# Attacking Problems Related to the Subset Sum Problem

#### 11/30/2023

This document covers the topic of solving problems related to the subset sum problem with RcppAlgos. We have already covered integer partitions, which is a special case of the subset sum problem, in Constraints and Integer Partitions and it is highly encouraged to read that vignette first.

## Building on Integer Partitions

The integer partition problem presents the question “how can we write n as a sum of positive integers?” There are well-known algorithms for enumerating all partitions of an integer n. We even have algorithms for generating partitions of a specific length or with distinct parts only. But how do we enumerate partitions of n with a specific set of numbers? What about enumerating partitions of a specific length m of n given a specific set of numbers?

For example, using only the numbers 3:18, find all partitions of 50 of length 5.

With RcppAlgos, this is easily achieved. We simply use the same template as we did in Constraints and Integer Partitions. Observe (We continue to use the ht function defined in the Combination and Permutation Basics vignette):

library(RcppAlgos)
options(width = 90)

ht <- function(d, m = 5, n = m) {
## print the head and tail together
cat("--------\n")
cat("tail -->\n")
print(tail(d, n))
}

## Each element can only occur once
ht(partitionsGeneral(3:18, 5, target = 50))
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    3    4    8   17   18
#> [2,]    3    4    9   16   18
#> [3,]    3    4   10   15   18
#> [4,]    3    4   10   16   17
#> [5,]    3    4   11   14   18
#> --------
#> tail -->
#>        [,1] [,2] [,3] [,4] [,5]
#> [180,]    7    8    9   12   14
#> [181,]    7    8   10   11   14
#> [182,]    7    8   10   12   13
#> [183,]    7    9   10   11   13
#> [184,]    8    9   10   11   12

ht(partitionsGeneral(3:18, 5, TRUE, target = 50))
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    3    3    8   18   18
#> [2,]    3    3    9   17   18
#> [3,]    3    3   10   16   18
#> [4,]    3    3   10   17   17
#> [5,]    3    3   11   15   18
#> --------
#> tail -->
#>        [,1] [,2] [,3] [,4] [,5]
#> [507,]    9    9    9   11   12
#> [508,]    9    9   10   10   12
#> [509,]    9    9   10   11   11
#> [510,]    9   10   10   10   11
#> [511,]   10   10   10   10   10

## Even works on multisets
ht(partitionsGeneral(3:18, 5, freqs = rep(1:4, 4), target = 50))
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    3    4    7   18   18
#> [2,]    3    4    8   17   18
#> [3,]    3    4    9   16   18
#> [4,]    3    4    9   17   17
#> [5,]    3    4   10   15   18
#> --------
#> tail -->
#>        [,1] [,2] [,3] [,4] [,5]
#> [401,]    8   10   10   10   12
#> [402,]    9    9    9   10   13
#> [403,]    9    9    9   11   12
#> [404,]    9    9   10   10   12
#> [405,]    9   10   10   10   11

In fact, these optimized algorithms can be applied when the vector passed has the quality that if you were to sort them, the difference of each element with it’s neighbor is constant (E.g. c(121, 126, 131, 136, ..., 221)).

even_time <- system.time({
genParts <- partitionsGeneral(seq(121, 221, 5), 13,
TRUE, target = 2613)
})
even_time
#>    user  system elapsed
#>   0.002   0.001   0.003

ht(genParts)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [1,]  121  121  161  221  221  221  221  221  221   221   221   221   221
#> [2,]  121  121  166  216  221  221  221  221  221   221   221   221   221
#> [3,]  121  121  171  211  221  221  221  221  221   221   221   221   221
#> [4,]  121  121  171  216  216  221  221  221  221   221   221   221   221
#> [5,]  121  121  176  206  221  221  221  221  221   221   221   221   221
#> --------
#> tail -->
#>           [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [119542,]  196  196  196  201  201  201  201  201  201   201   206   206   206
#> [119543,]  196  196  201  201  201  201  201  201  201   201   201   201   211
#> [119544,]  196  196  201  201  201  201  201  201  201   201   201   206   206
#> [119545,]  196  201  201  201  201  201  201  201  201   201   201   201   206
#> [119546,]  201  201  201  201  201  201  201  201  201   201   201   201   201

