Questions related to the (computational) complexity of solving problems

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9
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1answer
113 views

Can an NP-hard problem be polynomial on average?

I'm wondering if there are any $NP$-hard problems which are ``polynomial" in the average case. I think there are two ways to interpret this? If $P \neq NP$, can there be an algorithm solving an ...
3
votes
1answer
59 views

A Reduction from XORSAT to 2-SAT

Does anyone know of a non-trivial reduction from XORSAT to 2-sat since they are both in P? (By non-trivial I mean one that does not just solve the instance of XORSAT and map it to a fixed instance of ...
1
vote
1answer
30 views

Is random access allowed in the Bit Complexity model, or is it just expensive?

In the RAM model, you're allowed to do unbounded indirect access (pointers can be arbitrarily large and still fit in a single machine word). In the Bit Complexity model (no wiki article, sorry), ...
5
votes
1answer
78 views

How hard is this constrained $n$-rooks problem?

I asked this over on math.stackexchange.com, then I found out about this forum. Suppose you have an $(n\times n)$-chessboard, together with a constraining function $C : n \times n \to 2$ where ...
3
votes
1answer
60 views

What happens to quantum algorithms such as BB84 if P=NP

Under the hypothesis that P=NP, many cryptographic protocols are no longer secure (i.e. attacks are feasible). The BB84 algorithm (http://en.wikipedia.org/wiki/BB84) is based on the idea that by ...
1
vote
1answer
29 views

Why is there only a polynomial number of provers in multi-prover interactive protocols?

The paper On The Power of Multi-prover Interactive Protocols by Fortnow, Rompel, Sipser states the following: There are provers $P_1, P_2, \dots, P_k$ in a multi-prover interactive proof system such ...
1
vote
1answer
38 views

A detail on variant of Mahaney's theorem about reductions of sparse languages vs P/NP

Wikipedia states on sparse languages that There is a Turing reduction (as opposed to the Karp reduction from Mahaney's theorem) from a NP-complete language to a sparse language iff NP $\subseteq$ ...
5
votes
1answer
61 views

Easy infinite subclass of cubic graphs for Hamiltonian cycle problem

I know that Hamiltonian cycle problem is $NP$-complete for 2-connected planar bipartite cubic graphs. I'm interested in non-trivial infinite subclass of cubic graphs where the Hamiltonian cycle ...
2
votes
2answers
132 views

Does this mean $P = NP$

I am not a formally trained guy on Complexity theory, but due to interest I am learning it. Based on different feedbacks, I have started my journey with Micheal Sipser's "Theory of Computation" (2013 ...
4
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1answer
35 views

First Order interpretation of arbitrary structures as a graph

I am currently trying to get some intuition on the concept of First Order reductions, and have come across this exercise question by Immerman, dubbed "Everything is a Graph". Given some arbitrary ...
7
votes
2answers
155 views

Generating a set of minimal-length strings that, together, invoke every production of a context free language

Problem (tl;dr) Given a context free grammar, $G$, find a set of strings that take $G$ through every production it has at least once. How and how fast can it be done? Background I'm working on a ...
2
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0answers
53 views

Are there online available solved homeworks on complexity theory?

I have never seen this subject before but certain things I read got me curious. I found various online lecture notes on complexity theory and theoretical CS but almost no where do I see solved ...
2
votes
1answer
69 views

Is FNP = FEXPTIME if and only if NP = EXPTIME?

It is very well known that if the classes $\sf FP$ and $\sf FNP$ are equal, then also the classes $\sf P$ and $\sf NP$ are equal (see e.g. FNP on Wikipedia). Is it also true that if $\sf ...
3
votes
1answer
61 views

Can quantified renamable Horn formulas be identified using the same procedure as unquantified formulas?

Definition: A renamable Horn formula is a Boolean formula that can be transformed into a Horn formula by flipping the polarity of every instance of one of more of its variables. Example: $\qquad ...
4
votes
1answer
58 views

Why is $BPP$ closed under complement?

