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 and 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.


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).


Weapon-target assignment problem

The decision version of the Weapon-Target Assignment (WTA) problem is as follows:

There are a number of weapons and a number of targets. The weapons are of type $i = 1, \ldots, m$ . There are $W_{i}$ available weapons of type $i$. Similarly, there are $j = 1, \ldots, n$ targets, each with a value of $V_{j}$ . Any of the weapons can be assigned to any target. Each weapon type has a certain probability of destroying each target, given by $p_{i,j}$ . Does there exist a weapon-target assignment such that:

$$D = \sum_{j=1}^n V_j \prod_{i=1}^m q_{i,j}^{x_{i,j}} \leq k$$

where $k$ is a given integer; $x_{i,j} = 1$ if weapon $i$ is assigned to target $j$, $x_{i,j} =0$ otherwise; $q_{i,j} = 1 – p_{i,j}$ is the survival probability of target $j$ when hit by weapon $i$. No more than $W_i$ weapons of type $i$ can be used, so we must add the following constraint:

$$\sum_{j=1}^n x_{i,j} \leq W_i \quad \mbox{ for } i = 1,\ldots,m$$

WTA problem is NP complete (see L. Chen, C. J. Ren, and S. Deng, “An efficient approximation for weapon-target assignment,” vol. 1, Computing, Communication, Control, and Management. ISES International Colloquium, 2008, pp. 764-767), we give an alternate hardness proof using a reduction from the NP-complete SUBSET PRODUCT problem. Continue reading

The Crazy Frog Puzzle and Permutation Reconstruction from Differences

The Crazy Frog Puzzle (CFP) is the following: we have a $n \times n$ partially filled board, a crazy frog placed on an empty cell and a sequence of horizontal, vertical, and diagonal jumps; each jump has a fixed length and two possible opposite directions. The crazy frog must follow the sequence of jumps, and at each step it can only choose among the two available directions. Can the frog visit every empty cell of the board exactly once (it cannot jump on a blocked cell and on the same cell two times)?

 Crazy Frog PuzzleFigure 1: an example of the Crazy Frog Puzzle on the left and its solution on the right.

 We prove that the Crazy Frog Puzzle is $\sf{NP}$-complete even if restricted to 1 dimension and even without blocked cells. The 1-D CFP without blocked cells corresponds to the problem of reconstructing a permutation from its differences:

Permutation Reconstruction from Differences problem:
Input: a set of $n-1$ distances $a_1,a_2,…,a_{n-1}$
Question: does exist a permutation $\pi_1,…,\pi_n$ of the integers $[1..n]$ such that $| \pi_{i+1} – \pi_i| = a_i$, $i=1,…,n-1$ ?

For example, given the differences $(2,1,2,1,5,3,1,1)$ a valid permutation of $[1..9]$ is $(5,7,6,8,9,4,1,2,3)$

Update 2013-12-19: a new version of the paper is available.

The Complexity of Crazy Frog Puzzle and Permutation Reconstruction from Differences click here to download a draft of the paper


Complexity of the Hidden Polygon Puzzle

The Hidden Polygon Puzzle (for brevity HPP) decision problem  is:

Input: a set $P$ of $m$ integer points on a $n \times n$ square grid and an integer $k \leq m$;

Question: does exist a simple rectilinear polygon with $k$ or more vertices $(v_1,v_2, …, v_t), \; v_i \in P, t \geq k$?

The following figure shows an example of a HPP puzzle.

Hidden Polygon PuzzleFigure 1: Given the 21 points on the right, can we find
a simple rectilinear polygon with at least 16 vertices?
A possible solution is shown on the right.

The problem is a slight variant of the $\sf{NP}$-complete puzzle game Hiroimono; we prove that the Hidden Polygon Puzzle is  $\sf{NP}$-complete, too using a completely different reduction.

click here to download the paper

Hidato is NP-complete

Hidato (also known as Hidoku) is a logic puzzle game invented by Dr. Gyora Benedek, an Israeli mathematician. The rules are simple: given a grid with $n$ cells some of which are already filled with a number between $1$ and $n$ (the first and the last number are circled), the player must completely fill the board with consecutive numbers that connect horizontally, vertically, or diagonally.


Figure 1: An Hidato game (that fits on a $8 \times 8$ grid) and its solution on the right.

We prove that the corresponding decision problem $\sf{HIDATO}$ : “Given a Hidato game that fits in a $m \times n$ grid, does a valid solution exist?” is $\sf{NP}$-complete.

