# NPC Pill #1: Single Overlap RX3C

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 #1: SINGLE OVERLAP RX3C

The EXACT COVER BY 3-SETS (X3C) problem is:

Instance: Set $X = \{x_1,x_2,…,x_{3q}\}$ and a collection $C = \{C_1,…,C_m\}$ of 3-element subsets of $X$.
Question: Does $C$ contain an exact cover for $X$, i.e. a subcollection $C’ \subseteq C$ such that every element of $X$ occurs in exactly one member of $C’$?

X3C is NP-complete [1], and as shown in [2] it remains NP-complete even if every element $x_i$ contained in exactly 3 subsets of $C$ (Restricted Exact Cover by 3-Sets – RX3C).

We proved that it remains NP-complete even if every pair of subsets in $C$  share at most one element; i.e. for all $i \neq j,\; | C_i \cap C_j | \leq 1$ and we call this restricted version SINGLE OVERLAP RX3C.

[1] M. R. Garey, David S. Johnson: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman 1979, ISBN 0-7167-1044-7.
[2] Teofilo F. Gonzalez: Clustering to Minimize the Maximum Intercluster Distance. Theor. Comput. Sci. 38: 293-306 (1985).

## One thought on “NPC Pill #1: Single Overlap RX3C”

1. hi MdB nice work on all this, must say though liked your older blog name “fractalmuse” a bit more than “nearly 42″… is that a douglas adams reference? the blog is neat & esp like the ref to the computational complexity of video games, think have seen 1-2 of your answers on tcs.se relating to that… have you ever heard of kozas work on GA/GPs applied to video games? you might enjoy it. [havent found a really good ref on subj by him though]…. hey maybe drop by chat sometimes more than 1/yr wink… ps wordpress has some nice features [eg latex etc, although looks like youve got that worked out], you might consider it for your blog, will follow your blog for 1 if you ever go that route….