Structural graph parameters PART 3 APPLICATIONS Bart M

Structural graph parameters PART 3: APPLICATIONS Bart M. P. Jansen Networks Training Week 2017 January 31 st – February 2 nd 2017, Doorn, Netherlands

Plan for today Tackling a party planning problem Algorithms on tree decompositions Provably effective preprocessing Independent Set reconfiguration The PLANAR DISJOINT PATHS problem

ALGORITHMS EXPLOITING GRAPH STRUCTURE 3

Algorithms using tree structure Many problems that are NP-complete in general, can be solved efficiently on graphs of bounded treewidth INDEPENDENT SET, DOMINATING SET, 3 -COLORING, HAMILTONIAN CYCLE, CLIQUE, FEEDBACK VERTEX SET, DISJOINT PATHS, PARTITION INTO TRIANGLES, … Tree-like structure can be exploited by dynamic programming Two-step presentation: 1. Dynamic programming on trees 2. Dynamic programming on tree decompositions 4

The party problem We are planning a party in a hierarchical company Each employee has a fun factor that they would contribute Goal: invite people that maximize the total fun factor Constraint: a party is no fun when your direct boss is present 5

The party crowd forms an independent set • 6

Solving the party problem • 7

Subproblems for subtrees • 8

An expression using child values • 9

General recurrence • 10

Solving the party problem using the recurrence • 11

General algorithm for INDEPENDENT SET • 12

Tree decompositions • 13

Rooted tree decompositions BCE BEG BFG • 14 CDE EGH ABC

The tree decomposes the graph BCE BEG BFG • 15 CDE EGH ABC

The tree decomposes the graph BCE BEG BFG • 16 CDE EGH ABC

Subproblems for nodes of the decomposition BCE BEG BFG • 17 CDE EGH ABC

Nice tree decompositions BCE BEG BFG • 18 CDE EGH ABC

Making a tree decomposition nice BCE BEG BFG • 19 CDE EGH ABC

Making a tree decomposition nice BCE BEG BFG • 20 CDE EGH ABC

Making a tree decomposition nice BCE BEG BFG • 21 BCE BC CDE ABC EGH

BCE Making a tree decomposition nice BCE BFG • 22 BCE BEG CDE EGH BC ABC

BCE Structure of nice decompositions BCE BFG • 23 BCE BEG CDE EGH BC ABC

BCE BFG • 24 BCE BEG CDE EGH BC ABC

BCE BFG • 25 BCE BEG CDE EGH BC ABC

BCE BFG • 26 BCE BEG CDE EGH BC ABC

BCE BFG • 27 BCE BEG CDE EGH BC ABC

Solving INDEPENDENT SET using the recurrence • 28

Algorithms on bounded-treewidth graphs • 29

A meta-theorem • 30

Monadic second-order logic on graphs • 31

Examples of MSO expressions for graphs • 32

Examples of MSO expressions for graphs • 33

Examples of MSO expressions for graphs • 34

Exercise: Formulating properties in MSO • 35

Courcelle’s theorem Extension to optimization problems: 36

A combinatorial application RECONFIGURING INDEPENDENT SETS 37

Reconfiguring between independent sets • Buffer 1 38 2

Token addition/removal reconfiguration • Question: How does the required buffer capacity depend on the structure of the graph? 39

Results on buffer capacity 40 [de Berg, J, Mukherjee, FSTTCS ‘ 16]

A hierarchy of graph parameters Vertex-deletion distance to edgeless Vertex-deletion distance to paths Length of longest cycle Vertex-deletion distance to acyclic Degeneracy 41

Relation between buffer capacity & treewidth? the ``complete double binary tree’’: Simplest graphs with large treewidth & deletion-distance to acyclic? treewidth: 2 42 [de Berg, J, Mukherjee, FSTTCS ‘ 16]

STRUCTURAL PARAMETERS FOR PROVABLE PREPROCESSING 43

Kernelization: data reduction with a guarantee • A kernelization guarantees that instances that are large with respect to their complexity parameter can be shrunk 44

Dist. to edgeless Distance to linear forest Feedback Vertex Set Distance to Split graph components Distance to Cograph Distance to Interval Treewidth Distance to Chordal Odd Cycle Transversal Chromatic Number 45 Distance to Perfect

A WIN/WIN APPROACH FOR PLANAR DISJOINT PATHS 46

The PLANAR DISJOINT PATHS problem • 47

A WIN/WIN approach If the treewidth is larger: • Dynamic programming to find a solution if one exists • • Courcelle’s theorem applies WIN 48 WIN

Irrelevant vertices for PLANAR DISJOINT PATHS 49 [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCT B‘ 12]

50 [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, ICALP ‘ 11]

CLOSING REMARKS 51

Structural aspects of random graphs: last words arxiv. org/abs/1406. 2587 http: //tcs. rwth-aachen. de/~reidl/slides/Bergen 2015. pdf 52

53 [Demaine, Reidl, Rossmanith, Sanchez Villaamil, Sikdar, Sullivan ‘ 15] Figure by Felix Reidl

Summary Graph parameters have algorithmic & combinatorial applications Solve problems on treelike graphs by dynamic programming Courcelle’s theorem links this to monadic second-order logic Treewidth shows up in problems where you don’t expect it Use the parameter hierarchy to guide the attack on a problem! 54