Questions related to the (computational) complexity of solving problems

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0answers
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Minimum edge deletion partitioning of a planar graph

I'm interested in the time complexity of the following problem: Given an undirected planar graph $G=(V,E)$ and a weight function $w:E \rightarrow \mathbb{Z}$ (so weights can be negative, too), color ...
3
votes
1answer
23 views

Prove or disprove that $NL$ is closed under polynomial many-one reductions

If $B \in NL$ and there exists a Karp reduction (polynomial-time many-one reduction) from $A$ to $B$, then $A \in NL$. Prove that the above claim is correct, incorrect, or equivalent to an open ...
4
votes
2answers
101 views

Minimum edge deletion partitioning

I'm interested in the time complexity of the following problem: Given an undirected graph $G=(V,E)$ and a weight function $w: E \rightarrow \mathbb{Z}$ (so weights can be negative, too), color the ...
1
vote
1answer
44 views

NP-hardness reductions

Suppose I have two generic problems $A_{1}$ and $A_{2}$: the instance of $A_{i}$ is a graph $G$ and a number $t$, and the question is whether a certain parameter $P_{i}(G)$ is at least $t$. Suppose ...
1
vote
1answer
26 views

Prove that $coRP \subseteq RP^{RP}$

I've read in an article that $coRP = RP$ is an open question, but that it is obvious that $coRP \subseteq RP^{RP}$. If $L \in coRP$, I don't understand how access to the oracle helps to build a ...
3
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1answer
68 views

Are NP-complete sets formed from two other sets only if at least one is NP-hard?

This question is somewhat of a converse to a previous question on sets formed from set operations on NP-complete sets: If the set resulting from the union, intersection, or Cartesian product of two ...
2
votes
1answer
40 views

Example of reduction in communication complexity

Let us assume the standard situation in communication complexity with two players $P_1,P_2.$ We have a function $f:[n] \times [n] \mapsto \{0,1\}$ that both players known in advance. They wish to ...
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0answers
19 views

Inherent complexity of testing line segment intersections with aligned and oriented bounding boxes?

It is well known that in practice, a substantial difference in run-time between algorithms for testing intersection of a line segment with aligned or oriented bounding boxes (in computer graphics ...
5
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2answers
139 views

Complexity of “given a graph $G$ with vertex $v$, is there a maximum clique containing $v$”?

The usual way of translating the maximum clique problem into a decision problem is to ask "does there exist a clique of size $\ge k$ in $G$?" Clearly this problem is in NP (and is NP-hard). Another ...
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1answer
40 views

How can I efficiently find the optimal order to apply special offers to a shopping cart?

Given a list of items which represent items in a shopping cart, and a list of available special offers which replace one or more regular items to lower the cost of those items, how can I decide the ...
2
votes
1answer
53 views

Implications of $NP = \Sigma_2 P$ for PH collapse

A simple fact is that $P = NP \to P = coNP$, which follows from the observation that $P$ is closed under complement. I am having trouble seeing that an analogous statement is true at higher levels of ...
12
votes
3answers
918 views

Why is the class NP-Complete important compared to NP-hard?

I'm studying computational complexity and I was wondering why the NP-Complete (NPC) problems is an important class at all. I find it obvious why we're interested in showing a given NP problem is ...
3
votes
2answers
98 views

Is my theorem about $P \neq NP$ correct? [closed]

It is known that there are problems in P that, provably, are not solvable in less than $O(N^k)$, for some $k$. Now consider some infinite set $K \subseteq \mathbb{R}^+_0$ such as K is unbounded from ...
1
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1answer
28 views

Are non-trivial sets formed by set operations on NPC sets still in NPC?

I know that from this answer to a question on the class NPC, that NPC is not in general closed under intersection and union. However, the answer used languages which form trivial languages under ...
2
votes
0answers
33 views

Relations between P^#P, NP^#P and (CO-NP)^#P

I was wondering if there were relation between the complexity classes $P^{\#P}$, $NP^{\#P}$, $(Co-NP)^{\#P}$ ?(except the trivial inclusions) I've the feeling that when taking a $NP^{\#P}$ machine, ...
1
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1answer
119 views

Complexity of edit distance with block operations

Consider the following problem. I have a pattern $P$ of length $100$ and a text $T$ of length $n$. I want to find the minimum number of operations to transform $P$ into $T$. The operations are: ...
0
votes
1answer
64 views

Consequence of $\mathsf{NP\subseteq BPP}$ to $\mathsf{NP\subseteq ZPP}$?

