Questions tagged [cc.complexity-theory]
P versus NP and other resource-bounded computation.
2,621
questions
1
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0answers
97 views
Proof of SAT is complete for NP via first-order reductions
So I have been reading this: https://people.cs.umass.edu/~immerman/book/ch7.pdf
I do not understand the proof of theorem 7.16, which says that SAT is complete for NP via first-order reductions. My ...
2
votes
1answer
525 views
What is best known space requrement for solving SATISFIABILITY problem in exp time
I searched a lot for finding best space requirement algorithm for SATISFIABILITY problem but I didn't find any thing better than brute force that is in DSPACE(n). is there exists better bound? and ...
2
votes
0answers
42 views
Arithmetic circuits with restrictions on occurrence of pairs of variables
I am curious if the following model was studied or has some obvious lower bounds:
We want to compute a polynomial $P(x_1,x_2, \dots , x_n)$. Suppose we have a graph G on $n$ nodes that we are going ...
2
votes
3answers
114 views
Reference request: Arithmetic circuit complexity
I am completely new to this field. I want to read the prelims and unfortunately, I don't see any book written for a beginner in this area. Can anyone give me some basic starting points/references(text/...
3
votes
2answers
157 views
What is the best approximation and exact algorithm for vertex cover on cubic graphs?
"Best" = best performing in terms of run-time for exact algorithm and approximation ratio for an approximation algorithm.
1
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0answers
633 views
BQNC and Abelian Hidden Subgroup Problem
We know integer factorization is in $BPP^{BQNC}$ from Cleve and Watrous.
Is Abelian Hidden Subgroup Problem also in $BPP^{BQNC}$?
In particular is Discrete Logarithm in $BQNC$ or at least in $BPP^{...
2
votes
0answers
197 views
Primality in $NC$ hierarchy?
AKS primality testing solves whether a given integer is prime in $P$. AKS algorithm is following:
Input: integer n > 1.
Check if $n$ is a perfect power: if $n = a^b$ for integers $a > 1$ and $b &...
0
votes
1answer
100 views
Is Prime Bounded Quadratic Congruence NP-complete?
Bounded Quadratic Congruence:
Instance: Three positive integers $a$, $b$ and $c$.
Question: Is there a positive integer $x<c$ such that $x^{2} \equiv a \, (mod \ b)$?
Bounded Quadratic ...
1
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0answers
57 views
An alternative characterization of some NExp-Time Turing machine with oracles
Let me denote by $\Sigma_i^P$ be a class from i-th level of polynomial time hierarchy (see eg. PH).
I'm interested in the following type of a Turing Machine $\mathcal{M}$:
$\mathcal{M}$ is ...
5
votes
0answers
139 views
How hard is it to generate a set of relatively prime numbers between two given bounds?
Informal Question
How hard is it to generate a set of relatively prime numbers between two given bounds?
Decision Problem
Given $a$, $b$, and $k \in \mathbb{N}$. Does there exist a set $S \...
-1
votes
1answer
53 views
NP-Complete graph problems where a special vertex is given as input?
I am currently working on a graph theory problem where the instance includes a graph and a special vertex in the graph. I am trying to prove the NP-completeness of the problem as well as explore ...
0
votes
0answers
112 views
What definition for $FPT$ algorithm for $KSUM$ gives $W[P]=FPT\implies KSUM$ is $FPT$?
In the definition on $KSUM$ problem we are given $n$ input integers and we have to decide if $K$ of them sum to $0$.
$KSUM$ is $FPT$ if there is a $O(f(K)poly(n))$ algorithm for it. However Downey ...
3
votes
1answer
51 views
Complexity of distributively verifying that the diameter is small
Consider a graph $G=(V,E)$ and an integer parameter $k$.
I'm interested in the round complexity, in the CONGEST model, of checking if the diameter of the graph is "much larger" or "much smaller" than ...
5
votes
2answers
108 views
A stronger multiplexing rule for soft linear logic?
In (intuitionistic) linear logic the usual rules for the storage modality $!$ are promotion, dereliction, contraction, and weakening:
$$\frac{!\Gamma\vdash A}{!\Gamma\vdash !A}(prom) \qquad \frac{\...
4
votes
0answers
77 views
Circuits computing functions of inputs smaller than $n$
The usual circuit complexity concerns circuits where circuit $C_n$ computes function $f_n$. I am interested in circuits such that $C_n$ can compute $f_i$ for all $i \leq n$. I am assuming that the ...
1
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0answers
164 views
Is there any NC-complete problem with respect to logspace reduction?
The question is on the title.
We all know that $\text{NL}$ and $\text{P}$ have such problems. So I wonder the same thing about $\text{NC}$. More interestingly, is there any $k \ge 2$ and any $\text{...
