Two queries

AbstractWe consider the question whether two queries SAT are as powerful as one query. We show that if PNP[1]=PNP[2] then: Locally either NP=coNP or NP has polynomial-size circuits; PNP=PNP[1]; Σp2⊆Πp2/1; Σp2=UPNP[1]∩RPNP[1]; PH=BPPNP[1]. Moreover, we extend the work of Hemaspaandra, Hemaspaandra, and Hempel to show that if PΣp2[1]=PΣp2[2] then Σp2=Πp2. We also give a relativized world, where PNP[1]=PNP[2], but NP≠coNP

Lower bounds for kernelizations

"Vegeu el resum a l'inici del document del fitxer adjunt"

Complexity of Stability

Graph parameters such as the clique number, the chromatic number, and the independence number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the chromatic number of a graph (i.e., the smallest number of colors needed to color all vertices such that no two adjacent vertices are of the same color) can be applied in solving practical tasks as diverse as pattern matching, scheduling jobs to machines, allocating registers in compiler optimization, and even solving Sudoku puzzles. Typically, however, the underlying graphs are subject to (often minor) changes. To make these applications of graph parameters robust, it is important to know which graphs are stable for them in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs for such parameters in terms of their computational complexity. We show that, for various central graph parameters, the problem of determining whether or not a given graph is stable is complete for \Theta_2^p, a well-known complexity class in the second level of the polynomial hierarchy, which is also known as "parallel access to NP.

On Computing Boolean Connectives of Characteristic Functions

We study the existence of polynomial time Boolean connective functions for languages. A language $L$ has an AND function if there is a polynomial time $f$ such that $f(x,y) \in L \Longleftrightarrow x \in L$ and $y \in L$. $L$ has an OR function if there is a polynomial time $g$ such that $g(x,y) \in L \Longleftrightarrow x \in L$ or $y \in L$. While all NP-complete sets have these functions, we show that Graph Isomorphism, which is probably not complete, also has them. We characterize the complete sets for the classes $D^{P}$ and $P^{NP[O(log n)]}$ in terms of AND and OR, and we relate these functions to the structure of the Boolean hierarchy and the query hierarchies. We show that the sets that are complete for levels above the second level of the Boolean hierarchy do not have AND and OR unless the polynomial hierarchy collapses. We show that most of the structural properties of the Boolean hierarchy and query hierarchies depend only on the existence of AND and OR functions for NP-complete sets