4 fix notation

# Is #k-XORSAT #P-complete?

k-XORSAT is the problem of deciding whether a Boolean formula $$\bigwedge_{i \in I} \oplus_{j=1}^k l_{s_{ij}}$$ is satisfiable. Here $\oplus$ denotes the binary XOR operation, $I$ is some index set, and each clause has $k$ distinct literals $l_{s_{ij}}$ each of which is either a variable $x_{s_{ij}}$ or its negation.

Equivalently, $k$-XORSAT requires deciding whether a set of linear equations, each of the form $\sum_{j=1}^k x_{s_{ij}}\equiv 1\; (\mod 2)$, has a solution over $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$.

Every decision problem Q has an associated counting problem , #Q, which (informally speaking) requires establishing the number of distinct solutions. Such counting problems form the complexity class #P. The "hardest" problems in #P are #P-complete, as any problem in #P can be reduced to a #P-complete problem.

The counting problem associated with any NP-complete decision problem is #P-complete. However, many "easy" decision problems (some even solvable in linear time) also lead to #P-complete counting problems. For instance, Leslie Valiant's original 1979 paper The Complexity of Computing the Permanent shows that computing the permanent of a 0-1 matrix is #P-complete. As a second example, the list of #P-complete problems in the companion paper The Complexity of Enumeration and Reliability Problems includes #MONOTONE 2-SAT; this problem requires counting the number of solutions to Boolean formulas in conjunctive normal form, where each clause has two variables and no negated variables are allowed. (MONOTONE 2-SAT is of course rather trivial as a decision problem.)

Andrea Montanari has written about the partition function of $k$-XORSAT in some lecture notes, and his book with Marc Mézard apparently discusses this (unfortunately I do not have a copy available to hand, and the relevant Chapter 17 is not included in Montanari's online draft).

These considerations lead to the following question:

Is $k$-XORSAT #$k$-XORSAT #P-complete?

Note that the formula in Montanari's notes does not obviously appear to answer this question. Just because there is a nice closed form solution, doesn't mean we can evaluate it efficiently: consider the Tutte polynomial.

Some difficult counting problems in #P can still be approximated in a certain sense, by means of a scheme called an FPRAS. Jerrum, Sinclair, and collaborators have linked the existence of an FPRAS for #P problems to the question of whether $NP = RP$. I would therefore also be interested in the subsidiary question

Does $k$-XORSAT #$k$-XORSAT have an FPRAS?

Edit: clarified second question as per comment by Tsuyoshi Ito. Note that Peter Shor's answer renders this part of the question moot.

k-XORSAT is the problem of deciding whether a Boolean formula $$\bigwedge_{i \in I} \oplus_{j=1}^k l_{s_{ij}}$$ is satisfiable. Here $\oplus$ denotes the binary XOR operation, $I$ is some index set, and each clause has $k$ distinct literals $l_{s_{ij}}$ each of which is either a variable $x_{s_{ij}}$ or its negation.

Equivalently, $k$-XORSAT requires deciding whether a set of linear equations, each of the form $\sum_{j=1}^k x_{s_{ij}}\equiv 1\; (\mod 2)$, has a solution over $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$.

Every decision problem has an associated counting problem, which (informally speaking) requires establishing the number of distinct solutions. Such counting problems form the complexity class #P. The "hardest" problems in #P are #P-complete, as any problem in #P can be reduced to a #P-complete problem.

The counting problem associated with any NP-complete decision problem is #P-complete. However, many "easy" decision problems (some even solvable in linear time) also lead to #P-complete counting problems. For instance, Leslie Valiant's original 1979 paper The Complexity of Computing the Permanent shows that computing the permanent of a 0-1 matrix is #P-complete. As a second example, the list of #P-complete problems in the companion paper The Complexity of Enumeration and Reliability Problems includes #MONOTONE 2-SAT; this problem requires counting the number of solutions to Boolean formulas in conjunctive normal form, where each clause has two variables and no negated variables are allowed. (MONOTONE 2-SAT is of course rather trivial as a decision problem.)

Andrea Montanari has written about the partition function of $k$-XORSAT in some lecture notes, and his book with Marc Mézard apparently discusses this (unfortunately I do not have a copy available to hand, and the relevant Chapter 17 is not included in Montanari's online draft).

These considerations lead to the following question:

Is $k$-XORSAT #P-complete?

Note that the formula in Montanari's notes does not obviously appear to answer this question. Just because there is a nice closed form solution, doesn't mean we can evaluate it efficiently: consider the Tutte polynomial.

Some difficult counting problems are not in #P-complete, but instead P can still be approximated in a certain sense, by means of a scheme called an FPRAS. Jerrum, Sinclair, and collaborators have linked the existence of an FPRAS for some #P problems to the question of whether $NP = RP$.

These considerations lead to I would therefore also be interested in the following subsidiary question:

Is

Does $k$-XORSAT #P-complete? If not, does it have an FPRAS?

Edit: clarified second question as per comment by Tsuyoshi Ito. Note that the formula in Montanari's notes does not obviously appear to Peter Shor's answer renders this question. Just because there is a nice closed form solution, doesn't mean we can evaluate it efficiently: consider part of the Tutte polynomialquestion moot.

2 fixed wording as suggested in comments

k-XORSAT is the problem of deciding whether a Boolean formula $$\bigwedge_{i \in I} \oplus_{j=1}^k l_{s_{ij}}$$ is satisfiable. Here $\oplus$ denotes the binary XOR operation, $I$ is some index set, and each clause has $k$ distinct literals $l_{s_{ij}}$ each of which is either a variable $x_{s_{ij}}$ or its negation.

Equivalently, $k$-XORSAT requires deciding whether a set of linear equations, each of the form $\sum_{j=1}^k x_{s_{ij}}\equiv 1\; (\mod 2)$, has a solution over $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$.

Every decision problem has an associated counting problem, which (informally speaking) requires establishing the number of distinct solutions. Such counting problems form the complexity class #P. The "hardest" problems in #P are #P-complete, as any problem in #P can be reduced to a #P-complete problem.

The counting problem associated with any NP-complete decision problem is #P-complete. However, many "easy" decision problems (some even solvable in linear time) also lead to #P-complete counting problems: . For instance, Leslie Valiant's original 1979 paper The Complexity of Computing the Permanent shows that computing the permanent of a 0-1 matrix is #P-complete. There is As a second example, the list of #P-complete problems in the companion paper The Complexity of Enumeration and Reliability Problems. These include includes #MONOTONE 2-SAT, which ; this problem requires counting the number of solutions to Boolean formulas in conjunctive normal form, where each clause has two variables (and no negated variables are allowed)allowed. (MONOTONE 2-SAT is of course rather trivial as a decision problem.)

Andrea Montanari has written about the partition function of $k$-XORSAT in some lecture notes, and his book with Marc Mézard apparently discusses this (unfortunately I do not have a copy available to hand, and the relevant Chapter 17 is not included in Montanari's online draft).

Some counting problems are not #P-complete, but instead can be approximated in a certain sense, by means of a scheme called an FPRAS. Jerrum, Sinclair, and collaborators have linked the existence of an FPRAS for some #P problems to the question of whether $NP = RP$.

These considerations lead to the following question:

Is $k$-XORSAT #P-complete? If not, does it have an FPRAS?

Note that the formula in Montanari's notes does not obviously appear to answer this question. Just because there is a nice closed form solution, doesn't mean we can evaluate it efficiently: consider the Tutte polynomial.

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