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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?
Matthias's user avatar
  • 1,668
41 votes
2 answers
6k views

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 ...
Jeffε's user avatar
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38 votes
3 answers
1k views

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 ...
Andras Farago's user avatar
36 votes
6 answers
6k views

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 ...
Huck Bennett's user avatar
  • 4,908
32 votes
2 answers
2k views

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 ...
Henry Yuen's user avatar
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29 votes
7 answers
2k views

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 ...
Denis's user avatar
  • 8,903
28 votes
1 answer
889 views

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. ...
joro's user avatar
  • 1,955
26 votes
6 answers
2k views

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" ...
Joey Eremondi's user avatar
26 votes
2 answers
6k views

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. ...
William Macrae's user avatar
25 votes
5 answers
923 views

Curious about computer-assisted NP-completeness proofs

In the paper "THE COMPLEXITY OF SATISFIABILITY PROBLEMS" by Thomas J. Schaefer, the author has mentioned that ...
hengxin's user avatar
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24 votes
2 answers
983 views

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 ...
Derrick Stolee's user avatar
23 votes
9 answers
1k views

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 ...
22 votes
5 answers
35k views

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 ...
Mikola's user avatar
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22 votes
1 answer
1k views

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 ...
Simd's user avatar
  • 3,902
21 votes
1 answer
1k views

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 ...
András Salamon's user avatar
20 votes
3 answers
1k views

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 ...
Peter Shor 's user avatar
17 votes
2 answers
3k views

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 ...
Magnus Lie Hetland's user avatar
17 votes
2 answers
755 views

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. ...
Matej Konecny's user avatar
16 votes
2 answers
884 views

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 $B \leq_p A$?
Mal's user avatar
  • 355
16 votes
3 answers
421 views

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 ...
Regularity's user avatar
16 votes
3 answers
5k views

Subset sum vs. Subset product (strong vs. weak NP hardness)

I was hoping that some one might be able to explain to me why exactly the subset product problem is strongly NP-hard while the subset sum problem is weakly NP-hard. Subset Sum: Given $X = \{x_1,...,...
RDN's user avatar
  • 325
16 votes
0 answers
169 views

Is it possible to boost the error probability of a Consensus protocol over dynamic network?

Consider the binary consensus problem in a synchronous setting over dynamic network (thus, there are $n$ nodes, and some of them are connected by edges that may change round to round). Given a ...
Irvan's user avatar
  • 211
15 votes
2 answers
6k views

What does 'gadget' mean in NP-hard reduction?

This question may not be technical. As a non-native speaker and a TA for algorithm class, I always wondered what gadget means in 'clause gadget' or 'variable gadget'. The dictionary says a gadget is a ...
Federico Magallanez's user avatar
15 votes
1 answer
537 views

Sampling a uniformly random satisfying assignment

Problem: Given $\phi : \{0,1\}^n \to \{0,1\}$ represented by a boolean circuit, generate a uniformly random $x \in \{0,1\}^n$ such that $\phi(x)=1$ (or output $\perp$ if no such $x$ exists). Clearly ...
Thomas's user avatar
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15 votes
1 answer
475 views

Validity of exponentiation in a polynomial time reduction

I asked this question 10 days ago on cs.stackexchange here but I didn'y have any answer. In a very famous paper (in the networking community), Wang & Crowcroft present some $\mathsf{NP}$-...
Lamine's user avatar
  • 1,137
15 votes
1 answer
266 views

What is the minimum required depth of reductions for NP-hardness of SAT?

As everyone knows, SAT is complete for $\mathsf{NP}$ w.r.t. polynomial-time many-one reductions. It is still complete w.r.t. $\mathsf{AC^0}$ many-one reductions. My questions is what is the minimum ...
Kaveh's user avatar
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15 votes
1 answer
952 views

Integer relation detection for Subset Sum or NPP?

Is there a way to encode an instance of Subset Sum or the Number Partition Problem so that a (small) solution to an integer relation yields an answer? If not definitely, then in some probabilistic ...
user834's user avatar
  • 2,806
14 votes
5 answers
738 views

Should reductions make us more or less optimistic for the tractability of a problem?

It seems to me that most complexity theorists generally believe the following philosophical rule: If we can't figure out an efficient algorithm for problem $A$, and we can reduce problem $A$ to ...
GMB's user avatar
  • 2,403
14 votes
1 answer
866 views

Is there a list of canonical problems in distributed systems?

