The complexity of recognizing minimally tough graphs

A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. A graph is minimally $t$-tough if the toughness of the graph is $t$ and the deletion of any edge from the graph decreases the toughness. The complexity class DP is the set of all languages that can be expressed as the intersection of a language in NP and a language in coNP. In this paper, we prove that recognizing minimally $t$-tough graphs is DP-complete for any positive rational number $t$. We introduce a new notion called weighted toughness, which has a key role in our proof

Query Order and the Polynomial Hierarchy

Hemaspaandra, Hempel, and Wechsung [cs.CC/9909020] initiated the field of query order, which studies the ways in which computational power is affected by the order in which information sources are accessed. The present paper studies, for the first time, query order as it applies to the levels of the polynomial hierarchy. We prove that the levels of the polynomial hierarchy are order-oblivious. Yet, we also show that these ordered query classes form new levels in the polynomial hierarchy unless the polynomial hierarchy collapses. We prove that all leaf language classes - and thus essentially all standard complexity classes - inherit all order-obliviousness results that hold for P.Comment: 14 page

Weighted graphs defining facets: a connection between stable set and linear ordering polytopes

A graph is alpha-critical if its stability number increases whenever an edge is removed from its edge set. The class of alpha-critical graphs has several nice structural properties, most of them related to their defect which is the number of vertices minus two times the stability number. In particular, a remarkable result of Lov\'asz (1978) is the finite basis theorem for alpha-critical graphs of a fixed defect. The class of alpha-critical graphs is also of interest for at least two topics of polyhedral studies. First, Chv\'atal (1975) shows that each alpha-critical graph induces a rank inequality which is facet-defining for its stable set polytope. Investigating a weighted generalization, Lipt\'ak and Lov\'asz (2000, 2001) introduce critical facet-graphs (which again produce facet-defining inequalities for their stable set polytopes) and they establish a finite basis theorem. Second, Koppen (1995) describes a construction that delivers from any alpha-critical graph a facet-defining inequality for the linear ordering polytope. Doignon, Fiorini and Joret (2006) handle the weighted case and thus define facet-defining graphs. Here we investigate relationships between the two weighted generalizations of alpha-critical graphs. We show that facet-defining graphs (for the linear ordering polytope) are obtainable from 1-critical facet-graphs (linked with stable set polytopes). We then use this connection to derive various results on facet-defining graphs, the most prominent one being derived from Lipt\'ak and Lov\'asz's finite basis theorem for critical facet-graphs. At the end of the paper we offer an alternative proof of Lov\'asz's finite basis theorem for alpha-critical graphs

Knowledge Refinement via Rule Selection

In several different applications, including data transformation and entity resolution, rules are used to capture aspects of knowledge about the application at hand. Often, a large set of such rules is generated automatically or semi-automatically, and the challenge is to refine the encapsulated knowledge by selecting a subset of rules based on the expected operational behavior of the rules on available data. In this paper, we carry out a systematic complexity-theoretic investigation of the following rule selection problem: given a set of rules specified by Horn formulas, and a pair of an input database and an output database, find a subset of the rules that minimizes the total error, that is, the number of false positive and false negative errors arising from the selected rules. We first establish computational hardness results for the decision problems underlying this minimization problem, as well as upper and lower bounds for its approximability. We then investigate a bi-objective optimization version of the rule selection problem in which both the total error and the size of the selected rules are taken into account. We show that testing for membership in the Pareto front of this bi-objective optimization problem is DP-complete. Finally, we show that a similar DP-completeness result holds for a bi-level optimization version of the rule selection problem, where one minimizes first the total error and then the size