I just bought the nice book “Problems with a POINT – Exploring Math and Computer Science” by William Gasarch and Clyde Kruskal. The book explains many nice mathematical and theoretical computer science problems that are “easy-to-understand-but-not-so-easy-to-solve”, and most of them solicit the reader’s curiosity and invite him to futher explore the related topics (plenty of references are given at the end of each chapter).

Chapter 23 explains a sane reduction between COL4 (the problem of deciding whether a graph is 4-colorable) to COL3 (the problem of deciding whether a graph is 3-colorable). Both are well known NP-complete problems. I started reading the chapter, but suddenly stopped and tried to figure out a reduction by myself.

The idea presented in the book is to replace each node of the graph $G$ of the COL4 problem with four nodes, force them to be colorable with colors F and T (the third color R is not allowed), force exactly one of them to be T, and finally adding a gadget that prevents two adjacent quadruples from having two corresponding nodes colored with T.

I came up with a very similar reduction, but the gadgets involved are slightly smaller: indeed a 4 colorable node can be represented “in binary” with two nodes each one colorable with T or F (two bits).

Recently A. Amarilli (a3nm) posted a question on cs.stackexchange.com about the computational complexity of a Test Round problem of the Google France #Hash Code 2015: the “Pizza Regina” problem (March 27th, 2015):

Definition [Pizza Regina problem]

Input: A grid $M$ with some marked squares, a threshold $T\in \mathbb{N}$, a maximal area $A \in\mathbb{N}$

Output: The largest possible total area of a set of disjoint rectangles with integer coordinates in $M$ such that each rectangle includes at least $T$ marked squares and each rectangle has area at most $A$.

The problem can be converted to a decision problem adding a parameter $k \in \mathbb{N}$ and asking:

Question: Does there exist a set of disjoint rectangles satisfying the conditions (each rectangle has integer coordinates in $M$, includes at least $T$ marked squares and has area at most $A$) whose total area is at least $k$ squares?

The problem is clearly in $\mathsf{NP}$, and after struggling a little bit I found that it is $\mathsf{NP}$-hard (so the Pizza Regina problem is $\mathsf{NP}$-complete). This is a sketch of a reduction from MONOTONE CUBIC PLANAR 1-3 SAT:
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While thinking about possible answers to the question “Listing of strongly NP-complete problems” I came up with this simple numeric strongly NP-complete problem:

[SQUARE-FREE SUBSET PRODUCT] Given $3N$ integers, find $N$ of them whose product is square free.

I didn’t find it anywhere, so it can be somewhat “original”.

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NPC Pills (NP-Complete pills) is a collection of some NP-completeness proofs that I posted on the Q&A sites cstheory.stackexchange.com and cs.stackexchange.com. Up to my knowledge they are original and some of them are not so trivial. At present, most of them are just sketches that contain the NP-hardness reduction idea, but for some of them I wrote (or I’m going to write) a more formal paper. If you are particularly interested in one of them write me an email.

NPC Pill #3: Pick 3 or Pick 2 … another NP-complete 3DM variant

The following variant of the 3-Dimensional Matching problem (3DM) was posted on cstheory.stackexchange.com, a question and answer site for professional researchers in theoretical computer science and related fields:

Definition. 3-DIMENSIONAL MATCHING VARIANT
Input: Set $M \subseteq X \times Y \times Z$ where $X,Y,Z$ are disjoint sets; we call $M_{XY}= \{(x,y)\mid \exists z \text{ s.t. } (x,y,z)\in M \}$ the set of pairs of $X \times Y$ that appear in the triples of $M$, and $M_{XZ}= \{(x,z)\mid \exists y \text{ s.t. } (x,y,z)\in M \}$ the set of pairs of $X \times Z$ that appear in the triples of M.
Question: Does there exist a set $M’ \subseteq M \cup M_{XY} \cup M_{XZ} $ such that every element of $X \cup Y \cup Z$ is included in a triple or a pair of $M’$ exactly once?

Informally we want to build an exact cover of $X \cup Y \cup Z$ using the triples of $M$ or one of the two pairs $(x,y), (x,z)$ that are contained in a triple $(x,y,z) \in M$.

The problem is NP-complete, click the following link to download the technical report with the reduction (we also discuss some other variants of the problem).

Click here to download the technical report

If you have some comments or find some errors, contact me via email.