prettyNum(comboCount(seq(121, 221, 5), 13, TRUE), big.mark = ",")
#> [1] "573,166,440"

system.time(genMultiParts <- partitionsGeneral(seq(121, 221, 5), 13,
freqs = rep(1:7, 3),
targe = 2613))
#>    user  system elapsed
#>   0.004   0.001   0.005

ht(genMultiParts)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [1,]  121  126  171  216  216  216  221  221  221   221   221   221   221
#> [2,]  121  126  176  211  216  216  221  221  221   221   221   221   221
#> [3,]  121  126  176  216  216  216  216  221  221   221   221   221   221
#> [4,]  121  126  181  206  216  216  221  221  221   221   221   221   221
#> [5,]  121  126  181  211  211  216  221  221  221   221   221   221   221
#> --------
#> tail -->
#>          [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [70291,]  186  186  191  196  196  201  201  206  206   211   211   211   211
#> [70292,]  186  186  191  196  196  201  206  206  206   206   211   211   211
#> [70293,]  186  186  191  196  201  201  201  206  206   206   206   211   216
#> [70294,]  186  186  191  196  201  201  201  206  206   206   211   211   211
#> [70295,]  186  186  196  196  201  201  201  206  206   206   206   211   211

prettyNum(comboCount(seq(121, 221, 5), 13, freqs = rep(1:7, 3)), big.mark = ",")
#> [1] "256,047,675"

### Working with Negative Numbers

Generally, integer partition algorithms are restricted to positive integers. However, with the generalized partition algorithms in RcppAlgos, we can make light work of vectors with negative numbers (again, the sorted vector has to have the property that the difference of each element with it’s neighbor is constant).

system.time({
genDistParts <- partitionsGeneral(seq(-173L, 204L, 13L),
11, target = -460)
})
#>    user  system elapsed
#>   0.002   0.000   0.002

all(rowSums(genDistParts) == -460L)
#> [1] TRUE

ht(genDistParts)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#> [1,] -173 -160 -147 -134 -121 -108  -95  -82  165   191   204
#> [2,] -173 -160 -147 -134 -121 -108  -95  -69  152   191   204
#> [3,] -173 -160 -147 -134 -121 -108  -95  -69  165   178   204
#> [4,] -173 -160 -147 -134 -121 -108  -95  -56  139   191   204
#> [5,] -173 -160 -147 -134 -121 -108  -95  -56  152   178   204
#> --------
#> tail -->
#>           [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#> [108940,] -121 -108  -82  -69  -56  -43  -30  -17   -4    22    48
#> [108941,] -121 -108  -82  -69  -56  -43  -30  -17    9    22    35
#> [108942,] -121  -95  -82  -69  -56  -43  -30  -17   -4     9    48
#> [108943,] -121  -95  -82  -69  -56  -43  -30  -17   -4    22    35
#> [108944,] -108  -95  -82  -69  -56  -43  -30  -17   -4     9    35

## Partitions with no Restrictions

With the examples illustrated above, we had the restriction that the sorted input vector had to have the property that the difference of each element with it’s neighbor is constant. If this requirement is broken, it only means that we cannot use a particular algorithm and we must fall back to a more general algorithm. Fret not!! These general algorithms are extremely efficient and very flexible. We can use them with random input vectors, random targets, as well as over ranges.

Let us revisit the example above but with a slight variation that breaks the requirement.

inpVec <- c(116, seq(126, 221, 5))

## Non-constant difference... The specialized algo can't be used
diff(inpVec)
#>  [1] 10  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5

uneven_time <- system.time({
genParts2 <- partitionsGeneral(inpVec, 13, TRUE, target = 2613)
})
uneven_time    ## out of a possible 573 million in under a second
#>    user  system elapsed
#>   0.050   0.002   0.052

ht(genParts2)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [1,]  116  116  171  221  221  221  221  221  221   221   221   221   221
#> [2,]  116  116  176  216  221  221  221  221  221   221   221   221   221
#> [3,]  116  116  181  211  221  221  221  221  221   221   221   221   221
#> [4,]  116  116  181  216  216  221  221  221  221   221   221   221   221
#> [5,]  116  116  186  206  221  221  221  221  221   221   221   221   221
#> --------
#> tail -->
#>           [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [118556,]  196  196  196  201  201  201  201  201  201   201   206   206   206
#> [118557,]  196  196  201  201  201  201  201  201  201   201   201   201   211
#> [118558,]  196  196  201  201  201  201  201  201  201   201   201   206   206
#> [118559,]  196  201  201  201  201  201  201  201  201   201   201   201   206
#> [118560,]  201  201  201  201  201  201  201  201  201   201   201   201   201

Although the above was about 17 times slower than the first example dealing with 573 million combinations (0.052 milliseconds vs. 0.003 milliseconds), we are still dealing in milliseconds!!! For reference, version 2.3.4 takes about 18 seconds to find all 118,560 solutions.

