**Note:**This blog entry is a cross-post from Digital Operatives' blog. You can read the original post here.

### Hard Problems

Computer Scientists and Software Engineers deal with
computationally hard problems all the time. But those challenges
don’t solely apply to those people *creating* the software;
they are equally relevant to those people *analyzing* it. For
cybersecurity researchers, finding a hash collision…
determining the user inputs that can assign a certain value to a
tainted EIP… deciding whether a black-box binary is
malicious… they’re all really hard! As our work becomes more
and more automated with the help of program analysis techniques, these
computationally hard problems become more and more prevalent. But
what makes one problem harder than another? In the remainder of this
post I’ll answer that question, and introduce some surprising
new results of my research that demonstrate how good solutions to
inherently very hard problems can be quickly generated. Read on for more.

In order to get there, we’re first going to have to review the concepts of computational complexity and approximation algorithms. If you are already familiar with the former, you can skip to the third section. If you are already familiar with both, you can skip to the fourth section.

It’s clear that certain computational tasks are harder than others.
For example, finding the smallest value in an $n$-element array
can be completed in about $n$ operations (*i.e.*, simply
iterate over the array once, keeping track of the smallest). Sorting
that array, on the other hand, will always take at
least $n \log(n)$ operations in the worst case (for an
explanation of why that
is, here is a great article). Sure enough, if we look
at the common sorting algorithms (quicksort, heapsort, mergesort,
bubblesort, *etc.*), it’s always pretty easy to construct
$n$-element arrays that will require those algorithms to
execute at least $n \log(n)$ operations. In fact,
when quicksort is presented with an array that is already nearly in
sorted order, it will require $n^2$ operations, which
is much slower. Therefore, we can say that the problem of sorting is
inherently *harder* than finding the infimum (smallest element)
of an array.

There are in fact many different classes of problems
that have known “hardness.” Some classes are easier to solve than
others. Certain classes also have other interesting properties; for
example, a group of problems that are called
“**P**-complete” are all very likely to be
difficult to parallelize effectively. The common compression
algorithm LZW (used in GIF) solves one such **P**-complete
problem, so I’m sorry to report that it’s very unlikely that you’re
going to get a speedup by running your automated animated GIF meme
generator on a GPU.

In the remainder of this post we’re going to be focusing on what are
called **NP**-hard problems. These are a class of problems for
which it is very likely that there is no way to solve them
efficiently. The interesting thing is that if someone were to find a
single efficient way to solve *one of them*, then we’d
immediately have a solution for *all of them*.

### Really Hard Problems

One of the most famous examples of an **NP**-hard problem is
the *traveling salesman problem*:

What is the shortest possible route to visit each city in a map, without visiting any city more than once, and returning to the originating city at the end?Variants of this problem crop up in the real world all the time, particularly in logistical networks (

*e.g.*, how companies like UPS and FedEx decide routes for their delivery vehicles and locations for their distribution centers). Another famous

**NP**-hard problem is the

*Steiner network problem*:

What is the shortest set of roads that connects a given set of cities to each other?That problem is very important in efficient networking protocol design. The games Sudoku, Minesweeper, and Battleship, as well as many others, are all also

**NP**-hard. And, as you may have guessed, many of the cybersecurity tasks and program analysis problems we encounter on a daily basis are also

**NP**-hard. That’s why things like static taint analysis, constraint solving, and concolic execution that many people use in vulnerability research are so slow.

The important part here is that there is no known *polynomial
time* algorithm that solves any **NP**-hard problem. I’ll get
to what “polynomial time” means in a minute; for now you can just
assume that it means “solvable in a reasonable amount of computation
time.” Discovering a polynomial time algorithm that solves
an **NP**-hard problem or, conversely, proving no such algorithm
can exist, is one of the most important open problems in theoretical
computer science. If you can devise such an algorithm, xor prove that
none can exist, then you’ll win the million
dollar Millenium Prize. More importantly: If such an
algorithm *does* exist, then it has very many significant
consequences. For example, such an algorithm would break the
assumptions of many cryptographic schemes and mean that you could
efficiently break most cryptography very quickly. Most computer
scientists and mathematicians believe that it is impossible to create
a polynomial time algorithm to solve an **NP**-hard problem, but so
far no one has been able to prove it.

So what does “polynomial time” mean, and why do we want to achieve it?
Computer scientists classify the *computational complexity* of
an algorithm based on the order of magnitude of the number of
operations the algorithm would have to perform in the worst case.
Let’s say we create an algorithm to find the optional solution to the
traveling salesman problem simply by brute-force enumerating all
possible paths through the map. In the worst case, the input map is
going to have a road between each pair of cities. That means our
algorithm, in the worst case, is going to need to test every possible
path (these are what are
called *Hamiltonian paths*), of which there are
$\frac{(n - 1)!}{2}$, where $n$ is the total
number of cities. That means our algorithm *does not* run in
worst-case polynomial time, because the equation
$\frac{(n - 1)!}{2}$ is not
a polynomial. For it to be a polynomial, it would
have to have a worst-case runtime equation that behaves asymptotically
similar to $n^c$ for any constant $c$.

