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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 ...

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How can we prove hardness for small complexity classes? [closed]

If we want to show that a problem is hard for a small complexity class, like NL, L, NC... (any complexity class contained in P), how can we do this? We can't use a logspace reduction to another ...
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30 views

How should a reduction to the Cardinality Constrained Quadratic Knapsack Problem work?

in Polyhedral Study of the Cardinality Constrained Knapsack Problem the authors prove that the Cardinality Constrained Knapsack Problem is NP-Hard by reducing PARTITION to it. Besides, it's easy to ...
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48 views

Reduction of irregular graphs, to regular graphs, while preserving hamiltonicity

I am wondering if this is a topic that has had research done... If I could reduce irregular graphs to regular graphs (including replacing redundant node clusters with dummy nodes), while ensuring ...
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29 views

Reduce maximum capacity simple cycle to maximum capacity minimum cost cycle

Consider the following problems $(A)$: Let $G=(V,E)$ a non-directed graph, a capacity function $u:V\to\mathbb Q_+\cup\{0\}$ and $m \in \mathbb Q_+$. Decide if there exists a simple cycle such that ...
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98 views

Reduction between functions that preserves time and space-complexity

Under which reduction(s) is the class $\mathsf{FTISP}(t(n), s(n))$ closed? Let $\mathsf{FTISP}(t(n), s(n))$ the class of functions from $\{0,1\}^*$ to itself that are computable by a Turing machine ...
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2answers
706 views

Isn't it trivial to represent any classical Physics problem in a Spin-Glass format which is NP-Complete?

In the late 80's there were several efforts to use Spin-Glass models to formulate other computational problems such as: Protein Folding and Neural Networks. Wouldn't it be straight forward to reduce ...
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112 views

Reduce $m$-clause 3SAT to PLANAR-3SAT in $O(m^{2-\varepsilon})$

The classic reduction from 3SAT to PLANAR-3SAT requires a removal of $O(m^2)$ crossings from a rectilinear representation of 3SAT with $m$ clauses. However, the crossing number inequality suggests ...
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127 views

On sparse complete sets and P vs L

Mahaney's Theorem tells us that if there is a sparse $NP$-complete set under polynomial-time many-one reductions, then $P = NP$. (See "Sparse complete sets for NP: Solution of a conjecture of Berman ...
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1answer
159 views

Verifying that a reduction is correct

Alice has a function $f: \{0,1\}^* \to \{0,1\}^*$ which can be computed in polynomial time. She claims that $x \in \mathrm{SAT} \iff f(x) \in \mathrm{CLIQUE}$. Alice sends the circuit computing $f$ on ...
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40 views

Digraph game reduction problem

Let A and B be 2 players and D=(V,E) a digraph and v0 an fixed vertex. A chose v0 v1 B chose v1 v2 and so on.The player who can't chose anymore loses.(they are not allowed to chose an edge adjacent ...
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87 views

Random self reducibility and NP

I was reading the Wikipedia page https://en.m.wikipedia.org/wiki/Random_self-reducibility and it states: "If an NP-complete problem is non-adaptively random self-reducible the polynomial hierarchy ...
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61 views

On reduction between two classes?

https://link.springer.com/article/10.1007/s00153-013-0351-x gives seven reductions $m,c,d,p,btt(1),\ell,tt$. What does norm $1$ mean in $btt(1)$? Is there illustrative examples that help understand ...
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1answer
102 views

What is conjunctive truth table reduction?

What are conjunctive/disjunctive truth table reductions and how do they compare with other reductions?
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60 views

How does an NP-complete continuous optimization problem map to Boolean SAT?

I looking at this paper where the authors mention that the following problem: "Given a neural network and a set of training examples, does there exist a set of edge weights for the network so that ...
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1answer
49 views

Certainty of mutual confirmation over faulty channels?

