Balanced subset sum problem
$begingroup$
Suppose I have a set of $2N$ items with weights $w_0, w_1, ldots, w_{2N-1}$. I want to identify the two most "balanced" sets of $k$ items each, where $k$ is not given but limited to the range $k_min leq k leq N$.
More specifically: Identify the two disjunct subsets $A = {a_0, a_1, ldots a_{k-1}}$ and $B = {b_0, b_1, ldots a_{k-1}}$ subject to the following conditions:
- all the $a_i, b_i in [0,2N-1]$
- $A cap B = varnothing$
- both sets have equal cardinality $k$ where $k_min leq k leq N$
- the following quantity $epsilon$ is minimized:
$$ begin{align}
epsilon &= frac{|W_A-W_B|}{W_A+W_B} cr
W_A &= sumlimits_{i=0}^{k-1}w_{a_i} cr
W_B &= sumlimits_{i=0}^{k-1}w_{b_i} cr
end{align}$$
Is this a known problem? (I found Is there a "balanced knapsacks" problem with a known result? which is similar)
In my case the value of $N$ is not very large (under 50) so I don't really care about proving NP-hard, I just want to know a good strategy.
combinatorics
asked Dec 21 '18 at 16:44
Jason SJason S
2,04811617
$endgroup$
add a comment |
$begingroup$
Suppose I have a set of $2N$ items with weights $w_0, w_1, ldots, w_{2N-1}$. I want to identify the two most "balanced" sets of $k$ items each, where $k$ is not given but limited to the range $k_min leq k leq N$.
More specifically: Identify the two disjunct subsets $A = {a_0, a_1, ldots a_{k-1}}$ and $B = {b_0, b_1, ldots a_{k-1}}$ subject to the following conditions:
- all the $a_i, b_i in [0,2N-1]$
- $A cap B = varnothing$
- both sets have equal cardinality $k$ where $k_min leq k leq N$
- the following quantity $epsilon$ is minimized:
$$ begin{align}
epsilon &= frac{|W_A-W_B|}{W_A+W_B} cr
W_A &= sumlimits_{i=0}^{k-1}w_{a_i} cr
W_B &= sumlimits_{i=0}^{k-1}w_{b_i} cr
end{align}$$
Is this a known problem? (I found Is there a "balanced knapsacks" problem with a known result? which is similar)
In my case the value of $N$ is not very large (under 50) so I don't really care about proving NP-hard, I just want to know a good strategy.
combinatorics
asked Dec 21 '18 at 16:44
Jason SJason S
2,04811617
$endgroup$
add a comment |
$begingroup$
Suppose I have a set of $2N$ items with weights $w_0, w_1, ldots, w_{2N-1}$. I want to identify the two most "balanced" sets of $k$ items each, where $k$ is not given but limited to the range $k_min leq k leq N$.
More specifically: Identify the two disjunct subsets $A = {a_0, a_1, ldots a_{k-1}}$ and $B = {b_0, b_1, ldots a_{k-1}}$ subject to the following conditions:
- all the $a_i, b_i in [0,2N-1]$
- $A cap B = varnothing$
- both sets have equal cardinality $k$ where $k_min leq k leq N$
- the following quantity $epsilon$ is minimized:
$$ begin{align}
epsilon &= frac{|W_A-W_B|}{W_A+W_B} cr
W_A &= sumlimits_{i=0}^{k-1}w_{a_i} cr
W_B &= sumlimits_{i=0}^{k-1}w_{b_i} cr
end{align}$$
Is this a known problem? (I found Is there a "balanced knapsacks" problem with a known result? which is similar)
In my case the value of $N$ is not very large (under 50) so I don't really care about proving NP-hard, I just want to know a good strategy.
combinatorics
asked Dec 21 '18 at 16:44
Jason SJason S
2,04811617
$endgroup$
Suppose I have a set of $2N$ items with weights $w_0, w_1, ldots, w_{2N-1}$. I want to identify the two most "balanced" sets of $k$ items each, where $k$ is not given but limited to the range $k_min leq k leq N$.
More specifically: Identify the two disjunct subsets $A = {a_0, a_1, ldots a_{k-1}}$ and $B = {b_0, b_1, ldots a_{k-1}}$ subject to the following conditions:
- all the $a_i, b_i in [0,2N-1]$
- $A cap B = varnothing$
- both sets have equal cardinality $k$ where $k_min leq k leq N$
- the following quantity $epsilon$ is minimized:
$$ begin{align}
epsilon &= frac{|W_A-W_B|}{W_A+W_B} cr
W_A &= sumlimits_{i=0}^{k-1}w_{a_i} cr
W_B &= sumlimits_{i=0}^{k-1}w_{b_i} cr
end{align}$$
Is this a known problem? (I found Is there a "balanced knapsacks" problem with a known result? which is similar)
In my case the value of $N$ is not very large (under 50) so I don't really care about proving NP-hard, I just want to know a good strategy.
combinatorics
combinatorics
asked Dec 21 '18 at 16:44
Jason SJason S
2,04811617
asked Dec 21 '18 at 16:44
Jason SJason S
2,04811617
asked Dec 21 '18 at 16:44
Jason SJason S
2,04811617
asked Dec 21 '18 at 16:44
Jason SJason S
2,04811617
asked Dec 21 '18 at 16:44
Jason SJason S
2,04811617
2,04811617
add a comment |
add a comment |
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