reduction from 3-SAT to Subset Sum problem Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?NAE SAT reduction to weighted MAX CUTHow to reduce from subset-sum problem?Constructing a promise problem equivalent to XSAT from subset sumQuestion on SAT reductionReduction of SUBSET-SUM to SET-PARTITIONSubset-sum variation, multiple sumsCNF-SAT reduction problem variantSubset sum NP-CompletnessKarp reduction from PARTITION to SUBSET SUMHow can I reduce Subset Sum to Almost Subset Sum?
Recursive calls to a function - why is the address of the parameter passed to it lowering with each call?
Pointing to problems without suggesting solutions
Why these surprising proportionalities of integrals involving odd zeta values?
Can I take recommendation from someone I met at a conference?
Determine the generator of an ideal of ring of integers
Why are two-digit numbers in Jonathan Swift's "Gulliver's Travels" (1726) written in "German style"?
Does GDPR cover the collection of data by websites that crawl the web and resell user data
Are there any AGPL-style licences that require source code modifications to be public?
How to leave only the following strings?
Can this water damage be explained by lack of gutters and grading issues?
Compiling and throwing simple dynamic exceptions at runtime for JVM
When does Bran Stark remember Jamie pushing him?
How can I introduce the names of fantasy creatures to the reader?
Can 'non' with gerundive mean both lack of obligation and negative obligation?
false 'Security alert' from Google - every login generates mails from 'no-reply@accounts.google.com'
Why aren't these two solutions equivalent? Combinatorics problem
Etymology of 見舞い
Can gravitational waves pass through a black hole?
Why doesn't the university give past final exams' answers?
Who's this lady in the war room?
Like totally amazing interchangeable sister outfit accessory swapping or whatever
Continue tikz picture on next page
Should man-made satellites feature an intelligent inverted "cow catcher"?
Are Flameskulls resistant to magical piercing damage?
reduction from 3-SAT to Subset Sum problem
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)
Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?NAE SAT reduction to weighted MAX CUTHow to reduce from subset-sum problem?Constructing a promise problem equivalent to XSAT from subset sumQuestion on SAT reductionReduction of SUBSET-SUM to SET-PARTITIONSubset-sum variation, multiple sumsCNF-SAT reduction problem variantSubset sum NP-CompletnessKarp reduction from PARTITION to SUBSET SUMHow can I reduce Subset Sum to Almost Subset Sum?
$begingroup$
How to reduce 3-SAT to subset sum problem?
complexity-theory automata reductions
New contributor
$endgroup$
add a comment |
$begingroup$
How to reduce 3-SAT to subset sum problem?
complexity-theory automata reductions
New contributor
$endgroup$
add a comment |
$begingroup$
How to reduce 3-SAT to subset sum problem?
complexity-theory automata reductions
New contributor
$endgroup$
How to reduce 3-SAT to subset sum problem?
complexity-theory automata reductions
complexity-theory automata reductions
New contributor
New contributor
New contributor
asked 6 hours ago
SiluPandaSiluPanda
666
666
New contributor
New contributor
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The trick to the reduction is to use numbers to encode statements about the 3CNF formula, crafting those numbers in such a way that you can later make an arithmetic proposition about the numbers that is only true if the original 3CNF formula is satisfiable. The reduction below is lifted directly from the lecture notes found at https://people.clarkson.edu/~alexis/PCMI/Notes/lectureB07.pdf .