Here are some more exotic examples demonstrating the power of these algorithms.

set.seed(42)
mySamp <- sample(-100:100, 50)

sort(mySamp)
#>  [1] -98 -97 -96 -95 -81 -77 -74 -65 -60 -59 -58 -54 -52 -43 -36 -33 -30 -27 -12  -9  -2
#> [22]  -1   3   8   9  10  13  21  27  30  33  35  42  45  49  52  53  57  61  63  64  70
#> [43]  73  76  83  84  88  91  96 100

system.time(exotic <- partitionsGeneral(mySamp, 8, freqs = rep(1:5, 10),
target = 496))
#>    user  system elapsed
#>   0.130   0.001   0.130

dim(exotic)
#> [1] 102241      8

## Over 1 billion total combinations
prettyNum(comboCount(mySamp, 8, freqs = rep(1:5, 10)), big.mark = ",")
#> [1] "1,343,133,680"

## Only getting a few (a thousand in this case) is much faster
system.time(partitionsGeneral(mySamp, 8, freqs = rep(1:5, 10),
target = 496, upper = 1e3))
#>    user  system elapsed
#>   0.001   0.000   0.002

The function permuteGeneral benefits from these optimized algorithms as well. However, just as we discussed in Output Order with permuteGeneral, the output will not be in lexicographical order.

## Taming Floating Point Numbers

Oftentimes when working with numerical vectors, it can be hard to find combinations that sum to a particular number because of floating point errors (See Using tolerance). In practice, we may not need an exact match and a close approximation will suffice. For example, let’s say we have a football team of 100 players (including practice squad) and we are interested in a trade involving 6 players and a total salary of 20 million dollars. We may not be able to find 6 players whose sum of salaries is exactly 20 million, but we can find many 6 player combinations whose sum of salaries is within a tolerance of 20 million.

set.seed(22213)
football_player_salaries <- 2e7 * rbeta(100, 2, 25)

summary(football_player_salaries)
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
#>   115308   768338  1261683  1612271  1950565 10895883

## Over 1 billion combinations...
## An exhaustive search will not be feasible
prettyNum(comboCount(football_player_salaries, 6), big.mark = ",")
#> [1] "1,192,052,400"

system.time(exactly20 <- partitionsGeneral(football_player_salaries, 6,
target = 2e7, tolerance = 0))
#>    user  system elapsed
#>   1.438   0.004   1.441

dim(exactly20)
#> [1] 0 6

What if we increase the tolerance to $1000 (Honestly… what’s $1000 when we are talking about 20 million dollars)? Our intent is to explore these options, so we take advantage of the upper argument in anticipation that we obtain many results that meet the criteria. If we obtain the upper bound, we decrease the tolerance (if needed) and repeat.

## N.B. This is much more efficient. Also, we set keepResults
## to TRUE so we can see the total sum of salaries.
system.time(almost20 <- comboGeneral(football_player_salaries, 6,
constraintFun = "sum", comparisonFun = "==",
limitConstraints = 2e7, tolerance = 1000,
upper = 1000, keepResults = TRUE))
#>    user  system elapsed
#>   0.063   0.000   0.062