If $\mathsf{NP\subseteq BPP}$, then we know that $\mathsf{NP\subseteq RP}$ (http://www.csie.ntu.edu.tw/~lyuu/complexity/2011/20120103s.pdf). Does $\mathsf{NP\subseteq BPP}$ also imply ...
3
votes
3answers
149 views

Understanding definition of NP

In my lecture notes, the definition of the class NP is given as: A language $L$ is in the class NP, if there exists a turing machine $M$ and polynomials $T$ and $p$ such that: For every input $x$, ...
5
votes
3answers
247 views

Why is the addition function exponential for k-bit integers providing only zero, equality and the successor functions?

I'm currently reading the elements of programming book and have come across a section I don't quite understand A computational basis for a type is a finite set of procedures that enable the ...
10
votes
1answer
280 views

Is determining if there is a prime in an interval known to be in P or NP-complete?

I saw from this post on stackoverflow that there are some relatively fast algorithms for sieving an interval of numbers to see if there is a prime in that interval. However, does this mean that the ...
4
votes
0answers
32 views

One way communication complexity lower bounds techniques

I have been teaching myself communication complexity. I am starting to understand the general methods for proving randomized lower bounds. However, I don't yet understand what techniques are ...
1
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2answers
90 views

If an NP-complete problem is shown to have a non-polynomial lower bound, would that prove that P != NP?

I understand that the Cook-Levin theorem proved that any NP problem is reducible to an NP-complete problem, which signifies that if a polynomial-time algorithm for an NP-complete problem is found, it ...
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votes
0answers
10 views

recurrence relation complexity analysis [duplicate]

I want to find time complexity of recurrence relation without using masters theorem : T(n) = 3T(n/2)+n^2.could you please help me to do so without using master theorem?
3
votes
0answers
53 views

Complexity for finding zeroes of sum of cosines

Consider the following equation with variable $n \in \mathbb{N}$: $$\sum \limits_{i=1}^{k} \cos(n\theta_{i}) = 0.$$ Given $\theta_1,\dots,\theta_k$, I'd like to determine whether there exists $n \in ...
7
votes
3answers
1k views

How fast can we find all Four-Square combinations that sum to N?

A question was asked at Stack Overflow (here): Given an integer $N$, print out all possible combinations of integer values of $A,B,C$ and $D$ which solve the equation $A^2+B^2+C^2+D^2 = N$. ...
0
votes
1answer
138 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 ...
2
votes
0answers
39 views

Hardness of approximation for Disjoint Group Steiner Tree

Does anyone know any constant factor approximation hardness results on Group Steiner Tree when the groups partition the terminals, i.e. every terminal belongs to exactly one group? The (intuitive) ...
2
votes
1answer
92 views

How are these problem variants that ask about the size of optimal solutions in NP?

I just started reading Vazirani's book "Approximation Algorithms". It is legally available online here. On page 5 (23 in the pdf), it says that the following decision problems are in NP: Is the ...
9
votes
2answers
602 views

Why do we say that polynomial time is easy? [duplicate]

For years, I've been told (and I've been advocating) that problems which could be solved in polynomial time are "easy". But now I realize that I don't know the exact reason why this is so. ...
-1
votes
0answers
45 views

How can I prove that scheduling problem F2//Lmax is NP-Hard?

I'm trying to solve it via reduction to the 2-Partion problem. All online resource are leading to a single solution, which is: http://i.imgur.com/mkPrCzb.png (taken from ...
1
vote
1answer
53 views

Problem in Papadimitriou's “Computational Complexity” seems odd

I am studying (on my own, this is not homework) Papadimitriou's "Computational Complexity" textbook, 1st edition. On page 66, we have: 3.4.1. Problem: For each of the following problems involving ...
2
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1answer
17 views

A question about the polynomial hierarchy

Why does $\Pi_i^p \subseteq \Sigma_i^p$ imply $\Pi_i^p = \Sigma_i^p$?
35
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8answers
4k views

What would be the real-world implications of a constructive $P=NP$ proof?