We reduce 3SAT to SUBSET-SUM. Consider a 3CNF formula with variables $x_1, . . . , x_n$ and clauses $c_1, . . . , c_r$. For each variable $x_i$, we will have two numbers $y_i$ and $z_i$ in the list. For each clause $c_j$, we will also have two numbers $s_j$ and $t_j$. We define all of these numbers by specifying their base 10 representations. The construction is best explained by an example and a picture. If the formula is $(x_1∨x_2∨overlinex_3)∧(overlinex_1∨x_2∨overlinex_3)$, then the base 10 representations of the numbers will look like this:
beginarrayc
& x_1 & x_2 & x_3 & c_1 & c_2 \
hline
y_1 & 1 & 0 & 0 & 1 & 0 \
z_1 & 1 & 0 & 0 & 0 & 1 \
y_2 & 0 & 1 & 0 & 1 & 1 \
z_2 & 0 & 1 & 0 & 0 & 0 \
y_3 & 0 & 0 & 1 & 0 & 0 \
z_3 & 0 & 0 & 1 & 1 & 1 \
hline
s_1 & 0 & 0 & 0 & 1 & 0 \
t_1 & 0 & 0 & 0 & 1 & 0 \
s_2 & 0 & 0 & 0 & 0 & 1 \
t_2 & 0 & 0 & 0 & 0 & 1 \
hline
k & 1 & 1 & 1 & 3 & 3 \
endarray
The number $y_i$ corresponds to the positive occurrences of $x_i$ in the formula while the number $z_i$ corresponds to its negative occurrences. It should be clear how to generalize this construction to an arbitrary 3CNF formula. And the list of numbers can clearly be constructed in polynomial time. We claim that a subset of these numbers adds to exactly $k$ if and only if the formula is satisfiable. A key point is that the sum of the numbers can be done column by column, independently, because carries will never occur.
The $s$ value is crafted the same way for each clause; put a one in the digit position corresponding to that clause, and zeroes everywhere else. The $t$ value will be the same as the $s$ value for each clause.
The $k$ value is always 1111... followed by 33333.... where the number of ones is the same as the number of distinct variables in the formula and the number of threes is the number of clauses in the formula. Note that the required sum $k$ has a one in each digit position corresponding to the variables. This means that any solution to the subset sum problem can include only encoded statements about either a positive instance of the variable or a negative instance in each clause, not both. Note also that sum $k$ has a three in the digit position corresponding to each clause. The $s$ and $t$ values for each clause will sum to two, but to complete the sum a third one will have to come from one of the $y$ or $z$ values. All three ones could come from the $y$ and $z$ values, but the fact that $s$ and $t$ will only sum to two for any clause guarantees that any empty clause in the 3CNF formula forces the subset sum problem to become unsatisfiable.
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "419"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
SiluPanda is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f107376%2freduction-from-3-sat-to-subset-sum-problem%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The trick to the reduction is to use numbers to encode statements about the 3CNF formula, crafting those numbers in such a way that you can later make an arithmetic proposition about the numbers that is only true if the original 3CNF formula is satisfiable. The reduction below is lifted directly from the lecture notes found at https://people.clarkson.edu/~alexis/PCMI/Notes/lectureB07.pdf .
We reduce 3SAT to SUBSET-SUM. Consider a 3CNF formula with variables $x_1, . . . , x_n$ and clauses $c_1, . . . , c_r$. For each variable $x_i$, we will have two numbers $y_i$ and $z_i$ in the list. For each clause $c_j$, we will also have two numbers $s_j$ and $t_j$. We define all of these numbers by specifying their base 10 representations. The construction is best explained by an example and a picture. If the formula is $(x_1∨x_2∨overlinex_3)∧(overlinex_1∨x_2∨overlinex_3)$, then the base 10 representations of the numbers will look like this:
beginarrayc
& x_1 & x_2 & x_3 & c_1 & c_2 \
hline
y_1 & 1 & 0 & 0 & 1 & 0 \
z_1 & 1 & 0 & 0 & 0 & 1 \
y_2 & 0 & 1 & 0 & 1 & 1 \
z_2 & 0 & 1 & 0 & 0 & 0 \
y_3 & 0 & 0 & 1 & 0 & 0 \
z_3 & 0 & 0 & 1 & 1 & 1 \
hline
s_1 & 0 & 0 & 0 & 1 & 0 \
t_1 & 0 & 0 & 0 & 1 & 0 \
s_2 & 0 & 0 & 0 & 0 & 1 \
t_2 & 0 & 0 & 0 & 0 & 1 \
hline
k & 1 & 1 & 1 & 3 & 3 \
endarray
The number $y_i$ corresponds to the positive occurrences of $x_i$ in the formula while the number $z_i$ corresponds to its negative occurrences. It should be clear how to generalize this construction to an arbitrary 3CNF formula. And the list of numbers can clearly be constructed in polynomial time. We claim that a subset of these numbers adds to exactly $k$ if and only if the formula is satisfiable. A key point is that the sum of the numbers can be done column by column, independently, because carries will never occur.
