1. 1 Partition Into Triangles on Bounded Degree Graphs Johan M. M. van Rooij Marcel E. van Kooten Niekerk Hans L. Bodlaender

2. 2 Problem Statement and Overview of Results Partition Into Triangles  Input:A graph G=(V,E).  Question:Can V be partitioned into 3-element sets S 1, S 2,..., S n/3 such that for each S i the graph G[S i ] is a triangle?  We consider this problem on bounded degree graphs.  Maximum degree three.  Linear time solvable.  Maximum degree four.  Equivalence with Exact SAT.  Hard under ETH.  O(1.02220 n ) algorithm.

3. 3 Some Simple Observations to Start with: Vertices of Degree One or Two  Degree zero and one:  Cannot be in any triangle.  No-instance.  Degree two:  Unique triangle.  Reduction rule: remove this triangle from the instance.  Resulting instance Yes-instance ⇔ original instance Yes-instance.  Simple reduction rules that allow us to assume minimum degree three.

4. 4 On Graphs of Maximum Degree Three  We can assume maximum and minimum degree three.  A vertex v can have four possible local neighbourhoods (different induced subgraphs G[N[v]]).  Linear time algorithm!  Reduce if vertices of degree ≤2, otherwise do as shown here! no triangles Removing any triangle leads to vertices of degree ≤1 unique triangle

5. 5 On Graphs of Maximum Degree Four: Overview  If any vertex has degree at most three.  Reduction rules.  11 possible local neighbourhoods for a vertex of degree four.  Direct reduction rules for 8 of them.  One can only form connected components in which every vertex has the same local neighbourhood - additional reduction rule.  Two remaining: they form ‘clouds’ and ‘fans’.  Equivalent to Exact 3-Satisfiability:  A cloud is a variable.  A fan is a clause.  Corollaries:  NP-Complete.  No subexponential-time algorithms under ETH.  Very fast exponential time algorithms.

6. 6 Reduction Rules for Vertices of Degree at Most Three  Degree at most two: No-instance or unique triangle.  Degree three: four possible local neighbourhoods. no triangles Any triangle leads to degree ≤1 vertices unique triangle Different in max degree four

7. 7 Reduction Rule for a Vertex of Degree Three  One local neighbourhood remaining.  Taking any of the two triangles cannot give degree ≤1 vertices.  Hence, top and bottom right vertex of degree four!  Additional edge incident to bottom left vertex irrelevant.  Some case analysis required to ensure that no new triangles are created, i.e., these are the only two possible partitionings.

8. 8 On Graphs of Maximum Degree Four: An Overview  If any vertex has degree at most three.  Reduction rules.  11 possible local neighbourhoods for a vertex of degree four.  Direct reduction rules for 8 of them.  One can only form connected components in which every vertex has the same local neighbourhood - additional reduction rule.  Two remaining: they form ‘clouds’ and ‘fans’.  Equivalent to Exact 3-Satisfiability:  A cloud is a variable.  A fan is a clause.  Corollaries:  NP-Complete.  No subexponential-time algorithms under ETH.  Very fast exponential time algorithms.

9. 9 Eleven Possible Local Neighbourhoods Eleven possible local neighbourhoods for a vertex of degree four. contain edges that are not in a triangle: remove and reduce (degree ≤ three rules) similar edge! Any triangle leads to degree ≤1 vertices next slide!

10. 10 Two Specific Local Neighbourhoods  Consider the red edge.  Any triangle containing this edge leads to vertices of degree ≤ 1.  Remove and use the reduction rule for the resulting degree ≤ 3 vertex.  Consider the red edge.  It can be in one or two triangles: with the yellow vertex or (possibly) the green vertex.  If in a triangle with the yellow vertex: other two vertices need a common neighbour.  All four symmetric edges: endpoints or opposite vertices need common neighbour.  By maximum degree four: there must be an edge that cannot be in a triangle.  Remove and use the reduction rule.

