Questions related to the (computational) complexity of solving problems

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Constructing solution to 3SAT formulas using oracle queries [duplicate]

I'm interested in 3SAT and querying an oracle. Suppose we had an oracle that can decide, on an input boolean formula $\phi$, whether there exists any assignment to the variables that makes the formula ...
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1answer
57 views

What can I deduce if an NP-complete problem is reducible to its complement?

Let's say I have a decision problem $D$ and its complement $D'$. I know D is poly-time reducible to $D'$ (its complement). Furthermore, I know $D$ is NP-complete. What is the strongest statement I ...
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210 views

Does NP-completeness require to find the solution?

In the paper "Computing Equilibria:A Computational Complexity Perspective" by Tim Roughgarden, they consider the problem: Problem 2.1 (Clique). Given a graph $G = (V, E)$ and an integer $k$: if ...
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74 views

Proof of Karp-Lipton theorem

I am trying to understand the proof of the Karp-Lipton theorem as stated in the book "Computational Complexity: A modern approach" (2009). In particular, this book states the following: ...
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0answers
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Difference between decision problem and solution [duplicate]

I understand the class NP as the decision problems (yes/no) that, when given a candidate solution, if that candidate is an yes instance, we have efficient algorithms to verify that. Consider problem ...
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Proof sketch that NP total search problems cannot be NP-complete [duplicate]

From a blog post, about proving that NP total search problems cannot be NP-complete unless NP=co-NP. It's possible to write a convincing proof sketch as follows. Consider what would it would mean ...
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From hypercontractivity norm bounds to small set expansion property

Consider these two theorems on this theme, Lemma 8 on page 10, http://www.boazbarak.org/sos/files/lec2d.pdf Lemma B.1 on page 63 of http://arxiv.org/pdf/1205.4484v3.pdf Aren't these two theorems ...
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Polynomial verifier in computation theory (Schedule Problem)

P is an integer and M a matrix such as M ∈ {0,1}^k×m, M(i,j) = 1 signifies that a student i is inscribed in the activity j question: is there a Schedule at most P period that allows us to place ...
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Proof of Circuit-Sat to Nand-Sat polynomial time many–one reducibility

Given a gate called Nand with the following truth table: A | B | A Nand B ------------------ 0 | 0 | 1 0 | 1 | 1 1 | 0 | 1 1 | 1 | 0 We can ...
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34 views

Poly-time reduction: D and D Comp

Looking at the IndependentSet problem and it's complement. I want to show that IS is poly-time reducible to it's complement, however I am struggling on coming up with the reduction function. I will ...
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How do I prove that 1 function is an upper bound of the other? [duplicate]

If for every $n > 0$ and some $b > 1$, $T(n) \le h(n)$ and $h(n) = O(h(n/b))$ then how can I prove that $T(n) = O(h(n))$, I understand that $T$ is bounded by $h$, so $h$ must be its upper bound, ...
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46 views

Optimization in multivalued logic. Optimal strings with given patterns

This question comes from an application in multivalued logic. Suppose, we are given an alphabet of three letters $A, B, C$ and a set of indices $1,2,3,4,5$. Consider items formed by subscripting the ...
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177 views

Travelling with the most efficient path

A friend of mine actually asked me a very interesting computer science related question, and I have been stuck on it for a long time. The problem is: you have to travel $1000$ km. The only gas ...
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4answers
179 views

Can finding a witness be NP-hard even if we already know there is one?

The common examples of NP-hard problems (clique, 3-SAT, vertex cover, etc.) are of the type where we don't know whether the answer is "yes" or "no" beforehand. Suppose that we have a problem in which ...
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25 views

Complexity if special case of SAT

I have the following problem: How to show that the special case of SAT, in which each clause has either exactly two literals or at most one negative literal, is NP-complete?
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What is a sufficiently complex system?

I have been reading about the AI approaches, and I came across the AI emergent approach that has the following definition: That is, the appearance of an entity with a sense of its own identity and ...
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61 views

Are there name and literature for this SAT-like problem?

