Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, September 15, 2020, 12:15 pm

Duration: 30 minutes

Location: Zoom: conference room

Speaker: Maxime Larcher

Solving Random Jigsaw Puzzles in Polynomial Time

In 2017, Mossel and Ross introduced the Random Jigsaw Puzzle Problem: if we create a puzzle by cutting all the edges of an n x n grid u.a.r. using one of q shapes, for which choice of q = q(n) can we guarantee w.h.p. that there is a unique solution to the puzzle?

In a series of paper, Mossel and Ross (2017), Nenadov, Pfister and Steger (2017), and Martinsson (2019) offered some answers to this question. Parts of the proofs are based on Martinsson's observation that for q small enough, the number of ways of cutting a board is greater than the number of possible collection.

A question that naturally arises from this observation is whether a typical collection of pieces (taken from the whole set of collections, not necessarily obtained by cutting a grid) has at least one solution. We show that, when q < O(n^1/2 / log^1/4 n), if the number of pieces of each type is close to its expectation then the puzzle can be solved in expected polynomial time. This is also true if we impose the shapes of the edges on the boundary of the grid, provided this satisfies some trivial condition.

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