Why is $\text{BPP}=\text{co-BPP}$? I tried to find a proof online but couldn't. Can anyone please provide a quick explanation (if it's trivial and I just can't see it) or a link to a proof?
-3
votes
1answer
65 views

exponential lower bound on boolean formula conjunctions, what complexity class? [closed]

this new paper A Lower Bound for Boolean Satisfiability on Turing Machines by Hsieh asserts an exponential lower bound for a TM time complexity on a problem of finding whether a solution exists to a ...
2
votes
2answers
49 views

Computational complexity theory books [closed]

I recently attended a lecture on an introduction to computation complexity and I am looking to find out more, I haven't studied computer science or discrete mathmateics at university and I was ...
6
votes
1answer
83 views

Computing the number of bits of a large power of integer

Given two integers $x$ and $n$ in binary representation, what is the complexity of computing the bit-size of $x^n$? One way to do so is to compute $1+\lfloor \log_2(x^n)\rfloor=1+\lfloor ...
1
vote
0answers
64 views

Is there an algorithm to compute the shortest Hamiltonian path in an undirected graph from one point to another in polynomial time?

Assumptions: given a graph with N nodes, and two specific nodes A and B the graph is undirected and no edge has a negative cost there exists at least one Hamiltonian path with A and B as an end ...
3
votes
2answers
103 views

Why does the NP completeness of the Hartree-Fock method not lead to difficulty in practical calculation?

I read Computational Complexity of interacting electrons and fundamental limitations of Density Functional Theory. In appendix, it is claimed that In the following, we show that approximating ...
-3
votes
1answer
41 views

Why is Mixed Quantified Horn SAT in PSPACE?

I want to prove that Mixed Quantified Horn SAT is a PSPACE-complete problem. I have proved that it is PSPACE-hard. How can I prove that it is in PSPACE? My study: To prove QSAT to be in PSPACE: ...
14
votes
1answer
186 views

NP-complete problems not “obviously” in NP

It occurred to many that in all the $\textbf{NP}$-completeness proofs I've read (that I can remember), it's always trivial to show that a problem is in $\textbf{NP}$, and showing that it is ...
5
votes
0answers
106 views

Longest Repeated (Scattered) Subsequence in a String

Informal Problem Statement: Given a string, e.g. $ACCABBAB$, we want to colour some letters red and some letters blue (and some not at all), such that reading only the red letters from left to right ...
0
votes
1answer
27 views

Intersection and partial quantity decidability [closed]

I'm still insecure in the section decidability (no proof needed, I want to divine it): X is decidable and Y is undecidable. Is the intersection of X and Y decidable or undecidable? X is decidable ...
8
votes
0answers
77 views

Would $\sf RP = NP$ imply $\sf NP = coNP$?

If $\sf RP = NP$ then the hierarchy collapses to its second level (by the Karp-Lipton theorem). But what about $\sf NP$ and $\sf coNP$? I tried to prove that $\sf BPP$ is contained in $\sf NP$ (the ...
-1
votes
1answer
36 views

Is finding negative cycle vertices NP complete?

I was trying to find all the negative cycle vertices using the Bellman–Ford algorithm using this paper solution 7.1(b) in $O(V)$ by tracing back the predecessor subgraph.It is also stated in ...
8
votes
1answer
71 views

What do complexity classes look like, if we use Turing reductions?

For reasoning about things like NP-completeness, we typically use many-one reductions (i.e., Karp reductions). This leads to pictures like this: (under standard conjectures). I'm sure we're all ...
0
votes
1answer
56 views

Explanation of the complexity of a loop [duplicate]

I am confused on the following: In the following trivial code: ...
3
votes
2answers
115 views

Need Help Reducing Subset Sum to Show a Problem is NP-Complete

I want to show that the following problem is NP-Complete: For a set of vectors $v_1,\ldots,v_n \in \mathbb{N}^d$ and an integer $k$, does there exist a subset $S \subseteq \{v_1,\ldots,v_n\}$, such ...
0
votes
1answer
72 views

Is the Berman-Hartmanis Conjecture Solved?

The Berman-Hartmanis conjecture more or less states that if one-way functions exist, there are some problems in $NP$ which cannot be polynomially reduced to $NP$-complete (cf. Ker-I Ko, A Note on ...
0
votes
0answers
25 views

A polynomial reduction from HAMPATH to LONG-PATH [duplicate]

$\text{HAMPATH} = \{(G=(V,E),s',t')| \text{ G has a Hamilton path from s' to t' } \}$ $\text{LONG-PATH} = \{(G,s,t,k) | \text{G has a simple path p from s to t, length(p) $\geq$ k} \}$ I'm trying ...
0
votes
2answers
112 views

If P != NP, then 3-SAT is not in P

I hope I'm in the right section: I know that if P = NP, then 3-SAT can be solved in P (Cook), but is the opposite valid, too? If P != NP, then 3-SAT is not in P? Thanks!
4
votes
1answer
71 views

Unary in $P$, binary not in $P$

I would like to know if there is a known decision problem with the following characteristics: Represented in unary, the problem is decidable in polynomial time. Represented in binary, the problem is ...
-1
votes
1answer
41 views

Is P^SAT subset of sum of NP and co-NP

I have a following problem: Let $P^{SAT}$ be a class of problems decidable by a deterministic polynomial Turing Machines with SAT oracle. (only one question to oracle). Assume that: $co-NP \neq NP ...
4
votes
1answer
104 views

How to challenge Hutter's algorithm?

