A Parameterized Study of Maximum Generalized Pattern Matching Problems

The generalized function matching (GFM) problem has been intensively studied starting with Ehrenfreucht and Rozenberg (Inf Process Lett 9(2):86–88, 1979). Given a pattern p and a text t, the goal is to find a mapping from the letters of p to non-empty substrings of t, such that applying the mapping to p results in t. Very recently, the problem has been investigated within the framework of parameterized complexity (Fernau et al. in FSTTCS, 2013). In this paper we study the parameterized complexity of the optimization variant of GFM (called Max-GFM), which has been introduced in Amir and Amihood (J Discrete Algorithms 5(3):514–523, 2007). Here, one is allowed to replace some of the pattern letters with some special symbols “?”, termed wildcards or don’t cares, which can be mapped to an arbitrary substring of the text. The goal is to minimize the number of wildcards used. We give a complete classification of the parameterized complexity of Max-GFM and its variants under a wide range of parameterizations, such as, the number of occurrences of a letter in the text, the size of the text alphabet, the number of occurrences of a letter in the pattern, the size of the pattern alphabet, the maximum length of a string matched to any pattern letter, the number of wildcards and the maximum size of a string that a wildcard can be mapped to

Beyond Worst-Case Analysis for Joins with Minesweeper

We describe a new algorithm, Minesweeper, that is able to satisfy stronger runtime guarantees than previous join algorithms (colloquially, `beyond worst-case guarantees') for data in indexed search trees. Our first contribution is developing a framework to measure this stronger notion of complexity, which we call {\it certificate complexity}, that extends notions of Barbay et al. and Demaine et al.; a certificate is a set of propositional formulae that certifies that the output is correct. This notion captures a natural class of join algorithms. In addition, the certificate allows us to define a strictly stronger notion of runtime complexity than traditional worst-case guarantees. Our second contribution is to develop a dichotomy theorem for the certificate-based notion of complexity. Roughly, we show that Minesweeper evaluates $\beta$-acyclic queries in time linear in the certificate plus the output size, while for any $\beta$-cyclic query there is some instance that takes superlinear time in the certificate (and for which the output is no larger than the certificate size). We also extend our certificate-complexity analysis to queries with bounded treewidth and the triangle query.Comment: [This is the full version of our PODS'2014 paper.

On the Complexity Landscape of Connected f-Factor Problems

Given an n-vertex graph G and a function f:V(G) -> {0, ..., n-1}, an f-factor is a subgraph H of G such that deg_H(v)=f(v) for every vertex v in V(G); we say that H is a connected f-factor if, in addition, the subgraph H is connected. A classical result of Tutte (1954) is the polynomial time algorithm to check whether a given graph has a specified f-factor. However, checking for the presence of a connected f-factor is easily seen to generalize Hamiltonian Cycle and hence is NP-complete. In fact, the Connected f-Factor problem remains NP-complete even when f(v) is at least n^epsilon for each vertex v and epsilon<1; on the other side of the spectrum, the problem was known to be polynomial-time solvable when f(v) is at least n/3 for every vertex v. In this paper, we extend this line of work and obtain new complexity results based on restricting the function f. In particular, we show that when f(v) is required to be at least n/(log n)^c, the problem can be solved in quasi-polynomial time in general and in randomized polynomial time if c 1, the problem is NP-intermediate

Parameterized Compilation Lower Bounds for Restricted CNF-formulas

We show unconditional parameterized lower bounds in the area of knowledge compilation, more specifically on the size of circuits in decomposable negation normal form (DNNF) that encode CNF-formulas restricted by several graph width measures. In particular, we show that - there are CNF formulas of size $n$ and modular incidence treewidth $k$ whose smallest DNNF-encoding has size $n^{\Omega(k)}$, and - there are CNF formulas of size $n$ and incidence neighborhood diversity $k$ whose smallest DNNF-encoding has size $n^{\Omega(\sqrt{k})}$. These results complement recent upper bounds for compiling CNF into DNNF and strengthen---quantitatively and qualitatively---known conditional low\-er bounds for cliquewidth. Moreover, they show that, unlike for many graph problems, the parameters considered here behave significantly differently from treewidth

Hypergraph Acyclicity and Propositional Model Counting

We show that the propositional model counting problem #SAT for CNF- formulas with hypergraphs that allow a disjoint branches decomposition can be solved in polynomial time. We show that this class of hypergraphs is incomparable to hypergraphs of bounded incidence cliquewidth which were the biggest class of hypergraphs for which #SAT was known to be solvable in polynomial time so far. Furthermore, we present a polynomial time algorithm that computes a disjoint branches decomposition of a given hypergraph if it exists and rejects otherwise. Finally, we show that some slight extensions of the class of hypergraphs with disjoint branches decompositions lead to intractable #SAT, leaving open how to generalize the counting result of this paper

Towards a Polynomial Kernel for Directed Feedback Vertex Set

In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph D and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen, Liu, Lu, O\u27Sullivan and Razgon [JACM 2008]; since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics. In this paper, we study DFVS parameterized by the feedback vertex set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus

Backdoors to q-Horn

The class q-Horn, introduced by Boros, Crama and Hammer in 1990, is one of the largest known classes of propositional CNF formulas for which satisfiability can be decided in polynomial time. This class properly contains the fundamental classes of Horn and Krom formulas as well as the class of renamable (or disguised) Horn formulas. In this paper we extend this class so that its favorable algorithmic properties can be made accessible to formulas that are outside but "close"\u27 to this class. We show that deciding satisfiability is fixed-parameter tractable parameterized by the distance of the given formula from q-Horn. The distance is measured by the smallest number of variables that we need to delete from the formula in order to get a q-Horn formula, i.e., the size of a smallest deletion backdoor set into the class q-Horn. This result generalizes known fixed-parameter tractability results for satisfiability decision with respect to the parameters distance from Horn, Krom, and renamable Horn

Model Counting for Formulas of Bounded Clique-Width

We show that #SAT is polynomial-time tractable for classes of CNF formulas whose incidence graphs have bounded symmetric clique-width (or bounded clique-width, or bounded rank-width). This result strictly generalizes polynomial-time tractability results for classes of formulas with signed incidence graphs of bounded clique-width and classes of formulas with incidence graphs of bounded modular treewidth, which were the most general results of this kind known so far.Comment: Extended version of a paper published at ISAAC 201

Group Activity Selection with Few Agent Types

The Group Activity Selection Problem (GASP) models situations where a group of agents needs to be distributed to a set of activities while taking into account preferences of the agents w.r.t. individual activities and activity sizes. The problem, along with its well-known variants sGASP and gGASP, has previously been studied in the parameterized complexity setting with various parameterizations, such as number of agents, number of activities and solution size. However, the complexity of the problem parameterized by the number of types of agents, a natural parameter proposed already in the first paper that introduced GASP, has so far remained unexplored. In this paper we establish the complexity map for GASP, sGASP and gGASP when the number of types of agents is the parameter. Our positive results, consisting of one fixed-parameter algorithm and one XP algorithm, rely on a combination of novel Subset Sum machinery (which may be of general interest) and identifying certain compression steps which allow us to focus on solutions which are "acyclic". These algorithms are complemented by matching lower bounds, which among others close a gap to a recently obtained tractability result of Gupta, Roy, Saurabh and Zehavi (2017). In this direction, the techniques used to establish W[1]-hardness of sGASP are of particular interest: as an intermediate step, we use Sidon sequences to show the W[1]-hardness of a highly restricted variant of multi-dimensional Subset Sum, which may find applications in other settings as well