I’ll get to why such a polynomial expression is so desirable in a minute, but first let’s look at an example of why slower-than-polynomial-time algorithms, like our traveling salesman one, are very bad. As we can see, our algorithm quickly becomes very slow as the number of cities, $n$, increases:

$n$ (number of cities) | Number of Paths to Test | |
---|---|---|

3 | $\frac{1}{2}(3 - 1)!$ = | 1 |

4 | $\frac{1}{2}(4 - 1)!$ = | 3 |

5 | $\frac{1}{2}(5 - 1)!$ = | 12 |

6 | $\frac{1}{2}(6 - 1)!$ = | 60 |

7 | $\frac{1}{2}(7 - 1)!$ = | 360 |

8 | $\frac{1}{2}(8 - 1)!$ = | 2520 |

9 | $\frac{1}{2}(9 - 1)!$ = | 20160 |

10 | $\frac{1}{2}(10 - 1)!$ = | 181440 |

11 | $\frac{1}{2}(11 - 1)!$ = | 1814400 |

12 | $\frac{1}{2}(12 - 1)!$ = | 19958400 |

13 | $\frac{1}{2}(13 - 1)!$ = | 239500800 |

14 | $\frac{1}{2}(14 - 1)!$ = | 3113510400 |

15 | $\frac{1}{2}(15 - 1)!$ = | 43589145600 |

16 | $\frac{1}{2}(16 - 1)!$ = | 653837184000 |

17 | $\frac{1}{2}(17 - 1)!$ = | 10461394944000 |

18 | $\frac{1}{2}(18 - 1)!$ = | 177843714048000 |

19 | $\frac{1}{2}(19 - 1)!$ = | 3201186852864000 |

20 | $\frac{1}{2}(20 - 1)!$ = | 60822550204416000 |

⋮ | ⋮ | |

30 | $\frac{1}{2}(30 - 1)!$ = | 4420880996869850977271808000000 |

⋮ | ⋮ | |

61 | $\frac{1}{2}(61 - 1)!$ = | 4160493556370695072138170591611682190377086303180622976224638848204800000000000000 |

So, with as few as 61 cities on the map, our
algorithm would have to consider *over four sexvigintillion*
possible paths! That’s about 42 times the number of atoms in the
visible universe! With only nine cities, It’s Over 9000!!!1 And that’s
not even taking into account the number of operations required to
enumerate each of those paths, which likely increases the overall
figure by at least a factor of $n$.

But what makes polynomial time algorithms *tractable* while
asymptotically slower algorithms are not? Why does that phase
transition occur specifically at the point when the worst-case number
of operations grows polynomially? It all has to do
with Moore’s law, which states that processor speed
doubles every two years or so. (More accurately, Moore’s law speaks
of the number of components per integrated circuit.) Now, let’s say
we have an algorithm for solving the traveling salesman problem that
can quickly solve problem instances up to $k$ cities on current
hardware. How many years will we have to wait before hardware will be
fast enough to solve a problem of size $k + 1$? If our
algorithm runs in roughly $n^c$ operations
(*i.e.*, it runs in polynomial time), then, by Moore’s law, we
will only have to wait about $\log_2\left(1 +
k^{-1}\right)c$ years before we can solve a problem of
size $k+1$. For most values of $k$ and $c$, that is a
very short time! If our algorithm’s runtime were
instead *exponential* (*e.g.*, $c^n$), then
the number of years we would have to wait would be at least linear
in $k$.

If we’re stuck solving a problem that we know
is **NP**-hard—which means that there is no known polynomial
time algorithm, and there will likely *never* be one—what
are we to do? Luckily, there is a subfield of computer science known
as *approximation algorithms* whose goal is to
find algorithms that can quickly find solutions that are not
necessarily optimal, but are within some known bound of optimal.
That’s what we are going to talk about in the next section.

### The “Bingo Problem”: Approximate Solutions to Really Hard Problems

We’re all likely familiar with the game of BINGO: Balls labeled with letters and numbers are randomly drawn…

…until one of any given subset of balls (as specified by the BINGO cards) is collected.

