Finding the chromatic number of a graph is an NP-Hard problem, so there isn't a fast solver 'in theory'. Is there any publicly available software that can compute the exact chromatic number of a graph quickly?
I'm writing a Python script that computes the chromatic number of many graphs, but it is taking too long for even small graphs. The graphs I am working with a wide range of graphs that can be sparse or dense but usually less than 10,000 nodes. I formulated the problem as an integer program and passed it to Gurobi to solve. Do you have recommendations for software, different IP formulations, or different Gurobi settings to speed this up?
import networkx as nx
from gurobipy import *
# create test graph
n = 50
p = 0.5
G = nx.erdos_renyi_graph(n, p)
# compute chromatic number -- ILP solve
m = Model('chrom_num')
# get maximum number of variables necessary
k = max(nx.degree(G).values()) + 1
# create k binary variables, y_0 ... y_{k-1} to indicate whether color k is used
y = []
for j in range(k):
y.append(m.addVar(vtype=GRB.BINARY, name='y_%d' % j, obj=1))
# create n * k binary variables, x_{l,j} that is 1 if node l is colored with j
x = []
for l in range(n):
x.append([])
for j in range(k):
x[-1].append(m.addVar(vtype=GRB.BINARY, name='x_%d_%d' % (l, j), obj=0))
# objective function is minimize colors used --> sum of y_0 ... y_{k-1}
m.setObjective(GRB.MINIMIZE)
m.update()
# add constraint -- each node gets exactly one color (sum of colors used is 1)
for u in range(n):
m.addConstr(quicksum(x[u]) == 1, name='NC_%d' % u)
# add constraint -- keep track of colors used (y_j is set high if any time j is used)
for u in range(n):
for j in range(k):
m.addConstr(x[u][j] <= y[j], name='SH_%d_%d' % (u,j))
# add constraint -- adjacent nodes have different colors
for u in range(n):
for v in G[u]:
if v > u:
for j in range(k):
m.addConstr(x[u][j] + x[v][j] <= 1, name='ADJ_%d_%d_COL_%d' % (u,v,j))
# update model, solve, return the chromatic number
m.update()
m.optimize()
chrom_num = m.objVal
I am looking to compute exact chromatic numbers although I would be interested in algorithms that compute approximate chromatic numbers if they have reasonable theoretical guarantees such as constant factor approximation, etc.
You might want to try to use a SAT solver or a Max-SAT solver. I expect that they will work better than a reduction to an integer program, since I think colorability is closer to satsfiability.
SAT solvers receive a propositional Boolean formula in Conjunctive Normal Form and output whether the formula is satisfiable. The following problem COL_k is in NP:
Input: Graph G and natural number k.
Output: G is k-colorable.
To solve COL_k you encode it as a propositional Boolean formula with one propositional variable for each pair (u,c) consisting of a vertex u and a color 1<=c<=k. You need to write clauses which ensure that every vertex is is colored by at least one color. You also need clauses to ensure that each edge is proper.
Then you just do a binary search to find the value of k such that G is k-colorable but not (k-1)-colorable.
There are various free SAT solvers. I have used Lingeling successfully, but you can find many others on the SAT competition website. They all use the same input and output format. Google "MiniSAT User Guide: How to use the MiniSAT SAT Solver" for an explanation on this format.
You can also use a Max-SAT solver, again consult the Max-SAT competition website. They can solve the Partial Max-SAT problem, in which clauses are partitioned into hard clauses and soft clauses. Here, the solver finds the maximal number of soft clauses which can be satisfied while also satisfying all of the hard clauses, see the input format in the Max-SAT competition website (under rules->details).
You can formulate the chromatic number problem as one Max-SAT problem (as opposed to several SAT problems as above). In this sense, Max-SAT is a better fit. On the other hand, I have the impression that SAT solvers generally perform better than Max-SAT solvers. I don't have any experience with this kind of solver, so cannot say anything more.
Related
I am working on solving the following problem and implement the solution in C++.
Let us assume that we have an oriented weighted graph G = (V, A, w) and P a set of persons.
We receive a number of queries such that every query gives a person p and two vertices s and d and asks to compute the minimum weighted path between s and d for the person p. One person can have multiple paths.
After the end of all queries I have a number k <= |A| and I should give k arcs such that the number of persons using at least one of the k arcs is maximal (this is a maximum coverage problem).
To solve the first part I implemented the Djikistra algorithm using priority_queue and I compute the minimal weight between s and d. (Is this a good way to do ?)