Given $f : \{0,1\}^* \to \{0,1\}$ and $n \in \mathbb{N}$, we define $\textsf{Prob}(f,n)$ as the following problem: Find an $x \in \{0,1\}^n$ such that $f(x) = 1$. A machine solving ...
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How can we prove this using proof by induction? CFG [duplicate]

Let G be the following grammar: S → T⊣ T → TaTb | TbTa | ε Show that L(G) = {w⊣| w contains equal numbers of a’s and b’s} using proof by induction on the length of w. What can be assumed in this ...
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1answer
26 views

Complexity of recognizing whether two $\omega$-regular expressions represent the same language

If the complexity of recognizing whether two regular expressions represent different languages is EXPSPACE-complete, then what can be said for the complexity of recognizing whether two ...
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2answers
78 views

Why is Oracle Turing Machine important?

As you know, an Oracle Turing Machine (OTM) is a "black box" which somehow can tell us whether a given Turing machine with a given input eventually halts. By Church's Thesis it is impossible to design ...
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1answer
43 views

Does there exist a problem that is hard to do in parallel? [closed]

I am looking for a workload which is hard to paralellise/distribute between multiple machines. For example, integer factorization does not go 10 times faster if you have 10 machines to split the ...
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1answer
86 views

Polynominal reduction from unbounded knapsack problem to general integer programming

Given an oracle that can solve in polynominal time: $$a^Tx=b$$ $$x \geq 0$$ So it can solve the feasibility problem with one equality-constraint(a is here a vector and b is a constant, x is required ...
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A set of languages over {0, 1} which does not belong to Recursively Enumerable set are uncountable [closed]

I have a problem of Theory of Computation i.e. Prove that A set of languages over an alphabet Σ = {0, 1} which does not belong to Recursively Enumerable set, are uncountable. Anyone can ...
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4answers
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Why is linear programming in P but integer programming NP-hard?

Linear programming (LP) is in P and integer programming (IP) is NP-hard. But since computers can only manipulate numbers with finite precision, in practice a computer is using integers for linear ...
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Travelling salesman problem with detours

I am interested if there exists a following version of the travelling salesman problem: INSTANCE: A finite set $C = \{1,2,\dots,k\}$ of cities, a positive integer distance $\delta(i,j)$ for each ...
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1answer
56 views

Running time of partial algorithms

What is the correct term for the maximal running time of a given algorithm on all inputs of length bounded by given $n$, on which the algorithm halts? Assume, if necessary, that the halting problem ...
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205 views

Complexity classes where $C^C = C$

One possible motivation for studying computational complexity classes is to understand the power of different kinds of computational resources (randomness, non-determinism, quantum effects, etc.). If ...
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Is the weighted transitive reduction problem NP-hard?

The transitive reduction problem is to find the graph with the smallest number of edges such that $G^t = (V,E^t)$ has the same reachability as $G=(V,E)$. When $E^t \subseteq E$ it is NP-complete. ...
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P-Completeness and Reducibility

I am taking an algorithm analysis class and am stuck on one of my homework problems and would appreciate it if I could receive some guidance. The problem I'm stuck on is proving that the empty ...
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2answers
185 views

Some inference about NP

this is my first question on this site. I‌ recently, study on NP. I have some confusion about this Topic, and want to propose my inference and some one verify me. I) each NP problem can be ...
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3answers
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Could an NP-Hard problem be in P in after a basis transform? [closed]

I'm aware that there must be something wrong with my reasoning, but I'm not sure what and neither are a few other CS people I've asked. So here goes: Take the following problem for example: Let ...
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1answer
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How can TSP be an NP-optimization problem, when a feasible solution $s$ must be polynomial bounded in the instance size $|I|$?

How can TSP be an NP-optimization problem ? The definition of an NP-optimization problem $\Pi$ states that for each instance $I \in \Pi$ , the set of feasible solutions $S_\Pi(I)$ is non-empty and ...
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273 views

Length-preserving one-way functions

Unfortunately my background in computational complexity is still weak, but I am working on it. As I understand, the question of existence of one-way functions is very important in the field. Assume ...
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Totally unimodular <=> polynomial time?

