The 2x2 Simple Packing Problem André van Renssen Supervisor: Bettina Speckmann

/ Department of Mathematics and Computer Science Introduction Input: An axis-aligned polygon P drawn on a grid. P consists of n edges and contains N cells. Question: How many non-overlapping 2x2 squares can be packed into P?

/ Department of Mathematics and Computer Science State of the art NP-Complete for polygons with holes (Berman et al. 1981, Dulieu et al. 2009) Unknown for simple polygons PTAS (in N) for simple polygons and polygons with holes (Chan, 2004) Some special cases solvable in polynomial time (in n) (El-Khechen, 2009)

/ Department of Mathematics and Computer Science A simple technique

/ Department of Mathematics and Computer Science More ideas Maximizing the number of squares on boundary does not work Restricting to rectilinear convex does not help

/ Department of Mathematics and Computer Science Different approach Instead of looking at where to place squares, look at where NOT to place squares Needs some way to keep track of usable locations This leads to a graph representation

/ Department of Mathematics and Computer Science Graph V: vertex for each possible location of a square E: edge between vertex u and v iff their squares overlap O(n log n + |V|) construction time |V| is proportional to OPT

Reduction / Department of Mathematics and Computer Science

Reduction / Department of Mathematics and Computer Science

Additional reduction rules Cycles (with and without connections)

/ Department of Mathematics and Computer Science Additional reduction rules

Cornered diamonds / Department of Mathematics and Computer Science Additional reduction rules

/ Department of Mathematics and Computer Science Construction scheme

Polygons revisited For some configurations of edges: Always same number of squares Always same part of polygon removed / Department of Mathematics and Computer Science

Polygons revisited Algorithm: While the polygon is not solved: −Find a configuration −Remove the configuration Finding a configuration: O(n) time Removing a configuration: O(1) time Every configuration removes some edge(s): at most O(n) times Total running time: O(n 2 ) / Department of Mathematics and Computer Science

Polygons revisited / Department of Mathematics and Computer Science

Not (yet) solvable / Department of Mathematics and Computer Science

Conclusion Reduction rules: Runs in time polynomial in |V| Configuration removal: Runs in O(n 2 ) time Not shown during presentation: An O(n log n)-time algorithm for polygons having only even length vertical edges Improved running time of the ½-approximation algorithm (Berman et al. 1982) from O(N) to O(n 2 ) All graph reduction techniques are implemented / Department of Mathematics and Computer Science

Open problems Is this problem NP-hard? Find better approximations or PTAS in n Solve class of rectilinear convex polygons Prove (if possible) that |V| is polynomial in n after modifying the polygon / Department of Mathematics and Computer Science