To solve the second part I store for every arc the set of persons that use this arc and I use a greedy algorithm to compute the set of arcs (at each stage, I choose an arc used by the largest number of uncovered persons). (Is this a good way to do it ?)
Finally, if my algorithms are goods how can I implement them efficiently in C++?
I have a set of signals S1, S2, ....,SN, for which I am numerically computing the Fourier transforms F1, F2, ,,,,FN. Where Si's and Fi's are C++ vectors (my computations are in C++).
My Computation objective is as follows:
compute product: F = F1*F2*...*FN
inverse Fourier transform F to get a S.
What I numerically observe is when I am trying to compute the product either I am running into situation where numbers become too small. Or numbers become too big.
I thought of scaling F1 by say a1 so that the overflow or underflow is avoided.
With the scaling my step 1 above will become
F' = (F1/a1)*(F2/a2)*...*(FN/aN) = F'1*F'2*...*F'N
And when I inverse transform my final S' will differ from S by a scale factor. The structural form of the S will not change. By this I mean only the normalization of S is different.
My question is :
Is my rationale correct.
If my rationale is correct then given a C++ vector "Fi" how can I choose a good scale "ai" to scale up or down the Fi's.
Many thanks in advance.
You can change the range of the Fi vector in a new domain, let's say [a, b]. So from your vector Fi, the minimum number will become a and the maximum number will become b. You can scale the vector using this equation:
Fnew[i] = (b-a)/(max-min) * (F[i] - min) + a,
where: min = the minimum value from F
max= the maximum value from F
The only problem now is choosing a and b. I would choose [1,2], because the values are small (so you won't get overflow easily), but not smaller than 0.
I have a few distance functions which return distance between two images , I want to combine these distance into a single distance, using weighted scoring e.g. ax1+bx2+cx3+dx4 etc i want to learn these weights automatically such that my test error is minimised.
For this purpose i have a labeled dataset which has various triplets of images such that (a,b,c) , a has less distance to b than it has to c.
i.e. d(a,b)<d(a,c)
I want to learn such weights so that this ordering of triplets can be as accurate as possible.(i.e. the weighted linear score given is less for a&b and more for a&c).
What sort of machine learning algorithm can be used for the task,and how the desired task can be achieved?
Hopefully I understand your question correctly, but it seems that this could be solved more easily with constrained optimization directly, rather than classical machine learning (the algorithms of which are often implemented via constrained optimization, see e.g. SVMs).
As an example, a possible objective function could be:
argmin_{w} || e ||_2 + lambda || w ||_2
where w is your weight vector (Oh god why is there no latex here), e is the vector of errors (one component per training triplet), lambda is some tunable regularizer constant (could be zero), and your constraints could be:
max{d(I_p,I_r)-d(I_p,I_q),0} <= e_j for jth (p,q,r) in T s.t. d(I_p,I_r) <= d(I_p,I_q)
for the jth constraint, where I_i is image i, T is the training set, and
d(u,v) = sum_{w_i in w} w_i * d_i(u,v)
with d_i being your ith distance function.
Notice that e is measuring how far your chosen weights are from satisfying all the chosen triplets in the training set. If the weights preserve ordering of label j, then d(I_p,I_r)-d(I_p,I_q) < 0 and so e_j = 0. If they don't, then e_j will measure the amount of violation of training label j. Solving the optimization problem would give the best w; i.e. the one with the lowest error.
If you're not familiar with linear/quadratic programming, convex optimization, etc... then start googling :) Many libraries exist for this type of thing.
On the other hand, if you would prefer a machine learning approach, you may be able to adapt some metric learning approaches to your problem.
Given a bit vector in Z3, I am wondering how can I sum up each individual bit of this vector?
E.g.,
a = BitVecVal(3, 2)
sum_all_bit(a) = 2
Is there any pre-implemented APIs/functions that support this? Thank you!
It isn't part of the bit-vector operations.
You can create an expression as follows:
def sub(b):
n = b.size()
bits = [ Extract(i, i, b) for i in range(n) ]
bvs = [ Concat(BitVecVal(0, n - 1), b) for b in bits ]
nb = reduce(lambda a, b: a + b, bvs)
return nb
print sub(BitVecVal(4,7))
Of course, log(n) bits for the result will suffice if you prefer.
The page:
https://graphics.stanford.edu/~seander/bithacks.html#CountBitsSetNaive
has various algorithms for counting the bits; which can be translated to Z3/Python with relative ease, I suppose.