The $s$ value is crafted the same way for each clause; put a one in the digit position corresponding to that clause, and zeroes everywhere else. The $t$ value will be the same as the $s$ value for each clause.
The $k$ value is always 1111... followed by 33333.... where the number of ones is the same as the number of distinct variables in the formula and the number of threes is the number of clauses in the formula. Note that the required sum $k$ has a one in each digit position corresponding to the variables. This means that any solution to the subset sum problem can include only encoded statements about either a positive instance of the variable or a negative instance in each clause, not both. Note also that sum $k$ has a three in the digit position corresponding to each clause. The $s$ and $t$ values for each clause will sum to two, but to complete the sum a third one will have to come from one of the $y$ or $z$ values. All three ones could come from the $y$ and $z$ values, but the fact that $s$ and $t$ will only sum to two for any clause guarantees that any empty clause in the 3CNF formula forces the subset sum problem to become unsatisfiable.
$endgroup$
add a comment |
$begingroup$
The trick to the reduction is to use numbers to encode statements about the 3CNF formula, crafting those numbers in such a way that you can later make an arithmetic proposition about the numbers that is only true if the original 3CNF formula is satisfiable. The reduction below is lifted directly from the lecture notes found at https://people.clarkson.edu/~alexis/PCMI/Notes/lectureB07.pdf .
We reduce 3SAT to SUBSET-SUM. Consider a 3CNF formula with variables $x_1, . . . , x_n$ and clauses $c_1, . . . , c_r$. For each variable $x_i$, we will have two numbers $y_i$ and $z_i$ in the list. For each clause $c_j$, we will also have two numbers $s_j$ and $t_j$. We define all of these numbers by specifying their base 10 representations. The construction is best explained by an example and a picture. If the formula is $(x_1∨x_2∨overlinex_3)∧(overlinex_1∨x_2∨overlinex_3)$, then the base 10 representations of the numbers will look like this:
beginarrayc
& x_1 & x_2 & x_3 & c_1 & c_2 \
hline
y_1 & 1 & 0 & 0 & 1 & 0 \
z_1 & 1 & 0 & 0 & 0 & 1 \
y_2 & 0 & 1 & 0 & 1 & 1 \
z_2 & 0 & 1 & 0 & 0 & 0 \
y_3 & 0 & 0 & 1 & 0 & 0 \
z_3 & 0 & 0 & 1 & 1 & 1 \
hline
s_1 & 0 & 0 & 0 & 1 & 0 \
t_1 & 0 & 0 & 0 & 1 & 0 \
s_2 & 0 & 0 & 0 & 0 & 1 \
t_2 & 0 & 0 & 0 & 0 & 1 \
hline
k & 1 & 1 & 1 & 3 & 3 \
endarray
The number $y_i$ corresponds to the positive occurrences of $x_i$ in the formula while the number $z_i$ corresponds to its negative occurrences. It should be clear how to generalize this construction to an arbitrary 3CNF formula. And the list of numbers can clearly be constructed in polynomial time. We claim that a subset of these numbers adds to exactly $k$ if and only if the formula is satisfiable. A key point is that the sum of the numbers can be done column by column, independently, because carries will never occur.
