Unique key Horn functions

Given a relational database, a key is a set of attributes such that a value assignment to this set uniquely determines the values of all other attributes. The database uniquely defines a pure Horn function $h$, representing the functional dependencies. If the knowledge of the attribute values in set $A$ determines the value for attribute $v$, then $A\rightarrow v$ is an implicate of $h$. If $K$ is a key of the database, then $K\rightarrow v$ is an implicate of $h$ for all attributes $v$. Keys of small sizes play a crucial role in various problems. We present structural and complexity results on the set of minimal keys of pure Horn functions. We characterize Sperner hypergraphs for which there is a unique pure Horn function with the given hypergraph as the set of minimal keys. Furthermore, we show that recognizing such hypergraphs is co-NP-complete already when every hyperedge has size two. On the positive side, we identify several classes of graphs for which the recognition problem can be decided in polynomial time. We also present an algorithm that generates the minimal keys of a pure Horn function with polynomial delay. By establishing a connection between keys and target sets, our approach can be used to generate all minimal target sets with polynomial delay when the thresholds are bounded by a constant. As a byproduct, our proof shows that the Minimum Key problem is at least as hard as the Minimum Target Set Selection problem with bounded thresholds.Comment: 12 pages, 5 figure

Approximating minimum representations of key Horn functions

Horn functions form a subclass of Boolean functions and appear in many different areas of computer science and mathematics as a general tool to describe implications and dependencies. Finding minimum sized representations for such functions with respect to most commonly used measures is a computationally hard problem that remains hard even for the important subclass of key Horn functions. In this paper we provide logarithmic factor approximation algorithms for key Horn functions with respect to all measures studied in the literature for which the problem is known to be hard

Generating clause sequences of a CNF formula

Given a CNF formula $\Phi$ with clauses $C_1,\ldots,C_m$ and variables $V=\{x_1,\ldots,x_n\}$, a truth assignment $a:V\rightarrow\{0,1\}$ of $\Phi$ leads to a clause sequence $\sigma_\Phi(a)=(C_1(a),\ldots,C_m(a))\in\{0,1\}^m$ where $C_i(a) = 1$ if clause $C_i$ evaluates to $1$ under assignment $a$, otherwise $C_i(a) = 0$. The set of all possible clause sequences carries a lot of information on the formula, e.g. SAT, MAX-SAT and MIN-SAT can be encoded in terms of finding a clause sequence with extremal properties. We consider a problem posed at Dagstuhl Seminar 19211 "Enumeration in Data Management" (2019) about the generation of all possible clause sequences of a given CNF with bounded dimension. We prove that the problem can be solved in incremental polynomial time. We further give an algorithm with polynomial delay for the class of tractable CNF formulas. We also consider the generation of maximal and minimal clause sequences, and show that generating maximal clause sequences is NP-hard, while minimal clause sequences can be generated with polynomial delay.Comment: 9 page

Generating all arcs in the transitive closure of a directed hypergraph

リサーチレポート（北陸先端科学技術大学院大学情報科学研究科）本文は図書館に配架されています。 / This material is stored in the JAIST library

Nonpreemtive flowshops with a dominant machine : reductions to single mac

リサーチレポート（北陸先端科学技術大学院大学情報科学研究科）本文は図書館に配架されています。 / This material is stored in the JAIST library

On the Complexity of Horn Minimization

Boolean functions given in disjunctive normal form representation are considered in this paper. For a given function such a representation is typically not unique and different representations may consist of different number of terms. The problem of minimizing a Boolean formula ammounts to finding its equivalent representation which has the minimum possible number of terms. This problem is a well-known hard problem, where the difficulty stems largely from the fact that the satisfiability problem for general Boolean formulae in disjunctive normal forms is a NP-complete problem. There are however, special classes of Boolean formulae for which satisfiability is known to be polynomial, like Horn formulae arising in artificial intelligence or in relational database applications. The complexity of the minimization problem of Horn formulae was not known so far. In this paper we show that, given a Horn formula, finding a disjunctive normal form equivalent to it and having the minimum possible ..

Just-in-Time Scheduling with Periodic Time Slots

リサーチレポート（北陸先端科学技術大学院大学情報科学研究科）本文は図書館に配架されています。 / This material is stored in the JAIST library

A Quadratic Time Algorithm to Maximize the Number of Just-in-Time Jobs on Identical Parallel Machines

リサーチレポート（北陸先端科学技術大学院大学情報科学研究科）本文は図書館に配架されています。 / This material is stored in the JAIST library