8
votes
1answer
98 views
$BPL$ with polylog random bits is in $L$
Consider a $BPL$ machine (namely, a probabilistic algorithm that uses logspace and polynomially many random bits). It is known (Saks-Zhou) that $BPL \subseteq DSPACE(log^{1.5}(n))$.
My question is ...
4
votes
0answers
75 views
2-hop distributed coloring in the CONGEST model
Consider a graph $G=(V,E)$ and let $d(u,v)$ denote the distance between $u$ and $v$ in $G$.
A 2-hop coloring is a mapping $c:V\to{1,\ldots, C}$ ($C$ is the number of colors) such that $d(u,v)\le 2 \...
2
votes
1answer
128 views
Best $\Pi_k \text{SAT}$ running time?
Define $\Pi_k \text{SAT}$ by 'Given a quantified boolean formula $\varphi = \forall y_1\exists y_2\dots Q_ky_k\mbox{ }\phi(y_1, \dots, y_k)$,
where $\phi(y_1, \dots, y_k)$ is boolean predicate with ...
1
vote
1answer
105 views
Can MONOTONE WSAT be in solved in polynomial time?
In the weighted monotone satisfiability problem (MONOTONE WSAT), the input is an n-variable MONOTONE CNF Boolean formula (when there is no a clause with a negated variable) and an integer k, and the ...
2
votes
0answers
226 views
Career advice needed: Switching from theoretical CS to pure math
I couldn't think of a better forum to ask this question. I am a first year cs graduate student. I couldn't really pursue theoretical cs during my undergrad, but as part of our PhD qualifiers, i had to ...
3
votes
0answers
206 views
Implications of resolving $BPP$ vs $PSPACE$
The relationship between the complexity classes $BPP$, $P$, and $NEXP$ is currently undetermined. We know that $P \neq EXP$ by the time hierarchy theorem, but we don't know if $BPP = P$ (as many ...
3
votes
0answers
132 views
Problems complete for non-deterministic PSPACE
We know that PSPACE = NPSPACE by savitch's theorem , but before that was proved , what problems were known to be complete for NPSPACE ?
6
votes
1answer
110 views
NP completeness of classes of spanning trees
I am teaching a complexity course, and I want to give some examples of similar looking problems such that one is in P, and the other is NP complete.
This made me think of the following problem: does ...
3
votes
1answer
102 views
Relation between different “complexity theories” and complex systems theory
I know of at least 4 fundamentally different uses of the term "complexity theory":
the study of how hard a problem is to solve using some sort of computing machine (I am ignoring divisions within ...
-3
votes
1answer
66 views
Is $P^{SAT} \subseteq NP$? [closed]
Is this known? Trivially, $NP \subseteq P^{SAT}$ as any problem in $NP$ is poly-time reducible to $SAT$. I am not quite sure about the other direction especially because the $NP$ machine would require ...
21
votes
4answers
827 views
Problems that are counter-intuitively solvable in practice?
Recently, I went through the painful fun experience of informally explaining the concept of computational complexity to a young talented self-taught programmer, who never took a formal course in ...
3
votes
0answers
132 views
What are the consequences if $W[i]=W[i-1]$?
$FPT=W[1]$ does not collapse the $W$ hierarchy however falsifies $ETH$ belief. Is there non-trivial consequence if $W[i]=W[i-1]$ and any other consequence at $W[1]$?
1
vote
0answers
111 views
Proof of Sipser-Lautmann Theorem
I have written the following answer as an attempt to prove a variation of Sispser-Lautmann theorem, but it was rejected without any comments. I would appreciate if anyone can find the flaws in this ...
4
votes
1answer
187 views
How hard is it to determine the chromatic number of a unit distance graph?
For example, is it NP-complete to decide whether a unit distance graph is 3-colorable?
1
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0answers
95 views
Best way to represent NP-Hardness result for decision problem with two decision parameters
What is the best way to represent a NP-Hardness result for a decision problem with two decision parameters? Suppose we have a problem $P$ which asks to minimize two parameters $x$ and $y$ and we show ...
1
vote
0answers
76 views
What is the motivation behind W[P]?
I've been researching Parameterized Complexity Theory, and I've been puzzled by the definition of W[P]. Why is the number of non-deterministic steps bounded by $log|n|$? What is the intuition I should ...
9
votes
0answers
180 views
Regular expressions of prefixes/suffixes
It is well-known that star-free regular expressions, which are defined by the grammar
$r::= a \mid r \cdot r \mid r \cup r \mid \neg r \mid \varepsilon \mid \emptyset$
where $a$ belongs to a finite ...
1
vote
0answers
37 views
Monotone complexity of PLP
Blum and Nisan show Positive Linear Programming could be done in $NC$ if we only ask for approximate solutions. This paper https://pdfs.semanticscholar.org/8dc7/5aa9d72864022d848c3e599c5f24d9d527e7....
