Towards the Complexity of Recognizing Pseudo-intents

Transcription

1 Towards the Complexity of Recognizing Pseudo-intents Barış Sertkaya Theoretical Computer Science TU Dresden, Germany July 30, 2009 Supported by the German Research Foundation (DFG) under grant BA 1122/12-1

2 Outline 1 FCA and Implications 2 Pseudo-intents 3 Hypergraphs 4 Hypergraph Problems

3 Formal Concept Analysis formal context K = (G, M, I ) objects G, attributes M, incidence relation I nuclear G8 EU UN weapons mem. mem. mem. USA Germany France UK Turkey Italy derivation operator: {nuclear, G8} = {USA, France, UK} second derivative: {nuclear, G8} = {nuclear, G8, UN} ( it is a closure operator)

4 Implications between attributes implication between attributes: {nuclear, UN} {G8} nuclear G8 EU UN weapons mem. mem. mem. USA Germany France UK Turkey Italy computing all implications that hold in a context possibly exponentially many (2 M 2 M ) of them, many of them redundant instead a small non-redundant set that generates them? Duquenne-Guigues Base [Guigues & Duquenne(1986)] non-redundant implication base from which all follow minimum cardinality premises are called pseudo-intents

5 Pseudo-intents A pseudo-intent P M is not closed (P P ), and contains the closure of all strictly smaller pseudo-intents ( Q P. Q a pseudo-intent Q P) premises of the DG Base key rôle in FCA computing them no efficient algorithm known, open problem at least as hard as computing minimal transversals [Sertkaya(2009)] complexity of recognizing pseudo-intents major open problem

6 Complexity of psi Upper bound in conp Problem: pseudo-intent (psi) Input: Formal context K = (G, M, I ) and P M Question: Is P a pseudo-intent of K? [Kuznetsov & Obiedkov(2006)] Lower bound is it conp-hard? not known! so far neither a proof of conp-hardness, nor a polynomial algorithm

7 Hypergraphs Generalization of graphs. Edges connect any number of vertices Hypergraph H = (V, E) V is the set of vertices E = {E 1... E m E i V } is the set of (hyper)edges Simple hypergraph: No edge properly contains another one Saturated hypergraph: every W V is either contained in an edge, or contains an edge

8 Hypergraph Problems Hypergraph Saturation (h-sat) check whether a given hypergraph is saturated conp-complete [Eiter & Gottlob(1995)] Simple Hypergraph Saturation (simple-h-sat) conp-hardness is open! many equivalent problems in logic, AI, data mining [Eiter & Gottlob(2002)] solvable in quasi-polynomial time, i.e., n o(log n) time [Fredman & Khachiyan(1996)]

9 psi is simple-h-sat-hard Proof Sketch: From simple H = (V, E) construct K H = (G, M, I ) where M = V {a, b}, and G and I are as follows: v 1 v 2 v 3 v 4 v 5 a b g 1 g 2 g 3 g 4 g 11 g 12 g 21 g 22 g 23 g 31 g 32 g 41 g 42 g 43 context K H constructed from H P = V {a} is a pseudo-intent iff H is saturated

10 psi is simple-h-sat-hard v 1 v 2 v 3 v 4 v 5 a b g 1 g 2 g 3 g 4 g 11 g 12 g 21 g 22 g 23 g 31 g 32 g 41 g 42 g 43 context K H constructed from H edges of H are pseudo-intents of K H ( ) if H is not saturated, then there is a pseudo-intent W P s.t. W P, thus P not a pseudo-intent ( ) if P is not a pseudo-intent, then there is a W V s.t. W E i and E i W for 1 i m, thus H is not saturated

11 What does this mean? It can be: conp-hard equivalent to simple hypergraph saturation i.e., simple-h-sat-complete polynomial?

12 I cannot find a polynomial algorithm, but... I can t find an efficient algorithm, but neither can all these famous people! Leonid Khachiyan Georg Gottlob Thomas Eiter

13 What s next? reduction from h-sat, thus conp-hard? reduction to simple-h-sat, thus simple-h-sat-complete? limited non-determinism algorithms that make at most logarithmically many non-deterministic steps β k p classes Thank you!

14 References Eiter, T. & Gottlob, G. (1995). Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6), Eiter, T. & Gottlob, G. (2002). Hypergraph transversal computation and related problems in logic and AI. In S. Flesca, S. Greco, N. Leone, & G. Ianni (Eds.), Proceedings of the European Conference on Logics in Artificial Intelligence (JELIA 2002). Springer-Verlag, vol of Lecture Notes in Computer Science, Fredman, M. L. & Khachiyan, L. (1996). On the complexity of dualization of monotone disjunctive normal forms. Journal of Algorithms, 21(3), Guigues, J.-L. & Duquenne, V. (1986). Familles minimales d implications informatives resultant d un tableau de données binaries. Mathématiques, Informatique et Sciences Humaines, 95, Kuznetsov, S. O. & Obiedkov, S. A. (2006). Counting pseudo-intents and #P-completeness. In R. Missaoui & J. Schmid (Eds.), Proceedings of the 4th International Conference on Formal Concept Analysis (ICFCA 2006). Dresden, Germany: Springer-Verlag, vol of Lecture Notes in Computer Science, Sertkaya, B. (2009). Some computational problems related to pseudo-intents. In S. Ferré & S. Rudolph (Eds.), Proceedings of the 7th International Conference on Formal Concept Analysis, (ICFCA 2009). Springer-Verlag, vol of Lecture Notes in Artificial Intelligence,