The Complexity of Holant Problems over Boolean Domain with Non-Negative Weights

Holant problem is a general framework to study the computational complexity of counting problems. We prove a complexity dichotomy theorem for Holant problems over the Boolean domain with non-negative weights. It is the first complete Holant dichotomy where constraint functions are not necessarily symmetric. Holant problems are indeed read-twice #CSPs. Intuitively, some #CSPs that are #P-hard become tractable when restricted to read-twice instances. To capture them, we introduce the Block-rank-one condition. It turns out that the condition leads to a clear separation. If a function set F satisfies the condition, then F is of affine type or product type. Otherwise (a) Holant(F) is #P-hard; or (b) every function in F is a tensor product of functions of arity at most 2; or (c) F is transformable to a product type by some real orthogonal matrix. Holographic transformations play an important role in both the hardness proof and the characterization of tractability

Complexity of Non-Monotonic Logics

Over the past few decades, non-monotonic reasoning has developed to be one of the most important topics in computational logic and artificial intelligence. Different ways to introduce non-monotonic aspects to classical logic have been considered, e.g., extension with default rules, extension with modal belief operators, or modification of the semantics. In this survey we consider a logical formalism from each of the above possibilities, namely Reiter's default logic, Moore's autoepistemic logic and McCarthy's circumscription. Additionally, we consider abduction, where one is not interested in inferences from a given knowledge base but in computing possible explanations for an observation with respect to a given knowledge base. Complexity results for different reasoning tasks for propositional variants of these logics have been studied already in the nineties. In recent years, however, a renewed interest in complexity issues can be observed. One current focal approach is to consider parameterized problems and identify reasonable parameters that allow for FPT algorithms. In another approach, the emphasis lies on identifying fragments, i.e., restriction of the logical language, that allow more efficient algorithms for the most important reasoning tasks. In this survey we focus on this second aspect. We describe complexity results for fragments of logical languages obtained by either restricting the allowed set of operators (e.g., forbidding negations one might consider only monotone formulae) or by considering only formulae in conjunctive normal form but with generalized clause types. The algorithmic problems we consider are suitable variants of satisfiability and implication in each of the logics, but also counting problems, where one is not only interested in the existence of certain objects (e.g., models of a formula) but asks for their number.Comment: To appear in Bulletin of the EATC

On the frontiers of polynomial computations in tropical geometry

We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as counting the number of connected components. We characterize the borderline between tractable and hard computations by proving $\mathcal{NP}$-hardness and #$\mathcal{P}$-hardness results under various strong restrictions of the input data, as well as providing polynomial time algorithms for various other restrictions.Comment: 17 pages, 5 figures. To appear in Journal of Symbolic Computatio

Reductions for Frequency-Based Data Mining Problems

Studying the computational complexity of problems is one of the - if not the - fundamental questions in computer science. Yet, surprisingly little is known about the computational complexity of many central problems in data mining. In this paper we study frequency-based problems and propose a new type of reduction that allows us to compare the complexities of the maximal frequent pattern mining problems in different domains (e.g. graphs or sequences). Our results extend those of Kimelfeld and Kolaitis [ACM TODS, 2014] to a broader range of data mining problems. Our results show that, by allowing constraints in the pattern space, the complexities of many maximal frequent pattern mining problems collapse. These problems include maximal frequent subgraphs in labelled graphs, maximal frequent itemsets, and maximal frequent subsequences with no repetitions. In addition to theoretical interest, our results might yield more efficient algorithms for the studied problems.Comment: This is an extended version of a paper of the same title to appear in the Proceedings of the 17th IEEE International Conference on Data Mining (ICDM'17

Holant Problems for Regular Graphs with Complex Edge Functions

We prove a complexity dichotomy theorem for Holant Problems on 3-regular graphs with an arbitrary complex-valued edge function. Three new techniques are introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue Shifted Pairs, which allow us to prove that a pair of combinatorial gadgets in combination succeed in proving #P-hardness; and (3) algebraic symmetrization, which significantly lowers the symbolic complexity of the proof for computational complexity. With holographic reductions the classification theorem also applies to problems beyond the basic model.Comment: 19 pages, 4 figures, added proofs for full versio