3
votes
0answers
121 views
Is 3-coloring bounded degree graphs subexponential: $O(\exp{(\sqrt{n}\log^2{n})})$? [closed]
We got an argument that 3-coloring bounded degree graphs is subexponential
with complexity $O(\exp{(\sqrt{n}\log^2{n})})$.
The treewidth of a planar graphs on $n$ vertices is $O(\sqrt{n})$
and 3-...
3
votes
1answer
127 views
On complexity of linear programming with quadratic equality/inequality constraints?
Feasibility test in Linear programming is in $P$ and in convex quadratic programming is in $P$. What is the maximum $k$ such that $n$-variable $m=poly(n)$ linear constraint feasibility test with $k$ ...
6
votes
1answer
179 views
0-partition number vs partition number
Denote by $\chi_0(f)$ the minimum number of 0 monochromatic rectangles needed to cover the 0-inputs of $f$, and by $\chi(f)$ the minimum number of monochromatic rectangles needed to cover the all ...
4
votes
0answers
141 views
Formalization of proofs and CC paradox? - Part II
This was the second part of my previous question. It is very similar, and probably it has a similar answer (as Emil said in a comment), but I thought it was worth to separate it and ask it as a new ...
5
votes
1answer
131 views
Path in a graph with durations [closed]
I have the following problem:
given
a directed graph $G=(V,E,d)$, where $d:V\to\mathcal{I}(\mathbb{Q}_0^+\cup\{+\infty\})$ (here $\mathbb{Q}_0^+$ denotes the set of non-negative rationals and $\...
3
votes
0answers
80 views
PTIME or NP-Hardness of stochastic objective function
I will begin by linking a previous post where I asked a general question for a stochastic setting which I describe below. It turns out that my "proof" for a restricted case had a mistake and there is ...
3
votes
0answers
89 views
Showing hardness of maximizing stochastic objective function over graph
Consider a graph $G = (V, E)$ with $n$ vertices and $m$ edges. Each vertex $v_i$ can take positive value $a_i$ with probability $p_i$ and value $0$ with probability $1-p_i$.
The challenge is to ...
3
votes
1answer
113 views
Results/concepts that also proved useful outside of their “home areas”
There are some results/concepts in TCS which are used in areas other than the "home area" where they emerged. For example, NP-completeness has complexity theory as its home area, but it is also used ...
2
votes
1answer
97 views
Clarification regarding Bounded Quadratic Congruence Problem
Given: 3 positive integers $a, b, L$.
Problem: Is there a positive integer $x < L$ such that $x^2 \equiv \ a (mod\ b)$?
The above problem is NP Complete (as mentioned in G&J) even if we ...
5
votes
0answers
191 views
How hard is APPROXIMATE-#SAT?
It is well known that the problem of counting the satisfying assignments of SAT, namely the problem #SAT, is #P-complete.
It is also suspected (somewhat less widely) that even deciding SAT should ...
6
votes
1answer
148 views
Is End-of-Monotone-Line PPAD-complete?
Consider the following problem from TFNP that is somewhere between End-of-the-Line (PPAD) and End-of-Metered-Line (CLS).
Input (via polynomial circuit): Graph on vertex set $0$ to $2^n-1$ such that ...
1
vote
0answers
95 views
Fixed dimension Integer programming minus LLL in fixed parameter $NC$?
If you remove LLL part then is remaining part of
a. Lenstra algorithm
b. Barvinok algorithm
in $O(f(n)(\log(mL))^c)$ time on $O(g(n)(mL)^c)$ processors with fixed $c>0$ in fixed $n$...
6
votes
2answers
587 views
a polynomial representation of boolean functions
I came up with this linear transformation to map boolean functions to polynomials and it seems to have some nice properties. I was wondering if there is any reference describing this (and/or similar) ...
1
vote
0answers
90 views
How hard is gapped satisfiability?
It is well known that general satisfiability (of polynomially many clauses) is NP-hard, and in fact, it is conjectured that an algorithm deciding instances of SAT takes time nearly $2^n$ on $n$ ...
2
votes
1answer
120 views
Is there any time efficient way of achieving the result of FKS hashing lemma?
FKS hashing lemma states.
Given a set of $n-$bit numbers $\{x_1,x_2,\dots,x_k\}$ there exist a
prime $p$ of $O(\log n + \log k)$-bit such that $x_i$ mod $p \neq $
$x_j$ mod $p$ if $x_i \neq x_j$...
4
votes
1answer
283 views
If a root||nonce Proof-of-Work certificate is prime, can it be used in any other interesting proofs?
Because Bitcoin and many other cryptocurrency mining certificates are "rare" in that their respective hash is less than a very small number, can we leverage their rarity in probabilistic proofs of ...