This is a very theoretical question, although I am sure the problem pops up in lots of IT and automation applications. Still, I prefer to formulate it in an action-movie scenario (a bit of the ...
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1answer
72 views

Proving properties about a compilation from one problem into another

Say I have two problems A and B. A is the shortest path problem with positive weights B is the shortest path problem (with potentially negative weights) I would like to show: There is no mapping m,...
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3answers
183 views

Calculus of Constructions: Compress expression to it's 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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72 views

Prove that finding set of $k$ vertices $S$, such that $G{\setminus}S$ is claw-free is NP-Complete

The claw in a graph $G(V,E)$ consists of a vertex $v\in V$, and it's three neighbours - $\{x_1,x_2,x_3\}\in V\setminus \{v\}$, if $\{x_1,x_2,x_3\}$ form an independent set in $G$. The problem asks us ...
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144 views

Reduction from k-Almost Independent Set to Independent Set

The problem of $k$-Almost Independent Set is to decide whether or not $(G,m)$ where $G$ is a graph and $m \in \mathbb{N}$ has a subset of $m$ vertices that induces a subgraph with at most $k$ edges. I ...
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55 views

Construct sets S and T such that S is polynomial-time Turing-reducible to T, but not many-one reducible to T

I am currently trying to self-study Descriptive Complexity from Neil Immermans book and spent hours thinking about exercise 2.23, stated in the title. The hint provided there is to construct S and T ...
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46 views

About reduction relation between $HP$ and $\mathcal{E}\mbox{*}$ [closed]

I'm studying Theory Of Computation and have some questions in the beginning: About reduction relation between $HP$ and $\mathcal{E}\mbox{*}$ $HP =$ {$<M,w>$ $|$ $M$ is a $TM$ and it halts on ...
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1answer
97 views

hardness of constant approximation of largest matching set

We say that $H$ is a matching graph if it contains $2n$ vertices and only $n$ vertex-disjoint edges, i.e. $H$ only contains those $n$ edges and no more. Given a graph $G=(V,E)$ a subset $U\subseteq V$...
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1answer
130 views

A conceptual question regarding hardness proofs by reduction [closed]

If we restrict the input domain of a known NP-hard problem P so that this restricted domain is equal to the input domain of another problem S, then show that we can reduce a solution to P given input ...
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385 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. ...
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310 views

A list of XP-hard problems

The class $XP$ is the class of problems parameterized by $k$ that can be solved in time $n^{f(k)}$ for some function $f$ (each $k$ may require a different algorithm). In their book on parameterized ...
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83 views

Many-one reducibility equivalent for more general computational problems?

Many-one reducibility, denoted by $\leq_m$, is a binary relation between 2 decision problems which is defined as follows: $L' \leq_m L$ iff there exists a computable function $f$ (called a reduction) ...
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244 views

What are the best known reductions from SAT to CNF-SAT?

Problems Let SAT denote the following problem: Given a boolean formula, does there exist a satisfying assignment? Let CNF-SAT denote the following problem: Given a ...
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1answer
166 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, $A$ ...
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2answers
478 views

Is iszero of the untyped lambda calculus sound and complete? [closed]

I am using the following definitions in the notation of Haskell. In case it matters, I would like to use only the $\alpha,\beta,\eta$ reductions rather than the Haskell evaluation rules. ...
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234 views

UnambiguousSAT reductions

Let $\Pi$ be an $\mathsf{NP}$-complete problem. It is standard that $3SAT$ and $\Pi$ are reducible from each other. Let UnambiguousSAT, or USAT for short, denote the promise problem which is 3SAT but ...
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142 views

Is graph automorphism Karp-reducible to graph isomorphism under hidden subgroup representation?