11. 11 Eleven Possible Local Neighbourhoods contain edges that are not in a triangle: remove and reduce (degree ≤ three rules) similar edge! Any triangle leads to degree ≤1 vertices also contain edges that cannot be in a solution next slide! Eleven possible local neighbourhoods for a vertex of degree four.

12. 12 Only Three Possible Local Neighbourhoods Remaining  Consider the local neighbourhood below.  What local neighbourhoods can the blue vertices have?  Consider the top right (green) vertex.  It can only have the same local neighbourhood as it neighbouring yellow vertex.  By induction: all vertices in the connected component must have this local neighbourhood!  Result, only components like this one exist: can be partitioned into triangles ⇔ number of vertices is a multiple of three

13. 13 Eleven Possible Local Neighbourhoods contain edges that are not in a triangle: remove and reduce (degree ≤ three rules) similar edge! Any triangle leads to degree ≤1 vertices also contain edges that cannot be in a solution done! Eleven possible local neighbourhoods for a vertex of degree four. these are the only two local neighbourhoods that we cannot reduce

14. 14 What Structures Can We Build Using These Two Local Neighbourhoods?  Yellow vertex and green vertex have the same local neighbourhood: they occur as pairs.  We call these pairs ‘fans’.  Example:  The second structure can be made into chains and loops.  We call these ‘clouds’.  Example: (blue vertices are fan vertices)

15. 15 On Graphs of Maximum Degree Four: An Overview  If any vertex has degree at most three.  Reduction rules.  11 possible local neighbourhoods for a vertex of degree four.  Direct reduction rules for 8 of them.  One can only form connected components in which every vertex has the same local neighbourhood - additional reduction rule.  Two remaining: they form ‘clouds’ and ‘fans’.  Equivalent to Exact 3-Satisfiability:  A cloud is a variable.  A fan is a clause.  Corollaries:  NP-Complete.  No subexponential-time algorithms under ETH.  Very fast exponential time algorithms.

16. 16 Equivalence to Exact 3-Satisfiability a Fan is a Clause! Exact 3-Satisfiability  Input:A set of variables X and a set of clauses C each of size at most three.  Question:Does there exists a truth assignment of the variables in X such that each clause contains exactly one literal that is set to true? A fan: pick exactly one of the tree triangles!

17. 17 Equivalence to Exact 3-Satisfiability a Cloud is a Variable!  One vertex cloud: variable with one positive and one negative occurrence.  Larger clouds: points adjacent to fans form the literals.

18. 18 Partition Into Triangles Interpreted as an Exact 3-Satisfiability Formula!  Consider the following graph:  Cloud vertices are yellow.  Fan vertices are blue.  We name the clouds w, x, y, z.  Each fan is a clause!  Formula: (w, z, ¬x) ⋀ (x, z, ¬y) ⋀ (y, z, ¬w)  Satisfying assignment:  True: w, x, y.  False: z. wyz z x z (w, z, ¬x) (x, z, ¬y) (y, z, ¬w)

19. 19 On Graphs of Maximum Degree Four: An Overview  If any vertex has degree at most three.  Reduction rules.  11 possible local neighbourhoods for a vertex of degree four.  Direct reduction rules for 8 of them.  One can only form connected components in which every vertex has the same local neighbourhood - additional reduction rule.  Two remaining: they form ‘clouds’ and ‘fans’.  Equivalent to Exact 3-Satisfiability:  A cloud is a variable.  A fan is a clause.  Corollaries:  NP-Complete.  No subexponential-time algorithms under ETH.  Very fast exponential time algorithms.

20. 20 NP-Completeness  Property of Exact 3-SAT instances obtained in this way:  For any variable: #positive literals = #negative literals (mod 3).  Any such variable can be represented by a cloud.  Every clause is represented by 2 vertices (fan).  Any cloud that represents a variable that occurs x times is represented by 2x-3 vertices.  The problem is NP-complete.  Exact 3-SAT is NP-complete.  Given an instance of Exact 3-SAT, we copy each clause 3 times.  #positive literals = #negative literals (mod 3), for any variable.  Hence, we can construct an equivalent Partition Into Triangles instance on graphs of maximum degree four.  New instance has size linear in the number of clauses.

