph measure called submodular width, and show that bounded submodular width of H (which is a strictly more general property than bounded fractional hypertree width) implies that CSP(H) is fixed-parameter tractable. In a matching hardness result, we show that if H has unbounded submodular width, then CSP(H) is not fixed-parameter tractable (and hence not polynomial-time solvable), unless the Exponential Time Hypothesis (ETH) fails. The algorithmic result uses tree decompositions in a novel way: instead of using a single decomposition depending on the hypergraph, the instance is split into a set of instances (all on the same set of variables as the original instance), and then the new instances are solved by choosing a different tree decomposition for each of them. The reason why this strategy works is that the splitting can be done in such a way that the new instances are "uniform" with respect to the number extensions of partial solutions, and therefore the number of partial solutions can be described by a submodular function. For the hardness result, we prove via a series of combinatorial results that if a hypergraph H has large submodular width, then a 3SAT instance can be efficiently simulated by a CSP instance whose hypergraph is H. To prove these combinatorial results, we need to develop a theory of (multicommodity) flows on hypergraphs and vertex separators in the case when the function b(S) defining the cost of separator S is submodular, which can be of independent interest. * A preliminary version of the paper was presented at STOC 2010.