I have a high-level understanding of the $P=NP$ problem and I understand that if it were absolutely "proven" to be true with a provided solution, it would open the door for solving numerous problems ...
5
votes
1answer
51 views

Complexity of black box satisfiablty

Say I have a black box $f : 2^n \to 2$ and I want to determine if it is satisfiable. That is, does there exist an input how which it returns true. I am for the purposes of this considering $n$ to be ...
7
votes
1answer
112 views

Find the central point in a metric-space point set, in less than $O(n^2)$?

I have a set of $n$ points which are defined in a metric space – so I can measure a 'distance' between points but nothing else. I want to find the most central point within this set, which I ...
4
votes
0answers
91 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 ...
3
votes
1answer
25 views

What can a quantum query to a function do?

The $n$-qubit Hadamard gate acts as, $$H (\otimes^n \vert 0 \rangle ) = \otimes ^n ( H | 0 \rangle ) = \otimes ^n ( \frac { |0\rangle + |1\rangle }{\sqrt{2} } ) = \frac{1}{\sqrt{2^n} } \sum_{x \in ...
1
vote
1answer
60 views

Good characterization of unsatisfiable Horn3SAT formula?

Horn3SAT is $P$-complete problem under logspace reductions. Since Horn3SAT is in $P$ its complement must have short witnesses. I am looking for natural short proof that a Horn3SAT formula is not ...
0
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1answer
20 views

Binary tree algorithm asymptotic analysis problem

Assume we have a perfectly balanced Binary tree. We have the following algorithm: For each passed node, traverse through all its ancestors and then do the same algorithm for the left and right child ...
1
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1answer
33 views

How to turing reduce equivalent languages $Q$ to infinite language $I$

Given two languages: $Q= \{(\langle M_1 \rangle , \langle M_2 \rangle ) \mid L(M_1) = L(M_2)\}$ $I= \{\langle M \rangle \mid \;\vert L(M) \vert = \infty \}$ I'm trying to Turing reduce $Q$ to $I$ ...
2
votes
1answer
68 views

Proof by Turing Reduction

I need to proof the following by turing reduction. Given two languages: $Q= \{(\langle M_1 \rangle , \langle M_2 \rangle ) \mid L(M_1) = L(M_2)\}$ $I= \{\langle M \rangle \mid \;\vert L(M) \vert = ...
7
votes
2answers
62 views

Is HORN-SAT in LIN, if so why is that not an indication that P=LIN?

The Complexity Zoo defines $LIN$ to be the class of decision problems solvable by a deterministic Turing machine in linear time. $$LIN \subseteq P$$ Since HORN-SAT is solvable in $O(n)$ (as ...
4
votes
1answer
292 views

Why do reductions to NP-complete problems in NTIME(n) not break the nondeterministic time hierarchy?

Let $\mathrm{L} \in \mathrm{NTIME}(n^3)$. Since $\mathrm{NTIME}(n^3) \subseteq \mathrm{NP}$, we have that $\mathrm{L} \le_p \mathrm{3SAT}$. However, $\mathrm{3SAT} \in \mathrm{NTIME}(n)$. Hence, ...
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1answer
84 views
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1answer
154 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 ...
0
votes
0answers
42 views

Existence of randomized reduction but no deterministic reduction

What is the consequence to complexity theory of having a randomized reduction from an NP-complete problem to problem $\Pi$ while there is no deterministic reduction from an NP-complete problem to ...
0
votes
1answer
94 views

Clique decision problem restricted to a subgraph [closed]

I know that the clique problem is NP-complete. However, what if we change the problem a little bit? For example, Given a graph $G(V,E)$, an integer $k$ and a subset $S$ of $m$ vertices, we are ...
23
votes
3answers
6k views

Knapsack problem — NP-complete despite dynamic programming solution?

Knapsack problems are easily solved by dynamic programming. Dynamic programming runs in polynomial time; that is why we do it, right? I have read it is actually an NP-complete problem, though, which ...
4
votes
0answers
43 views

Proof that P is closed against switching between polynomially related encodings

Lemma 34.1 Let $Q$ be an abstract decision problem on an instance set $I$, and let $e_1$ and $e_2$ be polynomially related encodings on $I$. Then, $e_1(Q)\in \mathrm{P}$ if and ...
2
votes
1answer
33 views

Are there other interpretations of |G| than |V|, that is |V(G)|?

This may be a basic question, but I'm hoping someone can settle this nagging doubt I'm having. I'm reading up on FPT complexity using a book by Downey and Fellows. It has some introductory examples ...