21. 21 No Subexponential-Time Algorithm Exist Under the Exponential Time Hypothesis  Exponential Time Hypothesis (Impagliazzo et al. 2001):  There is no algorithm for 3-SAT that runs in O(2 εn ) for all ε>0: no subexponential-time algorithm.  Assuming ETH, also no algorithm that runs in O(2 εm ) for all ε>0 by the Sparsificiation Lemma.  Assuming ETH, there is no subexponential-time algorithm for Partition Into Triangles on graphs of maximum degree four.  From a given 3-SAT formula to Exact 3-SAT:  Then, same linear transformation as for NP-completeness.  A, for all ε>0, O(2 εm ) time algorithm for Partition Into Triangles on graphs of maximum degree four implies a, for all ε>0, O(2 εn ) time algorithms for 3-SAT! SAT(x,y,z) = XSAT(x,a,b) ⋀ XSAT(y,b,c) ⋀ XSAT(a,c,d) ⋀ XSAT(¬z,b,e)

22. 22 Fast Exponential-Time Algorithms  First attempt: use the current fastest algorithms for Exact Satisfiability and Exact 3-Satisfiability.  Exact 3-SAT: O(1.0984 n ) due to Wahlström.  Exact SAT: O(1.1749 n ) due to Byskov, Madsen, and Skjernaa. This algorithm removes variables with only one positive and one negative occurrence by reduction rules: They are not counted in the time bound.  If many such variables use Exact SAT algorithm, otherwise use Exact 3-SAT algorithm.  Balancing gives an O(1.02445 n ) time algorithm.  Second attempt: algorithm for Exact 3-SAT measured by number of vertices used to create it.  Extensive case analysis gives an O(1.02220 n ) time algorithm.

23. 23 On Graphs of Maximum Degree Four: An Overview  If any vertex has degree at most three.  Reduction rules.  11 possible local neighbourhoods for a vertex of degree four.  Direct reduction rules for 8 of them.  One can only form connected components in which every vertex has the same local neighbourhood - additional reduction rule.  Two remaining: they form ‘clouds’ and ‘fans’.  Equivalent to Exact 3-Satisfiability:  A cloud is a variable.  A fan is a clause.  Corollaries:  NP-Complete.  No subexponential-time algorithms under ETH.  Very fast exponential time algorithms.

24. 24 Conclusion  Partition Into Triangles on graphs of maximum degree 3:  is linear time solvable.  Partition Into Triangles on graphs of maximum degree 4:  is closely related to Exact 3-Satisfiability  is NP-complete.  admits no subexponential-time algorithm under the ETH.  is solvable in O(1.02220 n ) time. Questions?  Open problem:  Can you find a problem that admits no subexponential-time algorithm and that is solvable faster than in O(1.02220 n ) time?  No constructions such as Independent Set in graphs where 99% of the vertices have maximum degree two allowed! (These instances can be reduced directly)

25. 25 Which Type of Variables Can a Cloud Form?  Any cloud that represents a variable has:  # positive occurrences = # negative occurrences (mod 3)  Why?  Cloud has fixed number of vertices.  A selected triangle that is internal to the cloud uses 3 vertices.  Remaining vertices are in triangles with the fans.  Remaining vertices in both configurations equal (mod 3)  Any variable with this property can be represented by a cloud.

26. 26  Adding a series of three triangles increases the number of positive or negative occurrences by three!  We can build any cloud with (a,b) such that a = b (mod 3). All Such Clouds Exist!  (1,1): single vertex  (3,0): one triangle  (2,2): two triangles  (3,3): chain of four triangles (+,-) = (1,1) (+,-) = (4,1)