dim(almost20)
#> [1] 1000    7

ht(almost20)
#>          [,1]     [,2]      [,3]    [,4]    [,5]     [,6]     [,7]
#> [1,] 115307.7 152563.4  809407.9 3163109 4863446 10895883 19999717
#> [2,] 115307.7 152563.4 1590746.9 2381655 4863446 10895883 19999602
#> [3,] 115307.7 152563.4 1669898.9 2302265 4863446 10895883 19999365
#> [4,] 115307.7 152563.4 1746659.2 2225285 4863446 10895883 19999145
#> [5,] 115307.7 152563.4 1853727.8 2850338 4132545 10895883 20000364
#> --------
#> tail -->
#>             [,1]     [,2]    [,3]    [,4]    [,5]     [,6]     [,7]
#>  [996,] 200278.8 550414.4 1751652 2984500 3618110 10895883 20000838
#>  [997,] 200278.8 550414.4 1855829 3163109 3334046 10895883 19999560
#>  [998,] 200278.8 550414.4 1884764 2850338 3618110 10895883 19999788
#>  [999,] 200278.8 550414.4 2013884 2850338 3489156 10895883 19999953
#> [1000,] 200278.8 550414.4 2051845 2984500 3316996 10895883 19999917

## decreasing the tolerance to $10 further we obtain 158 results system.time(superClose20 <- comboGeneral(football_player_salaries, 6, constraintFun = "sum", comparisonFun = "==", limitConstraints = 2e7, tolerance = 10, upper = 1000, keepResults = TRUE)) #> user system elapsed #> 1.435 0.002 1.438 ht(superClose20) #> head --> #> [,1] [,2] [,3] [,4] [,5] [,6] [,7] #> [1,] 115307.7 266606.5 695657.2 3163109 4863446 10895883 20000009 #> [2,] 115307.7 1117835.0 1318811.6 1688714 4863446 10895883 19999998 #> [3,] 152563.4 628078.8 1117835.0 3334046 3871591 10895883 19999997 #> [4,] 152563.4 695657.2 1635144.3 2984500 3636247 10895883 19999995 #> [5,] 200278.8 765448.4 1174496.9 1923219 5040664 10895883 19999990 #> -------- #> tail --> #> [,1] [,2] [,3] [,4] [,5] [,6] [,7] #> [154,] 1318812 1359188 1512157 1929459 2984500 10895883 19999997 #> [155,] 1318812 1371670 1771874 2225285 2416482 10895883 20000006 #> [156,] 1338303 1706514 1823915 1853728 2381655 10895883 19999998 #> [157,] 1371670 1512157 1516215 1853728 2850338 10895883 19999990 #> [158,] 1371670 1516215 1635144 2189663 2391419 10895883 19999994 ## prod and mean These optimized algorithms are also employed when constraintFun is "prod" or "mean". getAllThenFilter <- function(n, m, lim) { t <- comboGeneral(n, m, constraintFun = "prod") t[t[, m + 1] == lim, -(m+1)] } library(microbenchmark) microbenchmark(optimized = comboGeneral(25, 10, constraintFun = "prod", comparisonFun = "==", limitConstraints = 1037836800), brute = getAllThenFilter(25, 10, 1037836800), times = 20, unit = "relative", check = "equal") #> Warning in microbenchmark(optimized = comboGeneral(25, 10, constraintFun = "prod", : less #> accurate nanosecond times to avoid potential integer overflows #> Unit: relative #> expr min lq mean median uq max neval cld #> optimized 1.00000 1.0000 1.00000 1.00000 1.000 1.00000 20 a #> brute 21.60328 21.5596 22.11548 21.31989 22.093 26.20377 20 b ## What about cases when brute force isn't feasible set.seed(101) v <- runif(1000, 1, 2) prettyNum(comboCount(v, 100), big.mark = ",") #> [1] "63,850,511,926,305,130,236,698,511,142,022,274,281,262,900,693,853,331,776,286,816,221,524,376,994,750,901,948,920,974,351,797,699,894,319,420,811,933,446,197,797,592,213,357,065,053,890" system.time(prodAlmost100 <- comboGeneral(v, 100, constraintFun = "prod", comparisonFun = "==", limitConstraints = 100, tolerance = 0.0001, upper = 20)) #> user system elapsed #> 0.015 0.000 0.016 dim(prodAlmost100) #> [1] 20 100 apply(prodAlmost100, 1, prod) #> [1] 100.00008 100.00003 100.00003 100.00006 100.00010 100.00000 99.99993 99.99995 #> [9] 100.00002 99.99992 100.00004 99.99994 100.00002 100.00005 99.99992 99.99996 #> [17] 100.00006 