Last week, I was reading again Leslie's Lamport's 1982 trasncript of a conference he gave about Solved Problems, Unsolved Problems and Non-Problems in Concurrency. The paper is easily readable, but ...
marcmagransdeabril's user avatar
14 votes
0 answers
474 views

Is there a P-complete language X such that succinct-X is in P?

I came across a paper called "A Note on Succinct Representation of Graphs". It seems that in the discussion section they claim that for any problem $X$ that is $\mathrm{P}$-hard under projections, $\...
Michael Wehar's user avatar
13 votes
3 answers
345 views

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 ...
András Salamon's user avatar
13 votes
2 answers
1k views

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 ...
Thomas Klimpel's user avatar
13 votes
1 answer
944 views

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 ...
Naysh's user avatar
  • 686
13 votes
1 answer
3k views

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 ...
András Salamon's user avatar
13 votes
2 answers
625 views

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 ...
Daniil Musatov's user avatar
13 votes
1 answer
299 views

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 $L$. ...
Andras Farago's user avatar
12 votes
1 answer
1k views

Unique SAT vs Exactly $m$ models

Unique SAT is the well known problem : given a CNF formula $F$, is it true that $F$ has exactly one model ? I am interested in « Exactly $m$-SAT » problem : given a CNF formula $F$ and an integer $m&...
12 votes
1 answer
212 views

Reductions between languages of different densities?

The density of a language $X$ is a function $d_X \colon \mathbb{N} \to \mathbb{N}$ defined as $$d_X(n) = |\{x\in X \mid |x| \le n\}|.$$ Suppose $A$ and $B$ are languages over some finite alphabet, $A$ ...
András Salamon's user avatar
12 votes
1 answer
423 views

Which complexity class does this number theory problem belong to?

'Given $a,b,c\in\Bbb N$, is there $x,y\in\Bbb N$, $ax^2+by=c$' is $\mathsf{NP}$-complete. Which complexity class does 'Given $a,b,c\in\Bbb N$, is there $x,y\in\Bbb N$, $ax^2+by^2=c$' belong to?
user avatar
12 votes
3 answers
1k views

Transitive feedback arc set (TFAS): NP-complete?

Some time ago, I posted a reference request for graph problems where we want to find a 2-partition of the edges where both sets fulfill a property not related to their cardinality. I was trying to ...
G. Bach's user avatar
  • 431
12 votes
0 answers
394 views

Invisible electric fence even if P = NP?

Scott Aaronson has suggested that one argument in favor of $\mathsf{P} \ne \mathsf{NP}$ is that there seems to be an invisible electric fence separating $\mathsf{NP}$-complete problems from problems ...
Timothy Chow's user avatar
  • 7,560
12 votes
3 answers
343 views

Is there a reduction to "door and pressure plate" games that doesn't explode solution length?

This paper gives a proof that in a game with doors and pressure plates, it is PSPACE-hard to determine whether or not the (player's) avatar can reach a given location. This is proven by a reduction ...
user avatar
11 votes
6 answers
2k views

Do many-one reductions and Turing reductions define the same class NPC

I wonder if NPC classes defined by many-one reductions and Turing reductions are equal. Edit: Another question, are Turing reductions only collapsing C and co-C classes for some C or is there a class ...
Ludovic Patey's user avatar
11 votes
4 answers
909 views

Does NP-completeness/hardness have to be constructive?

Is there any $L\in {\bf NP}$ with the following properties: It is known that $L\in {\bf P}$ implies ${\bf P}={\bf NP}$. There is no (known) polynomial time Turing reduction of $SAT$ (or some other ${\...
Andras Farago's user avatar
11 votes
5 answers
917 views

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 ...
yzll's user avatar
  • 428
11 votes
2 answers
2k views

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 ...
Anonymous's user avatar
  • 4,041
11 votes
3 answers
386 views

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 ...
mdenil's user avatar
  • 113
11 votes
3 answers
320 views

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 ...
user47376's user avatar
  • 213
11 votes
2 answers
2k views

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 ...
Ian Gent's user avatar
  • 428
11 votes
1 answer
842 views

Why do NP-complete problems not have similar approximation ratios?

Since 2 NP-complete problems are by definition reducible to each other, so a solution to one of them can be obtained by using a black-box solving the other one, why don't they have similar ...
N27's user avatar
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