The $s$ value is crafted the same way for each clause; put a one in the digit position corresponding to that clause, and zeroes everywhere else. The $t$ value will be the same as the $s$ value for each clause.
The $k$ value is always 1111... followed by 33333.... where the number of ones is the same as the number of distinct variables in the formula and the number of threes is the number of clauses in the formula. Note that the required sum $k$ has a one in each digit position corresponding to the variables. This means that any solution to the subset sum problem can include only encoded statements about either a positive instance of the variable or a negative instance in each clause, not both. Note also that sum $k$ has a three in the digit position corresponding to each clause. The $s$ and $t$ values for each clause will sum to two, but to complete the sum a third one will have to come from one of the $y$ or $z$ values. All three ones could come from the $y$ and $z$ values, but the fact that $s$ and $t$ will only sum to two for any clause guarantees that any empty clause in the 3CNF formula forces the subset sum problem to become unsatisfiable.
$endgroup$
add a comment |
$begingroup$
The trick to the reduction is to use numbers to encode statements about the 3CNF formula, crafting those numbers in such a way that you can later make an arithmetic proposition about the numbers that is only true if the original 3CNF formula is satisfiable. The reduction below is lifted directly from the lecture notes found at https://people.clarkson.edu/~alexis/PCMI/Notes/lectureB07.pdf .
We reduce 3SAT to SUBSET-SUM. Consider a 3CNF formula with variables $x_1, . . . , x_n$ and clauses $c_1, . . . , c_r$. For each variable $x_i$, we will have two numbers $y_i$ and $z_i$ in the list. For each clause $c_j$, we will also have two numbers $s_j$ and $t_j$. We define all of these numbers by specifying their base 10 representations. The construction is best explained by an example and a picture. If the formula is $(x_1∨x_2∨overlinex_3)∧(overlinex_1∨x_2∨overlinex_3)$, then the base 10 representations of the numbers will look like this:
beginarrayc
& x_1 & x_2 & x_3 & c_1 & c_2 \
hline
y_1 & 1 & 0 & 0 & 1 & 0 \
z_1 & 1 & 0 & 0 & 0 & 1 \
y_2 & 0 & 1 & 0 & 1 & 1 \
z_2 & 0 & 1 & 0 & 0 & 0 \
y_3 & 0 & 0 & 1 & 0 & 0 \
z_3 & 0 & 0 & 1 & 1 & 1 \
hline
s_1 & 0 & 0 & 0 & 1 & 0 \
t_1 & 0 & 0 & 0 & 1 & 0 \
s_2 & 0 & 0 & 0 & 0 & 1 \
t_2 & 0 & 0 & 0 & 0 & 1 \
hline
k & 1 & 1 & 1 & 3 & 3 \
endarray
The number $y_i$ corresponds to the positive occurrences of $x_i$ in the formula while the number $z_i$ corresponds to its negative occurrences. It should be clear how to generalize this construction to an arbitrary 3CNF formula. And the list of numbers can clearly be constructed in polynomial time. We claim that a subset of these numbers adds to exactly $k$ if and only if the formula is satisfiable. A key point is that the sum of the numbers can be done column by column, independently, because carries will never occur.
The $s$ value is crafted the same way for each clause; put a one in the digit position corresponding to that clause, and zeroes everywhere else. The $t$ value will be the same as the $s$ value for each clause.
The $k$ value is always 1111... followed by 33333.... where the number of ones is the same as the number of distinct variables in the formula and the number of threes is the number of clauses in the formula. Note that the required sum $k$ has a one in each digit position corresponding to the variables. This means that any solution to the subset sum problem can include only encoded statements about either a positive instance of the variable or a negative instance in each clause, not both. Note also that sum $k$ has a three in the digit position corresponding to each clause. The $s$ and $t$ values for each clause will sum to two, but to complete the sum a third one will have to come from one of the $y$ or $z$ values. All three ones could come from the $y$ and $z$ values, but the fact that $s$ and $t$ will only sum to two for any clause guarantees that any empty clause in the 3CNF formula forces the subset sum problem to become unsatisfiable.
