The Landscape of Structural Graph Parameters Michael Lampis KTH Royal Institute of Technology November 18th, 2011 1 / 29
Introduction ❖ FPT theory in 30 seconds ❖ Vertex Cover ❖ Search tree algorithm ❖ So what? ❖ Parameterized Zoo Introduction ❖ Methodology ❖ What parameter to choose? Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions 2 / 29
FPT theory in 30 seconds Introduction ● Most problems are NP-hard → need exp time in the ❖ FPT theory in 30 seconds worst case ❖ Vertex Cover ❖ Search tree algorithm ● They may be easily solvable in some special cases ❖ So what? ❖ Parameterized ✦ Typically for graph problems, when the graph is a Zoo ❖ Methodology tree ❖ What parameter to choose? Graph Widths and ● What about the almost easy cases? Meta-Theorems Vertex Cover and ✦ We consider the concept of “distance from triviality” Max-Leaf Conclusions 3 / 29
FPT theory in 30 seconds Introduction ● Most problems are NP-hard → need exp time in the ❖ FPT theory in 30 seconds worst case ❖ Vertex Cover ❖ Search tree algorithm ● They may be easily solvable in some special cases ❖ So what? ❖ Parameterized ✦ Typically for graph problems, when the graph is a Zoo ❖ Methodology tree ❖ What parameter to choose? Graph Widths and ● What about the almost easy cases? Meta-Theorems Vertex Cover and ✦ We consider the concept of “distance from triviality” Max-Leaf Conclusions ● Examples: 3 / 29
FPT theory in 30 seconds Introduction ● Most problems are NP-hard → need exp time in the ❖ FPT theory in 30 seconds worst case ❖ Vertex Cover ❖ Search tree algorithm ● They may be easily solvable in some special cases ❖ So what? ❖ Parameterized ✦ Typically for graph problems, when the graph is a Zoo ❖ Methodology tree ❖ What parameter to choose? Graph Widths and ● What about the almost easy cases? Meta-Theorems Vertex Cover and ✦ We consider the concept of “distance from triviality” Max-Leaf Conclusions ● Examples: ✦ Satisfying 7 8 m of the clauses of a 3-CNF formula ✦ Satisfying 7 8 m + k of the clauses of a 3-CNF formula 3 / 29
FPT theory in 30 seconds Introduction ● Most problems are NP-hard → need exp time in the ❖ FPT theory in 30 seconds worst case ❖ Vertex Cover ❖ Search tree algorithm ● They may be easily solvable in some special cases ❖ So what? ❖ Parameterized ✦ Typically for graph problems, when the graph is a Zoo ❖ Methodology tree ❖ What parameter to choose? Graph Widths and ● What about the almost easy cases? Meta-Theorems Vertex Cover and ✦ We consider the concept of “distance from triviality” Max-Leaf Conclusions ● Examples: ✦ Euclidean TSP on a convex set of points ✦ Euclidean TSP when all but k of the points lie on the convex hull 3 / 29
FPT theory in 30 seconds Introduction ● Most problems are NP-hard → need exp time in the ❖ FPT theory in 30 seconds worst case ❖ Vertex Cover ❖ Search tree algorithm ● They may be easily solvable in some special cases ❖ So what? ❖ Parameterized ✦ Typically for graph problems, when the graph is a Zoo ❖ Methodology tree ❖ What parameter to choose? Graph Widths and ● What about the almost easy cases? Meta-Theorems Vertex Cover and ✦ We consider the concept of “distance from triviality” Max-Leaf Conclusions ● Examples: ✦ Vertex Cover on bipartite graphs ✦ Vertex Cover on graphs with small bipartization number 3 / 29
Vertex Cover Introduction ● Vertex Cover is NP-hard in general. ❖ FPT theory in 30 seconds ❖ Vertex Cover ● It is easy (in P) on bipartite graphs. ❖ Search tree algorithm ❖ So what? ✦ Maximum matching, K¨ onig’s theorem ❖ Parameterized Zoo ❖ Methodology ❖ What parameter to ● What about almost bipartite graphs? choose? Graph Widths and Meta-Theorems ✦ Is there an efficient algorithm for Vertex Cover on Vertex Cover and graphs where the number of vertices/edges one Max-Leaf needs to delete to make the input graph bipartite is Conclusions small? ● Assume for now that some small bipartizing set is given. 4 / 29
Search tree algorithm Introduction ● Suppose we have an almost bi- ❖ FPT theory in 30 seconds partite graph. We cannot use ❖ Vertex Cover ❖ Search tree K¨ onig’s theorem to find its mini- algorithm mum vertex cover. ❖ So what? ❖ Parameterized Zoo ● However, we can try to get rid of ❖ Methodology ❖ What parameter to the offending vertices/edges. choose? Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions 5 / 29
Search tree algorithm Introduction ❖ FPT theory in 30 seconds ❖ Vertex Cover ❖ Search tree algorithm ❖ So what? ❖ Parameterized ● Pick an offending edge. Either Zoo its first endpoint must be in the ❖ Methodology ❖ What parameter to optimal vertex cover . . . choose? Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions 5 / 29
Search tree algorithm Introduction ❖ FPT theory in 30 seconds ❖ Vertex Cover ❖ Search tree ● Pick an offending edge. Either algorithm ❖ So what? its first endpoint must be in the ❖ Parameterized optimal vertex cover . . . Zoo ❖ Methodology ❖ What parameter to ● So, we should remove it. . . choose? Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions 5 / 29
Search tree algorithm Introduction ❖ FPT theory in 30 seconds ❖ Vertex Cover ❖ Search tree algorithm ❖ So what? ❖ Parameterized Zoo ● . . . or its other endpoint is in the ❖ Methodology ❖ What parameter to optimal cover . . . choose? Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions 5 / 29
Search tree algorithm Introduction ❖ FPT theory in 30 seconds ❖ Vertex Cover ❖ Search tree algorithm ❖ So what? ● . . . or its other endpoint is in the ❖ Parameterized optimal cover . . . Zoo ❖ Methodology ❖ What parameter to ● So, we can remove it. choose? Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions 5 / 29
Search tree algorithm Introduction ❖ FPT theory in 30 seconds ● We have produced two in- ❖ Vertex Cover ❖ Search tree stances, one equivalent to the algorithm ❖ So what? original. ❖ Parameterized Zoo ● Both are closer to being bipar- ❖ Methodology ❖ What parameter to tite. choose? Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions 5 / 29
Search tree algorithm Introduction ❖ FPT theory in 30 ● Continuing like this, we produce seconds 2 k instances, where k is original ❖ Vertex Cover ❖ Search tree distance from bipartite-ness. algorithm ❖ So what? ❖ Parameterized ● These are all bipartite. → Use Zoo poly-time algorithm to find the ❖ Methodology ❖ What parameter to best. choose? Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions 5 / 29
Search tree algorithm Introduction ❖ FPT theory in 30 seconds ● This is known as a bounded- ❖ Vertex Cover depth search tree ❖ Search tree algorithm. algorithm It’s essentially a brute-force ap- ❖ So what? ❖ Parameterized proach, confined to k . Zoo ❖ Methodology ❖ What parameter to ● Total running time: 2 k n c . choose? Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions 5 / 29
So what? Introduction ● This is too easy! Hence, boring. . . ❖ FPT theory in 30 seconds ❖ Vertex Cover ● This is just a cooked-up example. . . ❖ Search tree algorithm ❖ So what? ● This isn’t really new. . . ❖ Parameterized Zoo ❖ Methodology ❖ What parameter to choose? Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions 6 / 29
So what? Introduction ● This is too easy! Hence, boring. . . ❖ FPT theory in 30 seconds ❖ Vertex Cover ✦ This doesn’t work for all problems! 3-coloring is ❖ Search tree algorithm NP-hard for k = 3 [Cai 2002] ❖ So what? ❖ Parameterized Zoo ● This is just a cooked-up example. . . ❖ Methodology ❖ What parameter to choose? ● This isn’t really new. . . Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions 6 / 29
So what? Introduction ● This is too easy! Hence, boring. . . ❖ FPT theory in 30 seconds ❖ Vertex Cover ● This is just a cooked-up example. . . ❖ Search tree algorithm ❖ So what? ✦ True. But we can work this way with countless other ❖ Parameterized problems/graph families. Some cases are bound to Zoo ❖ Methodology be interesting. ❖ What parameter to choose? Graph Widths and ● This isn’t really new. . . Meta-Theorems Vertex Cover and Max-Leaf Conclusions 6 / 29
So what? Introduction ● This is too easy! Hence, boring. . . ❖ FPT theory in 30 seconds ❖ Vertex Cover ● This is just a cooked-up example. . . ❖ Search tree algorithm ❖ So what? ● This isn’t really new. . . ❖ Parameterized Zoo ✦ Novelty here is the pursuit of upper/lower bounds ❖ Methodology ❖ What parameter to with respect to n and k . choose? Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions 6 / 29
Parameterized Zoo Introduction Classical Parameterized ❖ FPT theory in 30 seconds Running time Examples Running time Examples ❖ Vertex Cover Clique, DS, ❖ Search tree 2 O ( n ) algorithm TSP , VC ❖ So what? ❖ Parameterized Zoo ❖ Methodology ❖ What parameter to choose? Graph Widths and Meta-Theorems Vertex Cover and Max-Leaf Conclusions n c MM, MST 7 / 29
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