Crossposting due to recommendation. I formulated a MIP problem which I didn't expect to be unimodular. The problem is to find a minimum complete sequence in a strongly connected digraph. That is, ...
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242 views

Use minimum number of swaps so each bin contains balls of the same color

There are $n$ bins, the $i$th bin contain $a_i$ balls. The balls has $n$ colors, there are $a_i$ balls of color $i$. Let $m=\sum_{i=1}^n a_i$. A swap is take a ball from one bin and swap with a ball ...
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What's the complexity of the Bombe?

Now my knowledge of this comes through watching The Imitation Game, a glance of a wiki article, and a couple of computerphile videos, so forgive me if it's obvious. While watching the Imitation Game, ...
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25 views

Particular function communication complexity computation

Consider a boolean function $f:\{0,1\}^n\rightarrow\{0,1\}$. If $f$ satisfies $f(\bar{0})=0$ where $\bar{0}$ is vector of $0$, $f(x)=1$ with every $0/1$ vector of hamming weight $1$, then ...
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58 views

Array search NP completeness

Given an unsorted array of size n, it's obvious that finding whether an element exists in the array takes O(n) time. If we let h = log n then it takes O(2^h) time. Notice that if the array is ...
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3answers
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Why are NP-complete problems so different in terms of their approximation?

I'd like to begin the question by saying I'm a programmer, and I don't have a lot of background in complexity theory. One thing that I've noticed is that while many problems are NP-complete, when ...
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1answer
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Complexity of self-reducible set

I am trying to solve the following problem: A set $S$ is self-reducible if the following holds: $x \in S$ iff $x = 1$(Base case) or (recursively) $l(x) \in S$ and $r(x) \in S$ where ...
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Parallel time is sequential space

Studying for my qualifying exam, have a past exam here, which has the following question, verbatim: Give a proof of the Folklore statement: "sequential space is parallel time." In other words, ...
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1answer
135 views

Hardness of a constrained quadratic maximization

Consider the following quadratic maximization: \begin{align} \max_{\mathbf{x} \in \mathcal{X}} &\quad\mathbf{x}^{T}\mathbf{A}\mathbf{x} \end{align} with \begin{align} \mathcal{X} = \lbrace ...
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1answer
24 views

Complexity bound on $RP^{RP}$

This is a homework question, I'm wondering if anyone could help. Recall $RP$ is the set of languages recognized by randomized algorithms in polynomial time. The question is given an algorithm in ...
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Relationship between functions and formal languages?

PR is defined as "the complexity class of all primitive recursive functions" and also equivalently as "the set of all formal languages that can be decided by such a function". ...
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Problems that provably require quadratic time

I'm looking for examples of problem which has a lower bound of $\Omega(|x|^2$) for input $x$. The problem needs to have the following properties: $\Omega(n^2)$ runtime proof for any algorithm - ...
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2answers
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Restricted Integer Programming

The integer feasibility problem is NP-complete: $Ax=b, x \geq 0, x \mbox{ integer}$ $A$ contains elements in $\mathbb{R}$ If we restrict this: $A$ contains only elements in: $\{1,0\}$ ...
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Algorithm for a special case of SAT/#SAT

Does anyone know of an algorithm that can solve the following special case of SAT in polynomial time? Are there any algorithms that can solve the counting (#SAT) version of it in polynomial time? ...
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Applications of model counting

I have been reading about model counting, a.k.a. the #SAT problem. What are the practical applications, if any, of this problem, and how exactly do they reduce to it? I have been unable to find any, ...
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Why use languages in Complexity theory

I'm just starting to get into the theory of computation, which studies what can be computed, how quickly, using how much memory and with which computational model. I have a pretty basic question, but ...
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complexity of decision problems vs computing functions [closed]

This is an area that admittedly I've always found subtle about CS and occasionally trips me up, and clearly others. recently on tcs.se a user asked an apparently innocuous question about N-Queens ...