100.00003 100.00006 100.00002 ## Showcasing mean system.time(meanAlmost1.5 <- comboGeneral(v, 100, constraintFun = "mean", comparisonFun = "==", limitConstraints = 1.5, tolerance = 0.0001, upper = 20)) #> user system elapsed #> 0 0 0 dim(meanAlmost1.5) #> [1] 20 100 rowMeans(meanAlmost1.5) #> [1] 1.500000 1.499999 1.500001 1.500000 1.500000 1.500001 1.500000 1.500000 1.500000 #> [10] 1.500000 1.499999 1.500000 1.500001 1.500001 1.499999 1.499999 1.500001 1.500000 #> [19] 1.500001 1.500000 ## Using Iterators As of version 2.5.0 all of the above cases can be attacked with iterators (See Combinatorial Iterators in RcppAlgos). As mentioned in the suggested reading, iterators are very flexible and just as efficient as their “general” counterparts. They have the added benefit of allowing one to save the current state, allowing one to generate n results at a time. Below are a few demonstrations using some of the examples in earlier sections. ## The football salary example salary <- partitionsIter(football_player_salaries, 6, target = 2e7, tolerance = 1000) ## Or use comboIter: ## ## comboIter(football_player_salaries, 6, constraintFun = "sum", ## comparisonFun = "==", limitConstraints = 2e7, ## tolerance = 1000, upper = 1000, keepResults = TRUE)) system.time(almost20withIter <- salary@nextNIter(1e3)) #> user system elapsed #> 0.067 0.000 0.067 ## almost20 was generated above with comboGeneral all.equal(almost20[, 1:6], almost20withIter) #> [1] TRUE ## With iterators we can easily continue iterating. With the general ## functions if we wanted the next 1000 results, we would have to ## generate the first 1000 along with the next 1000 system.time(nextAlmost20withIter <- salary@nextNIter(1e3)) #> user system elapsed #> 0.067 0.000 0.066 ht(nextAlmost20withIter) #> head --> #> [,1] [,2] [,3] [,4] [,5] [,6] #> [1,] 200278.8 550414.4 2189662.6 2829267 3334046 10895883 #> [2,] 200278.8 628078.8 653771.4 3489156 4132545 10895883 #> [3,] 200278.8 628078.8 809407.9 3334046 4132545 10895883 #> [4,] 200278.8 628078.8 852693.7 2381655 5040664 10895883 #> [5,] 200278.8 628078.8 1165478.3 3239395 3871591 10895883 #> -------- #> tail --> #> [,1] [,2] [,3] [,4] [,5] [,6] #> [996,] 264972.3 663453 1318812 3239395 3618110 10895883 #> [997,] 264972.3 663453 1376878 3163109 3636247 10895883 #> [998,] 264972.3 663453 1688714 2850338 3636247 10895883 #> [999,] 264972.3 663453 1706514 2850338 3618110 10895883 #> [1000,] 264972.3 663453 1853728 2189663 4132545 10895883 salary@summary() #>$description
#> [1] "Combinations of 100 choose 6 where the sum is between 19999000 and 20001000"
#>
#> $currentIndex #> [1] 2000 #> #>$totalResults
#> [1] NA
#>
#> $totalRemaining #> [1] NA ## The prodAlmost100 example prodIter <- comboIter(v, 100, constraintFun = "prod", comparisonFun = "==", limitConstraints = 100, tolerance = 0.0001) system.time(prodAlmost100WithIter <- prodIter@nextNIter(20)) #> user system elapsed #> 0.017 0.000 0.017 all.equal(prodAlmost100, prodAlmost100WithIter) #> [1] TRUE ## Again, with iterators, we can continue iterating from ## where we left off system.time(nextAlmost100WithIter <- prodIter@nextNIter(20)) #> user system elapsed #> 0.012 0.000 0.012 dim(nextAlmost100WithIter) #> [1] 20 100 ## Use @ or$ to access methods. If one needs to access these methods
## often (e.g. nextIter inside a loop), it is recommended to use the
## @ accessor as it is much more efficient.
prodIter$summary() #>$description
#> [1] "Combinations of 1000 choose 100 where the prod is between 99.999899999999997 and 100.0001"
#>
#> $currentIndex #> [1] 40 #> #>$totalResults
#> [1] NA
#>
#> \$totalRemaining
#> [1] NA