Continue reading

Binary Puzzle is NP-complete

Net puzzleBinary Puzzle (also known as Binary Sudoku) is an addictive puzzle played on a $n \times n$ grid; intially some of the cells contain a zero or a one; the aim of the game is to fill the empty cells according to the following rules:

  • Each cell should contain a zero or a one and no more than two similar numbers next to or below each other are allowed
  • Each row and each column should contain an equal number of zeros and ones
  • Each row is unique and each column is unique

We prove that the decision version of Binary Puzzle is NP-complete.

click here to download the paper

IcoSoKu solver

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Faces example

I wrote a simple javascript IcoSoKu solver. You can enter the configuration of the 12 yellow pins in the text boxes and click “Solve IcoSoKu” to calculate the solution. For example if you have the yellow pin 5 on vertex A (the top vertex), you must type 5 in the Pin A text box. The faces are identified with their three yellow pins (see a small example on the right).

The 20 white tiles are identified with their black dots; for example (2,2,0) represents the tile with two black dots in the first and second corner and zero dots on the third.If the solution says put tile (1,2,3) on face [11,12,7], then you must put the white tile on the bottom-left face, and rotate it until 1 black dot is on yellow pin 11, 2 black dots are on yellow pin 12 and 3 black dots are on yellow pin 7. You can also generate a random yellow pins configuration clicking “Random IcoSoKu”.

Continue reading

Three easy deletion games on paths

The node deletion game NODE KAYLES is a two persons perfect information game played on a graph $G$. Players alternate picking a node from the graph $G$; the node and its adjacent nodes are deleted. The first player unable to move loses the game. Deciding the winner of a NODE KAYLES game is $\mathsf{PSPACE}$-complete [1]. Similarly in the EDGE KAYLES [1] game the two players must pick an edge; the edge is deleted along with the two endpoint nodes and the edges incident to its two endpoint nodes. The complexity of finding the winner of a EDGE KAYLES game is unknwon. On the Q&A site I submitted a mix of the two games; I call it the BRIDGES AND ISLANDS game: at each turn each player must pick an edge (a “bridge”) or an isolated node (an “island”); if a player picks a node the node is deleted along with its incident edges, if it picks an edge, the edge is deleted along with its two endpoint nodes and the edges incident to them. Like the other two games, the first player unable to move loses the game. I didn’t succeed in finding the complexity of BRIDGES AND ISLANDS.

However decideng the winner is polynomial time solvable when the three games are played on a path. Here I give a quick proof for EDGE KAYLES; the proof for the other two games is similar. Continue reading

Two bits are enough for a “hard” sum

On Jan 17 I asked the following question on

What is the complexity of the following variant of the SUBSET-SUM decision problem?

2-BIT SUBSET SUM: Given an integer $m \geq 0$, and a set of nonnegative integers $A = \{x_1, x_2, …, x_n\}$ such that every $x_i$ has at most $k=2$ bits set to $1$ ($x_i = 2^{b_{i_1}}+2^{b_{i_2}},\;\; b_{i_1},b_{i_2}\geq 0$); is there a subset $A’ \subseteq A$ such that the sum of its elements is equal to $m$ ?

Continue reading

The complexity of the puzzle game Net: rotating wires can drive you crazy

An amateur proof that the puzzle game Net is NP-complete.

Net puzzleAbstract
The puzzle game Net, also known as FreeNet or NetWalk, is played on a grid filled with terminals and wires; each tile of the grid can be rotated and the aim of the game is to connect all the terminals to the central power unit avoiding closed loops and open-ended wires. We prove that Net is NP-complete.

click here to download the paper

Rolling a cube can be tricky

An amateur proof that the rolling cube puzzle is NP-complete.

We settle two open problems related to the rolling cube puzzle: Hamil-
tonian cycles are not unique even in fully labeled boards and rolling
cube puzzle is NP-complete in labeled boards without free cells and with
blocked cells.

Rolling a cube can be tricky click here to download the paper

NOTE: another example of two distinct Hamiltonian cycles in a fully labeled board has also been found by Pálvölgyi Dömötör (see this post on mathoverflow).

Have fun with Boulder Dash

An amateur proof that the popular game is NP-hard.

Boulder Dash is a videogame created by Peter Liepa and Chris Gray in 1983 and released for many personal computers and console systems under license from First Star Software. Its concept is simple: the main character must dig through caves, collect diamonds, avoid falling stones and other nasties, and finally reach the exit within a time limit. In this report we show that the decision problem “Is an $N\times N$ Boulder Dash level solvable?” is NP-hard. The constructive proof is based on a simple gadget that allows us to transform the Hamiltonian cycle problem on a 3-connected cubic planar graph to a Boulder Dash level in polynomial time.

 click here to download the paper

NOTE: the same result has been proved by G. Viglietta in the paper: Gaming Is a Hard Job, But Someone Has to Do It! ; his proof, which is embedded in a more general and powerful framework that can be used to prove complexity of games, doesn’t require the Dirt element.