# 6.890 Algorithmic Lower Bounds: Fun with Hardness Proofs (Fall '14)

## Prof. Erik Demaine     TAs: Sarah Eisenstat, Jayson Lynch

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## Lecture 2 Video     [previous] [next]

 [+] 3-partition I. 2-partition vs. 3-partition; variations (Subset Sum, Numerical 3-dimensional matching, 3DM, X3C); weakly vs. strongly NP-hard; pseudopolynomial vs. polynomial. Multiprocessor scheduling, rectangle packing, edge-matching puzzles, jigsaw puzzles, polyform packing puzzles, packing squares into a square. Scribe Notes [src] This lecture introduces my favorite (and a generally lesser known) starting point for NP-hardness reductions, called 3-partition. This problem is particularly useful when you have a problem that involves adding up numbers, even when those numbers must be encoded in unary (a common feature of many puzzles). We'll discuss many variations of the problem: 2-partition: Partition integers into two sets of equal sum Subset Sum: Select integers to equal a target sum 3-partition: Partition n integers into n/3 triples of equal sum Numerical 3-dimensional matching: Integers are of three different types, and each triple must have all three types. 3-dimensional matching: A generalization to tripartite hypergraphs. Exact cover by 3-sets: A generalization to hypergraphs. 2-partition vs. 3-partition is an example of the weak vs. strong NP-hardness dichotomy, and on the algorithmic side, the pseudopolynomial vs. (weakly) polynomial dichotomy. We'll see weak and strong NP-hardness proofs, by reductions from 2-partition and 3-partition respectively, for two problems: multiprocessor scheduling packing rectangles into a rectangle Next we'll see a fun series of reductions between different puzzles, starting from 3-partition / rectangle packing to establish strong NP-hardness. edge-matching puzzles ("signed" like lizards, and "unsigned" like Eternity II) jigsaw puzzles polyomino packing puzzles (like Eternity) Finally, we'll see how to prove strong NP-hardness of packing squares into a square. This is a handy result that we'll use as the basis for another reduction next lecture.
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