Slides.

More documents

Recommendations

Info

STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 55/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Classes of Unbounded Tree-Width What we would like to show. If the tree-width of C is not bounded by log c n, for small constant c, then MC(MSO,C) is not FPT. Problem. Any such result would separate PTIME from PSPACE. For, MC(MSO 2 ) is in PSPACE. Hence, if PSPACE collapses to PTIME then MSO is fixed-parameter tractable on the class of all graphs. We will therefore show hardness of MC(MSO 2 ,C) by reducing a hard problem to it. For this to work we need to 1. understand what structural information we can draw from the fact that the tree-width of graphs is high obstructions 2. use this information to reduce a hard problem to MC(MSO 2 ,C). This requires some further technical conditions.
STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 56/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Classes of Unbounded Tree-Width Definition. Let f : N → N be a function and p(n) be a polynomial. The tree-width of a class C is (f, p)-unbounded if there is an ǫ < 1 such that for all n ∈ N there is a graph G n ∈ C with 1. n ≤ tw(G n ) ≤ p(n) and tw(G n ) ≥ f(|G n |) 2. given n (in unary), G n can be constructed in time 2 nǫ . The tree-width of C is f -unbounded if it is (f, p)-unbounded for some p(n). Theorem. (K., Tazari 10) Let C be a class of graph closed under sub-graphs and let p(n) be a polynomial of degree < γ. If the tree-width of C is (log 28+γ n, p)-unbounded then MC(MSO,C) is not fpt unless SAT can be solved in sub-exponential time. (fpt: with parameter |ϕ|)
Page 1 and 2:
Towards a Structural Characterisati
Page 3 and 4:
STEPHAN KREUTZER COMPLEXITY OF MODE
Page 5 and 6:
Page 7 and 8:
Page 9 and 10:
Page 11 and 12:
Page 13 and 14:
Page 15 and 16:
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54: STEPHAN KREUTZER COMPLEXITY OF MODE
Page 91 and 92: INTRODUCTION COMPLEXITY UPPER BOUND
Page 103: STEPHAN KREUTZER COMPLEXITY OF MODE
Page 155 and 156:
Page 157 and 158:
show all

Slides.

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?