# Questions tagged [reductions]

A reduction is the transformation of one problem into another problem. A example of using a reduction would be to be to show if a problem P is undecidable. This would be achieved by transforming or performing a reduction of a decision problem $P$ into an undecidable problem. If this can be achieved then we have shown that this problem P is undecidable.

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### Many-one reductions vs. Turing reductions to define NPC

Why do most people prefer to use many-one reductions to define NP-completeness instead of, for instance, Turing reductions?
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### Sum-of-square-roots-hard problems?

The sum of square roots problem asks, given two sequences $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ of positive integers, whether the sum $\sum_i \sqrt{a_i}$ less than, equal to, or greater ...
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### Why does randomness have stronger effect on reductions than on algorithms?

It is conjectured that randomness does not extend the power of polynomial time algorithms, that is, ${\bf P}={\bf BPP}$ is conjectured to hold. On the other hand, randomness seems to have a quite ...
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### Fast Reduction from RSA to SAT

Scott Aaronson's blog post today gave a list of interesting open problems/tasks in complexity. One in particular caught my attention: Build a public library of 3SAT instances, with as few variables ...
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### Derandomizing Valiant-Vazirani?

The Valiant-Vazirani theorem says that if there is a polynomial time algorithm (deterministic or randomized) for distinguishing between a SAT formula that has exactly one satisfying assignment, and an ...
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### Nontrivial membership in NP

Is there an example of a language which is in $NP$, but where we cannot prove this fact directly by showing that there exists a polynomial witness for membership in this language? Instead, the fact ...
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### Are there subexponential algorithms for PLANAR SAT known?

Some NP-hard problems which are exponential on general graphs are subexponential on planar graphs because the treewidth is at most $4.9 \sqrt{|V(G)|}$ and they are exponential in the treewidth. ...
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### Advanced techniques for determining complexity lower bounds

Some of you may have been following this question, which was closed due to not being research level. So, I'm extracting the part of the question which is at a research level. Beyond the "simpler" ...
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### Natural CLIQUE to k-Color reduction

There is clearly a reduction from CLIQUE to k-Color because they're both NP-Complete. In fact, I can construct one by composing a reduction from CLIQUE to 3-SAT with a reduction from 3-SAT to k-Color. ...
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### Curious about computer-assisted NP-completeness proofs

In the paper "THE COMPLEXITY OF SATISFIABILITY PROBLEMS" by Thomas J. Schaefer, the author has mentioned that ...
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### Is there a direct/natural reduction to count non-bipartite perfect matchings using the permanent?

Counting the number of perfect matchings in a bipartite graph is immediately reducible to computing the permanent. Since finding a perfect matching in a non-bipartite graph is in NP, there exists ...
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### Reductions from the book.

This is along the lines of "Algorithms from the Book". Although reductions are algorithms as well, I thought it doubtful that one would think of a reduction in response to the question about ...
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### Direct SAT to 3-SAT reduction

Here the goal is to reduce an arbitrary SAT problem to 3-SAT in polynomial time using the fewest number of clauses and variables. My question is motivated by curiosity. Less formally, I would like ...
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### Binary multiplication and parity convolution

This question is about the relationship between normal multiplication of binary numbers and polynomial multiplication mod 2. To make the question concrete, I would ideally like to know if there is a ...
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### How to prove that USTCONN requires logarithmic space?

USTCONN is the problem that requires deciding whether there is a path from the source vertex $s$ to the target vertex $t$ in a graph $G$, where these are all given as part of the input. Omer Reingold ...
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### Problems that are NP-complete under randomized or P/poly reductions.

In this question, we appear to have identified a natural problem that is NP-complete under randomized reductions, but possibly not under deterministic reductions (although this depends on which ...
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### Can strong NP-hardness really be shown using plain polytime reductions?

I recently read a proof that intended to show that a problem was strongly NP-hard, simply by reducing to it (in polynomial time) from a strongly NP-hard problem. This didn’t make any sense to me. I ...
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### Is intersection of $k \ge 3$ graphic matroids in P?

It is known that intersection of three general matroids is NP-hard (source), which is done via reduction from Hamiltonian cycle. The reduction uses one graphic matroid and two connectivity matroids. ...
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### Does Karp reducibility yield a total order?

Or with other words, do we have that for every language $A$ and $B$, $A \leq_p B$ or $B \leq_p A$?
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### Any references for techniques in FPT reductions?

As everyone knows, Garey and Johnson's famous book (and many others) provides an excellent reference for reduction technique in classical setting. Are there any surveys or books on the topic of ...
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### Can limit of hard languages be easy?

Can the following all hold simultaneously? $L_s$ is contained in $L_{s+1}$ for all positive integers $s$. $L = \bigcup_s L_s$ is the language of all finite words over $\{0,1\}$. There is some ...
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### Is ALogTime != PH hard to prove (and unknown)?

Lance Fortnow recently claimed that proving L != NP should be easier than proving P != NP: Separate NP from Logarithmic space. I gave four approaches in a pre-blog 2001 survey on diagonalization ...
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### Why is the reduction from 3-SAT to 3-dimensional Matching Parsimonious?

In this talk at the Simons Institute, Holger Dell notes that there is a parsimonious reduction from 3-SAT to the 3-dimensional Matching (3-DM) problem. In other words, there is a reduction between ...
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### Improving Cook's generic reduction for Clique to SAT?

I am interested in reducing $k$-Clique to SAT without making the instance much larger. Clique is in NP so it can be reduced to SAT using logarithmic space. The straightforward Garey/Johnson textbook ...
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### Does PPAD really capture the notion of finding another unbalanced vertex?

Complexity class PPAD was invented by Christos Papadimitriou in his seminal 1994 paper. The class is designed to capture the complexity of search problems where the existence of a solution is ...
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### Slowest many-one reduction?

When we want to prove that an $L\in \bf NP$ is $\bf NP$-complete, then the standard approach is to exhibit a polynomial time computable many-one reduction of a known $\bf NP$-complete problem to $L$. ...
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### Instance of FPT-reductions that is not a polynomial-time reduction

In parametrized complexity people use fixed-parameter-tractable (FPT) reduction to prove W[t]-hardness. Theoretically a FPT-reduction is not a polynomial-time reduction, since it can run exponentially ...
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### Worst case to average case reductions

Are there problems whose average case complexity is the same as their worst case complexity? What are the underlying properties of these problems that makes reducing the worst case to the average case ...
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### Reductions of hard problems to physical models

I am looking for examples of hard problems (in NP or harder) from computer science which can be reduced to models of physical processes. For example, max-2-sat can be reduced to energy minimization ...
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### Calculus of Constructions: compress expression to its smallest form

I'm aware that the Calculus of Constructions is strongly normalizing, meaning every expression has a normal for that cannot be beta,eta-reduced further. So in fact this is the most efficient ...
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### Limited number of variable occurrences in 1-in-3 SAT

Is there a known result on complexity class of 1-in-3-SAT with restricted number of variable occurrences? I've come up with the following parsimonious reduction with Peter Nightingale, but I want ...
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