My favorite is: https://graphics.stanford.edu/~seander/bithacks.html#CountBitsSetKernighan
which has the nice property that it loops as many times as there are set bits in the input. (But you shouldn't extrapolate from that to any meaningful complexity metric, as you do arithmetic in each loop, which might be costly. The same is true for all these algorithms.)
Having said that, if your input is fully symbolic, you can't really beat the simple iterative algorithm, as you can't short-cut the iteration count. Above methods might work faster if the input has concrete bits.
So you're computing the Hamming Weight of a bit vector. Based on a previous question I had, one of the developers had this answer. Based on that original answer, this is how I do it today:
def HW(bvec):
return Sum([ ZeroExt(int(ceil(log2(bvec.size()))), Extract(i,i,bvec)) for i in range(bvec.size())])
I have data set n x m where there are n observations and each observation consists of m values for m attributes. Each observation has also observed result assigned to it. m is big, too big for my task. I am trying to find a best and smallest subset of m attributes that still represents the whole dataset quite well, so that I could use only these attributes for teaching a neural network.
I want to use genetic algorithm for this. The problem is the fittness function. It should tell how well the generated model (subset of attributes) still reflects the original data. And I don't know how to evaluate certain subset of attributes against the whole set.
Of course I could use the neural network(that will later use this selected data anyway) for checking how good the subset is - the smaller the error, the better the subset. BUT, this takes a looot of time in my case and I do not want to use this solution. I am looking for some other way that would preferably operate only on the data set.
What I thought about was: having subset S (found by genetic algorithm), trim data set so that it contains values only for subset S and check how many observations in this data ser are no longer distinguishable (have same values for same attributes) while having different result values. The bigger the number is, the worse subset it is. But this seems to me like a bit too computationally exhausting.
Are there any other ways to evaluate how well a subset of attributes still represents the whole data set?
This cost function should do what you want: sum the factor loadings that correspond to the features comprising each subset.
The higher that sum, the greater the share of variability in the response variable that is explained with just those features. If i understand the OP, this cost function is a faithful translation of "represents the whole set quite well" from the OP.
Reducing to code is straightforward:
Calculate the covariance matrix of your dataset (first remove the
column that holds the response variable, i.e., probably the last
one). If your dataset is m x n (columns x rows), then this
covariance matrix will be n x n, with "1"s down the main diagonal.
Next, perform an eigenvalue decomposition on this covariance
matrix; this will give you the proportion of the total variability
in the response variable, contributed by that eigenvalue (each
eigenvalue corresponds to a feature, or column). [Note,
singular-value decomposition (SVD) is often used for this step, but
it's unnecessary--an eigenvalue decomposition is much simpler, and
always does the job as long as your matrix is square, which
covariance matrices always are].
Your genetic algorithm will, at each iteration, return a set of
candidate solutions (features subsets, in your case). The next task
in GA, or any combinatorial optimization, is to rank those candiate
solutions by their cost function score. In your case, the cost
function is a simple summation of the eigenvalue proportion for each
feature in that subset. (I guess you would want to scale/normalize
that calculation so that the higher numbers are the least fit
though.)
A sample calculation (using python + NumPy):
>>> # there are many ways to do an eigenvalue decomp, this is just one way
>>> import numpy as NP
>>> import numpy.linalg as LA
>>> # calculate covariance matrix of the data set (leaving out response variable column)
>>> C = NP.corrcoef(d3, rowvar=0)
>>> C.shape
(4, 4)
>>> C
array([[ 1. , -0.11, 0.87, 0.82],
[-0.11, 1. , -0.42, -0.36],
[ 0.87, -0.42, 1. , 0.96],
[ 0.82, -0.36, 0.96, 1. ]])
>>> # now calculate eigenvalues & eivenvectors of the covariance matrix:
>>> eva, evc = LA.eig(C)
>>> # now just get value proprtions of each eigenvalue:
>>> # first, sort the eigenvalues, highest to lowest:
>>> eva1 = NP.sort(eva)[::-1]
>>> # get value proportion of each eigenvalue:
>>> eva2 = NP.cumsum(eva1/NP.sum(eva1)) # "cumsum" is just cumulative sum
>>> title1 = "ev value proportion"
>>> print( "{0}".format("-"*len(title1)) )
-------------------
>>> for row in q :
print("{0:1d} {1:3f} {2:3f}".format(int(row[0]), row[1], row[2]))
ev value proportion
1 2.91 0.727
2 0.92 0.953
3 0.14 0.995
4 0.02 1.000
so it's the third column of values just above (one for each feature) that are summed (selectively, depending on which features are present in a given subset you are evaluating with the cost function).