Let’s imagine a variant in which all letters of the alphabet exist on the balls and they are each given arbitrary floating point values:

The object of this new type of BINGO game is:
Given *n* balls chosen randomly from the infinite set of all
possible letter/number combinations, find some subset of the balls
that spells out an English word. The goal is to maximize the sum of
the values of the balls used for that word. It’s a bit like
scrabble. If the word length is unbounded, this game is in
fact **NP**-hard (it is a variant
of the subset sum problem). As an example, say the
following thirteen balls are randomly chosen:

P
1 |
R
48.7 |
N
79.8 |
O
26 |
O
83.8 |
P
23.6 |
A
22.7 |
M
92.1 |
I
34.2 |
T
36.7 |
A
64.5 |
I
84.5 |
X
49.1 |

One solution would be to choose the word “MOTION” with a value of 402.9:

P
1 |
R
48.7 |
N
79.8 |
O
26 |
O
83.8 |
P
23.6 |
A
22.7 |
M
92.1 |
I
34.2 |
T
36.7 |
A
64.5 |
I
84.5 |
X
49.1 |

That isn’t the best we can do, though. A better solution would be to choose the word “APPROXIMATION”, which has a value of 646.7:

P
1 |
R
48.7 |
N
79.8 |
O
26 |
O
83.8 |
P
23.6 |
A
22.7 |
M
92.1 |
I
34.2 |
T
36.7 |
A
64.5 |
I
84.5 |
X
49.1 |

This is in fact the optimal solution, since it uses all of the possible letters. If we had originally chosen MOTION, then, in approximation algorithms parlance, we say that the “*constant of approximation”* is 646.7 ÷ 402.9 = 1.6051. In other words, MOTION is 1.6051 times “worse” than optimal.

In the case of the traveling salesman problem, there are well known
polynomial time approximation algorithms that can return a result with
a 1.5 constant of approximation. In other words, these algorithms can
return a result that is guaranteed to be no worse than 1.5 times the
value of the optimal result, *even though we don’t actually know
the value of the optimal solution!* For most **NP**-hard
problems, any constant of approximation that is 2 or lower is
considered good.

### Random Solutions to Really Hard Problems

I like to ask stupid questions. So I asked myself: “What if our
algorithm just chooses a solution randomly?” It’s usually pretty easy
to write a fast, dumb algorithm to return random solutions. In the
Bingo problem, it simply amounts to ignoring the ball values and
returning a random word whose letters match the balls. That can
certainly be implemented in polynomial time if we have an indexed
dictionary of words. For problems like minimum Steiner network, we can just randomly
generate any spanning tree, which will always be a
*feasible* (valid) solution, and which we can compute in linear
time. **On average, how bad are those random solutions likely to
be?**

First, let’s sort the balls in order of increasing value:

P
1 |
A
22.7 |
P
23.6 |
O
26 |
I
34.2 |
T
36.7 |
R
48.7 |
X
49.1 |
A
64.5 |
N
79.8 |
O
83.8 |
I
84.5 |
M
92.1 |

$\ell$ | $m$ |

If we know that the shortest word in our dictionary is $\ell$ letters
long and the longest word in our dictionary is $m$ letters long,
then the expected value for the constant of approximation of a
randomly selected word is *at most* equal to the expected value
of the sum of the $m$ highest-value balls divided by the expected
value of the $\ell$ lowest value balls. This is simplifying a lot of
details (*e.g.*, things get a lot more complex if the values can
be negative), and it requires a bit of advanced math, but luckily
we’ve already worked it all out. You can read about it
in our
recently published paper here.

If we distill the crufty mathematical incantations we needed to handle all of the edge cases for the formal proofs, our results can be summarized as follows:

That’s really surprising, especially realizing that this applies forUnder very reasonable assumptions, the expected value for the constant of approximation of a randomly selected feasible solution is almost always going to beat mosttwo!

*all*

**NP**-hard problems, if formulated correctly. So, simply choosing a random solution is often effectively as good as the best approximation algorithms that are currently known. In fact, in our paper we linked above we present some empirical evidence suggesting that the random solutions are often even closer to optimal than ones produced by state-of-the-art approximation algorithms.

### Conclusions

Certain computational problems are harder than others, and a class of
them, called **NP**-hard, are very likely to be intractable to
solve optimally. A great number of cybersecurity and vulnerability
research tasks fall into that class. Polynomial time algorithms are
desirable because if a problem instance is too large to be solved by
current hardware, we don’t have to wait very long until hardware will
become fast enough to solve that instance using the polynomial time
algorithm. Polynomial time approximation algorithms can be used on
intractable **NP**-hard problems to find solutions relatively
quickly that are within a known distance from the optimal solution
(even if we don’t actually know what the optimal solution is). For
most **NP**-hard problems, a polynomial time approximation
algorithm that can achieve a solution that is no worse than two times
the cost of the optimal solution is usually considered good. The
surprising result of our new research is that *randomly*
selected feasible solutions will on average be at least as good as
those complex approximation algorithms!

So the moral of the story is: Sometimes quickly and mindlessly choosing a random solution isn’t half bad!