$endgroup$
The trick to the reduction is to use numbers to encode statements about the 3CNF formula, crafting those numbers in such a way that you can later make an arithmetic proposition about the numbers that is only true if the original 3CNF formula is satisfiable. The reduction below is lifted directly from the lecture notes found at https://people.clarkson.edu/~alexis/PCMI/Notes/lectureB07.pdf .
We reduce 3SAT to SUBSET-SUM. Consider a 3CNF formula with variables $x_1, . . . , x_n$ and clauses $c_1, . . . , c_r$. For each variable $x_i$, we will have two numbers $y_i$ and $z_i$ in the list. For each clause $c_j$, we will also have two numbers $s_j$ and $t_j$. We define all of these numbers by specifying their base 10 representations. The construction is best explained by an example and a picture. If the formula is $(x_1∨x_2∨overlinex_3)∧(overlinex_1∨x_2∨overlinex_3)$, then the base 10 representations of the numbers will look like this:
beginarrayc
& x_1 & x_2 & x_3 & c_1 & c_2 \
hline
y_1 & 1 & 0 & 0 & 1 & 0 \
z_1 & 1 & 0 & 0 & 0 & 1 \
y_2 & 0 & 1 & 0 & 1 & 1 \
z_2 & 0 & 1 & 0 & 0 & 0 \
y_3 & 0 & 0 & 1 & 0 & 0 \
z_3 & 0 & 0 & 1 & 1 & 1 \
hline
s_1 & 0 & 0 & 0 & 1 & 0 \
t_1 & 0 & 0 & 0 & 1 & 0 \
s_2 & 0 & 0 & 0 & 0 & 1 \
t_2 & 0 & 0 & 0 & 0 & 1 \
hline
k & 1 & 1 & 1 & 3 & 3 \
endarray
The number $y_i$ corresponds to the positive occurrences of $x_i$ in the formula while the number $z_i$ corresponds to its negative occurrences. It should be clear how to generalize this construction to an arbitrary 3CNF formula. And the list of numbers can clearly be constructed in polynomial time. We claim that a subset of these numbers adds to exactly $k$ if and only if the formula is satisfiable. A key point is that the sum of the numbers can be done column by column, independently, because carries will never occur.
The $s$ value is crafted the same way for each clause; put a one in the digit position corresponding to that clause, and zeroes everywhere else. The $t$ value will be the same as the $s$ value for each clause.
The $k$ value is always 1111... followed by 33333.... where the number of ones is the same as the number of distinct variables in the formula and the number of threes is the number of clauses in the formula. Note that the required sum $k$ has a one in each digit position corresponding to the variables. This means that any solution to the subset sum problem can include only encoded statements about either a positive instance of the variable or a negative instance in each clause, not both. Note also that sum $k$ has a three in the digit position corresponding to each clause. The $s$ and $t$ values for each clause will sum to two, but to complete the sum a third one will have to come from one of the $y$ or $z$ values. All three ones could come from the $y$ and $z$ values, but the fact that $s$ and $t$ will only sum to two for any clause guarantees that any empty clause in the 3CNF formula forces the subset sum problem to become unsatisfiable.
answered 4 hours ago
Kyle JonesKyle Jones
5,81211840
5,81211840
add a comment |
add a comment |
SiluPanda is a new contributor. Be nice, and check out our Code of Conduct.
SiluPanda is a new contributor. Be nice, and check out our Code of Conduct.
SiluPanda is a new contributor. Be nice, and check out our Code of Conduct.
SiluPanda is a new contributor. Be nice, and check out our Code of Conduct.
Thanks for contributing an answer to Computer Science Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f107376%2freduction-from-3-sat-to-subset-sum-problem%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown