P versus NP and other resource-bounded computation.

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55 views

When is counting coins NP-complete? [on hold]

having a bit of an issue with this question and deciding which of these situations requires dynamic programming and which are NP-complete: All three (except the last one) ask how much goes to person ...
10
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1answer
122 views

What is this variant of set cover problem known as?

Input is a universe $U$ and a family of subsets of $U$, say, ${\cal F} \subseteq 2^U$. We assume that the subsets in ${\cal F}$ can cover $U$, i.e., $\bigcup_{E\in {\cal F}}E=U$. An incremental ...
7
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3answers
213 views

Are EXPSPACE-complete problems rare?

I am currently trying to find EXPSPACE-complete problems (mainly to find inspiration for a reduction), and I am surprised by the small number of results coming up. So far, I found these, and I have ...
2
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1answer
168 views

Ackermann Function Time Complexity

Are there any known problems that have an Ackermann function time complexity lower bound?
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0answers
86 views

Approximate Hard Problem

The Unique Game is easy to solve for exact solutions, but becomes extremely difficult for approximate solutions, with no exact solution available. It is quite against of our intuition. As it is known, ...
7
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2answers
373 views

Are there problems for which divide-and-conquer / recursion is provably useless?

When we try to construct an algorithm for a new problem, divide-and-conquer (using recursion) is one of the first approaches that we try. But in some cases, this approach seems fruitless as the ...
3
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0answers
108 views

Can we decide Red-blue cut problem in polynomial time?

Given a directed graph whose arcs are coloured red and blue and integers $r$ and $b$, can we decide in polynomial time whether the digraph has a cut with at most $r$ red arcs and at most $b$ blue ...
9
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1answer
130 views

Does ${\bf AC^0PAD} = {\bf PPAD}$?

What happens if we define ${\bf PPAD}$ such that instead of a polytime Turing-machine/polysize circuit, a logspace Turing-machine or an ${\bf AC^0}$ circuit encodes the problem? Recently giving ...
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32 views

NP-hardness of n queens problem [closed]

i have following questions 1> is this statemnt true or false: "If we could solve an NP-hard problem in polynomial time, this would prove P = NP." 2>is n queens problen np hard or just np??? 3> iam ...
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31 views

Oracle reduction to 3SAT [closed]

Show that there is an oracle A and a language L\in NP^A (NP TM with access to oracle A) such that L is not polynomial time reducible to 3SAT even when the machine computing the reduction is allowed ...
17
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3answers
399 views

What is the minimum size of a circuit that computes PARITY?

It is a classic result that every fan-in 2 AND-OR-NOT circuit that computes PARITY from the input variables has size at least $3(n-1)$ and this is sharp. (We define size as the number of AND and OR ...
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0answers
148 views

Determine the complexity class of a language [closed]

Let $L'$ be the language containing all the pairs $(G,v)$ where $G$ is a directed graph and $v$ is a vertex in $G$ such that $G$ contains a cycle (i.e. closed walk) that contains $v$ and the number of ...
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6answers
1k views

Statements that imply $\mathbf{P}\neq \mathbf{NP}$

This is sort of an open-ended question - for which I apologize in advance. Are there examples of statements that (seemingly) don't have anything to do with complexity or Turing machines but the ...
4
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0answers
108 views

Recognition problem of cycle permutation graphs

A cycle permutation graph is a cubic graph composed from two $n$-cycles each labeled $1, 2,..., n$, with additional edges connecting vertex $i$ in the first cycle to vertex $\pi(i)$ in the second, ...
8
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0answers
268 views

Is it possible to solve perfect matching in linear time

As we know matching can be solve in polynomial time. One classical and famous algorithm is designed by Karp and Hopcroft. Is it possible to solve perfect matching problem in linear time for given ...
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0answers
172 views

What if an $\mathsf L$-complete problem has $\mathsf{NC}^1$ circuits? More generally, what evidence is there against $\mathsf{NC}^1=\mathsf{L}$?

Edit: let me reformulate the question in a more specific way (and change the title accordingly). A slightly edited version of the original question follows. Is there a result comparable to the ...
4
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2answers
108 views

Variation on partial Set Cover with penalties

I'm looking at the following problem, which I'm hoping is perhaps a known variant of the Set Cover problem: Input: We're given a universe $U$ and two disjoint subsets $A$ and $B$ such that $A \cup ...
3
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1answer
336 views

Is DAG isomorphism NP-C

Given two directed acyclic graphs $G_1$ and $G_2$, is it NP-Complete to find a one-to-one mapping $f:V(G_1) \rightarrow V(G_2)$ such that $(v_i,v_j) \in D(G_1)$ if and only if $(f(v_i),f(v_j)) \in ...
4
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0answers
152 views

Is this graph polynomial known? Can it be efficiently computed?

Consider a connected simple graph $G$ with $n$ vertices and $m$ edges. View each edge $\ell$ as a transposition $t_{\ell}$ acting on the set of vertices. [To be more explicit, given an edge $\ell$ ...
6
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2answers
322 views

Complexity of finding even cuts for a graph

Given a graph $G=(V,E)$, what is known about the classical computational complexity of finding a non-trivial cut which partitions the vertices into two sets $V_a$ and $V_b$ such that every vertex in ...
4
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0answers
108 views

Problems in dynamic algorithms in computational geometry

The publication of Chiang and Tamassia's paper on dynamic algorithms in computational geometry included several algorithms used in solving dynamic computational geometry problems such as: Dynamic ...
16
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2answers
279 views

Complexity of the coset intersection problem

Given the symmetry group $S_n$ and two subgroups $G, H\leq S_n$, and $\pi\in S_n$, does $G\pi\cap H=\emptyset$ hold? As far as I know, the problem is known as the coset intersection problem. I am ...
9
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1answer
240 views

Berman-Hartmanis Isomorphism for NP$_{\mathbb{R}}$?

Using the real-RAM/BSS model, we have the class NP$_{\mathbb{R}}$, (where a BSS is the Blum-Shub-Smale model of a computer with operations over reals). We have NP$_{\mathbb{R}}$ complete problems. ...
3
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1answer
175 views

Can two strings be matched as disjoint subsequences of a string?

Consider a fixed finite alphabet $A$. I am given as input two strings $S_1$ and $S_2$ on $A$, and a string $S$ on $A$. It is of course possible in PTIME to determine whether $S_1$ is a ...
10
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3answers
303 views

Complexity of permutation related problems

Given a group $G$ of permutations on $[n]=\{1, \cdots, n\}$, and two vectors $u,v\in \Gamma^n$ where $\Gamma$ is a finite alphabet which is not quite relevant here, the question is whether there ...
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1answer
88 views

Finding an exactly weighted st-path in a digraph [closed]

I have a weighted digraph graph $G = (V,E)$ where the weights are positive and negative integers. The graph $G$ is not necessarily acyclic. The question is: given 2 nodes $v_1$ and $v_2$, is there a ...
18
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1answer
459 views

Is P equal to the intersection of all superpolynomial time classes?

Let us call a function $f(n)$ superpolynomial if $\lim_{n\rightarrow\infty} n^c/f(n)=0$ holds for every $c>0$. It is clear that for any language $L\in {\mathsf P}$ it holds that $L\in {\mathsf ...
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0answers
99 views

Complexity Book with Slides

Is there a good book on complexity theory (upper level undergraduate and/or graduate) that comes with slides prepared either by the author or by someone else (but that are specifically tailored to ...
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1answer
52 views

Is scheduling a set of tasks on single machine in P or in NP?

Given a set of tasks $T=\{t_1,\dots,t_n\}$ and execution times between the tasks $e(t_i,t_j)$ can we find a schedule $s$ for $T$ on a single machine with makespan $m_s < d$? Assume that the ...
4
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0answers
193 views

$NP \not\subseteq BPP \implies NP_{\mathbb{C}} \not\subseteq P_{\mathbb{C}}$

Stephen Smale claims in Mathematical Problems for the Next Century that $$NP \not\subseteq BPP \implies NP_{\mathbb{C}} \not\subseteq P_{\mathbb{C}}.$$ Can someone sketch the argument or provide a ...
2
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2answers
120 views

Complexity of smooth non-linear functions

EDIT: A more straightforward way of asking this question is: does evaluating a non-linear function require performing at least one multiplication? ORIGINAL QUESTION: I have an infinitely ...
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0answers
142 views

Lower bounds for the size of nondeterministic circuits

It is known that the minimum size of $U_2$-circuits computing the parity function exactly equals $3(n-1)$. The lower bound proof is based on the gate elimination method. Recently, I noticed that the ...
12
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2answers
248 views

Natural complete problems in higher levels of the $\mathsf{W}$-hierarchy

The $\mathsf{W}$-hierarchy is a hierarchy of complexity classes $\mathsf{W}[t]$ in parameterized complexity, see the Complexity Zoo for definitions. An alternative definition defines $\mathsf{W}[t]$ ...
8
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1answer
172 views

A purely graph-theoretic explanation of the reduction from Unique Label Cover to Max-Cut

I am studying the Unique Games Conjecture and the famous reduction to Max-Cut of Khot et al. From their paper and elsewhere on the internet, most authors use (what to me is) an implicit equivalence ...
8
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1answer
350 views

Problem that is in P only if P!=NP

Are there any problems that are solvable in polynomial time only if P!=NP, and otherwise solvable in (say) $O(2^n)$ time? A simple example would be: If P!=NP, compute a primality test for a random ...
6
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467 views

Consequences of $\oplus \mathbf{P} \subseteq \mathbf{NP}$

I have part of a proof attempt of $\oplus \mathbf{P} \subseteq \mathbf{NP}$. The proof attempt consists of a Karp reduction from the $\oplus \mathbf{P}$-complete problem $\oplus$3-REGULAR VERTEX COVER ...
4
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0answers
210 views

Is there a reason we haven't been able to prove that the existence of natural NPI problems even conditionally under assumption NPI is not empty?

We can write ${\mathsf {NP}}-{\mathsf P}= {\mathsf {NPC}}\cup {\mathsf {NPI}}$ where ${\mathsf {NPC}}$ is the set of ${\mathsf {NP}}$-complete languages (not in ${\mathsf {P}}$ by this partition), ...
22
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2answers
471 views

Is this variation of TQBF still PSPACE-complete?

Deciding if a quantified boolean formula such as $\forall x_1 \exists x_2 \forall x_3\cdots \exists x_n \varphi(x_1, x_2,\ldots , x_n),$ always evaluates to true is a classical PSPACE-complete ...
6
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1answer
213 views
1
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1answer
229 views

#P-complete problems are at least as hard as NP-complete problems

I just read J. Scott Provan, Michael O. Ball: The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected. SIAM J. Comput. 12(4): 777-788 (1983) and one of the first ...
12
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1answer
207 views

Large classes which contain LOGSPACE for which strict inclusions are unknown

The wikipedia page on PSPACE mentions that the inclusion $NL\subset PH$ is not known to be strict (unfortunately without references). Q1: What about $L\subset PH$ and $L\subset P^{\#P}$ - are these ...
8
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1answer
239 views

The exponential function over algebraic numbers

Given an algebraic number $\alpha$, I am interested in finding an approximation of $\Re(e^\alpha)$ up to a given precision, where $\Re()$ refers to the real part of the complex number. Formally, I ...
5
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0answers
119 views

Extensions of Affine Dispersers

A function $f\colon\{0,1\}^n\to\{0,1\}$ is called an affine disperser for dimension $d$, if for every affine subspace $S\subseteq \{0,1\}^n$ of dimension at least $d$, $f$ is not constant on $S$. This ...
4
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1answer
111 views

Inapproximability of $(\alpha, \beta)$ bi-criteria approximation

An $(\alpha, \beta)$ bi-criteria approximation algorithm for $k$-center is defined as an algorithm that returns a solution whose value is $\beta \cdot OPT$ ($OPT$ being the optimal solution for the ...
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1answer
81 views

Is this NP-Hard or does a known optimal polynomial time solution exist? [closed]

Suppose we have 10 items, each of a different cost Items: {1,2,3,4,5,6,7,8,9,10} Cost: {2,5,1,1,5,1,1,3,4,10} and 3 customers {A,B,C}. Each customer has a requirement for a ...
2
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1answer
153 views

Time complexity of d-dimensional convex hull

Consider the convex hull problem in $\Re^d$: Input: a list of $n$ points $S$ in $\Re^d$, Output: the vertices of the convex hull of $S$. What is the best lower bound on the time complexity of ...
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0answers
123 views

proving speedup phenomenon does not apply to any open complexity class separations

Aaronson recently wrote a blog refuting the idea that there could be some "glitch" in the formulation of the P vs NP conjecture[1] which reminds me of this following question. the Blum speedup ...
3
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0answers
122 views

How powerful are weak complexity classes with powerful oracles?

I am interested in complexity classes of the form $A^{B}$, where $A$ and $B$ are complexity classes such that $A \subsetneq \mathsf{P}$ and $\mathsf{NP} \subsetneq B$ are (believed to be) true. ...
4
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3answers
678 views

Factoring as a decision problem

I've seen in multiple places stating that factoring is in BQP and referencing Shor's algorithm, but Shor's algorithm is not solving a decision problem. How can factoring be restated in a decision ...
10
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1answer
256 views

Can affine lambda calculus solve every problem in P?

In Advanced Topics in Types and Programming Languages it is mentioned, in the chapter on sub-structural type systems, that a "carefully crafted" affine lambda calculus with a recursion combinator for ...