The classical representations of the graph automorphism problem is Karp-reducible to the classical representation of the graph isomorphism problem. The sketch of proof for this can be written as ...
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89 views

Reduction Unbounded Knapsack < k-Exact Unbounded Knapsack

I'd like to have an explicit reduction among these two problems: (1) Unbounded Knapsack: Given a set of $n$ item types with weight $w_i$ and quality $q_i$ solve: $$maximize \sum_{i=1}^n q_ix_i $$ ...
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808 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 $B \leq_p A$?
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60 views

one-to-k assignment problem with jobs capacities

update Mikhail Rudoy make me realize that I partially wrongly reduced the problem. I am interested in the following variation of the assignment problem: The problem instance has a number of agents ...
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1answer
354 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 ...
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404 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 $\mathsf{NP}$-...
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123 views

Graph Isomorphism Algorithm of Vertex Transistive Graphs and other

What are the best known Graph-Isomorphism algorithms for below graph classes- 1.vertex-transitive, 2. edge-transitive, 3.arc-transitive (or symmetric) 4.distance-transitive. Are they GI Complete?...
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180 views

Complexity of permanent modulo prime

Given $M\in\Bbb Z^{n\times n}$ with $O(n)$ bit entries (could be all in $\{0,1\}$), $p$ a prime of $O(n^\alpha)$ bits for some $\alpha\in(0,1]$ and a $c,d\in\Bbb Z$ with $0\leq c<d<p$, is 'Is $\...
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Can coNLogTime verification be used instead of PTime verification when characterising NP?

I recently read a paper that presented a proof calculus where the verification of whether a given proof is valid was NL-complete. The authors apparently decided that the checking procedure was not ...
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128 views

Could you explain to me the reduction? [closed]

I am looking at the following solved exercise: I haven't really understood at the reduction the part that we construct for each number $a_i$ a package of measurement $(\frac{4}{A}a_i, 5,3)$. Why do ...
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195 views

Reduction from independent set in hypergraphs to independent set in graphs

Let me introduce my notations. IS-H : Input : an hypergraph $G=(V,H)$, an integer $k$ Question : is there a (weak) independent set of size $k$, i.e. a set $S \subseteq V$ such that $|S| \ge k$ and ...
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87 views

Maximum weight triangles in dense graphs

There are multiple results (Vassilevska and Williams STOC09, for instance) on computing efficiently minimal-weight triangles (or more generally patterns) in node-weighted graphs. Several of these ...
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843 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 ...
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27 views

Satisfying two constraint with an oracle for satisfying one

Given an oracle to solve the knapsack feasibility problem: $$a^Tx=b, x \in \mathbb{N}^n$$ How can one solve in polynominal time the problem of satisfying two constraints at the same time? I already ...
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1answer
453 views

On reducing the hardness of CNF-SAT to k-Clique

CNF-SAT refers to the following problem: Given a boolean formula $\phi$ in conjunctive normal form, does there exist an assignment to the variables that satisfies $\phi$. There are several ...
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1answer
196 views

Verifying a subtlety of Karp's original proof that SAT has a polynomial time reduction to 3SAT

Stated briefly, my question is: is Karp's original proof reducing SAT to 3SAT unnecessarily elaborate? The details are as follows. In his 1972 paper Reducibility Among Combinatorial Problems, Karp ...
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148 views

Analogues of the Berman Hartmanis conjecture and the Creativity Hypothesis

The Berman Hartmanis conjecture which formally states that there is an isomorphism for two $NP$ complete languages $L_{1}$, and $L_{2}$, the isomorphism is a bijective function $f()$ such that $f()$ ...
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110 views

L-reduction From Matrix-Tiling To Minimal Dominating Set in Unit Disk Graph

Recently I read this paper which was published in FOCS2007. In section 4, just before Theorem 4.2, the author mentioned that the gadget construction of Minimum Dominating Set in UDG is similar to ...
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1answer
318 views

Is there a FNP problem that's NP-hard but not FNP-hard?

For the reductions, choose a class C such that [it's clear what FC means] and FC is not known to be able to solve the satisfaction search problem, and assume that FC indeed can't solve that search ...
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Is anything known about Sokoban with only 1 box?

This is intended to be a simpler version of my earlier question here. In this post, 1-Sokoban-search is Sokoban with only 1 box, 1-Sokoban-decision is the corresponding decision problem, and 1-Sokoban ...