For a given sufficiently strong formal axiomatic system $\mathsf{F}$ (like $\mathsf{PA}$ or $\mathsf{ZFC}$) and any given function $p^*(x)$ that can be specified within the formal system $\mathsf{F}$, ...
3
votes
1answer
22 views

Lower complexity bounds without mutation

Lower complexity bounds tend to be a very hard problem in general. Despite this, I was wondering if there are any theoretical results that relate lower complexity bounds for some class of problems in ...
2
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0answers
28 views

AC0 and first order logic equivalence

The page on descriptive complexity theory in Wikipedia states the following: "First-order logic defines the class FO, corresponding to AC0, the languages recognized by polynomial-size circuits of ...
3
votes
1answer
40 views

Could someone explain Fagin's theorem on the equivalence between NP and existential quantifier

This quote on descriptive complexity theory is given in Wikipedia: "The first main result of descriptive complexity was Fagin's theorem, shown by Ronald Fagin in 1974. It established that NP is ...
3
votes
1answer
46 views

Why do we assume that a nondeterministic Turing machine decides a language in NP in $n^k-3$ in Sipser's proof

At page 277 of Sipser's Introduction to the Theory of Computation, a proof of the NP-completeness of SAT is given. The following comment is made on the trace of some machine $N$ which can decide a ...
8
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1answer
87 views

Probability Distributions and Computational Complexity

This question is about the intersection of probability theory and computational complexity. One key observation is that some distributions are easier to generate than others. For example, the problem ...
1
vote
1answer
41 views

Can weighted problem have polynomial complexity if non-weighted problem is NP-complete: hitting set

I am confronted with task to find polynomial time complexity solution for weighted hitting set problem. I have found that usual hitting set problem is NP-complete and therefore the task seems to be ...
4
votes
3answers
79 views

number encoding effect on complexity

I started reading the book "Data Structures and Network Algorithms" by Robert Tarjan, which is a classic (but a bit outdated - 1983) and I am a bit perplexed by the paragraph in the first chapter, ...
3
votes
1answer
49 views

Reduction from PARTITION to MAX-CUT

I am trying to prove the NP-Hardness of the MAX-CUT problem. Other sources seem to reduce from the NAE-3SAT problem, however I have been trying to reduce from PARTITION because PARTITION and MAX-CUT ...
1
vote
2answers
55 views

Question about the definition of complexity class oracles

If $B$ is a complexity class, then the class $P^B$ (for example) is defined as the set of problems that can be run in polynomial time, given an oracle to every problem in $B$. That's what they told ...
2
votes
1answer
39 views

NP-complete problems and sub-expenential sized circuits

If one were to show that an NP-complete problem had $2^{n^{O(1)/\log{\log{n}}}}$ circuit complexity, what would the consequences of this be?
0
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1answer
74 views

Time complexity of Dynamic Array via repeated doubling

When we implement dynamic array via repeated doubling (if the current array is full) we simply create a new array that is double the current array size and copy the previous elements and then add the ...
-2
votes
1answer
56 views

NP hard: Mixed Q Horn SAT

Prove that Mixed Quantified Horn SAT problem is NP hard by reducing the Q3SAT problem to it. Q3SAT: 3SAT with possibly universally and existentially quantified variables. Mixed Quantified Horn ...
1
vote
1answer
55 views

To show that a graph-problem is in $L$ or $NL$

Consider the following problem: $$A=\left\{ (G(V,E),s,t)\mid\text{conditions 1, 2, 3 and 4 hold} \right\}$$ $G$ is a directed graph. $s,t\in V$. There is a simple path from $s$ to $t$ (a simple ...
3
votes
0answers
52 views

Graph canonization is not a decision problem. But what type of problem is it?

I noticed that the most convenient way to deal with quotient structures (like the rational numbers or other equivalence classes) within ZFC is to select a unique representant from each equivalence ...
0
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0answers
25 views

Is the following statement on NP reductions correct [duplicate]

Since NP is closed under polynomial-time reduction, any problem $A \in NP$ can be reduced to a problem $B \in NP$.