Questions about the hardest problems in NP, i.e. of those that can be solved in polynomial time by nondeterministic Turing machines.

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1answer
34 views

Why is Hamiltonian Path and graph coloring np complete and shortest path p when the former can also be solved using DFS recursively?

NP is a complexity class that represents the set of all decision problems for which the instances where the answer is "yes" have proofs that can be verified in polynomial time. But hamiltonian path ...
1
vote
1answer
92 views

Why doesn't subset sum solution violate Exponential Time Hypothesis?

The quickest algorithm for solving subset sum currently is $2^{n/2}$ (via Wiki). Why doesn't this violate the Exponential Time Hypothesis which states that “there is no family of algorithms that can ...
3
votes
2answers
91 views

What would NP-complete solution in O(2^N/B) mean?

Suppose we had an algorithm that solved an NP-complete problem (SAT, TSP, etc.) in time $O(2^{N/B})$ where $B>2$ is an input to the algorithm, along with the instance to be solved. So for $B < ...
0
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0answers
42 views

Complexity of reductions of NP-complete problems [closed]

Is there list of best known/possible algorithms (with minimal Big-O) for reductions of some of NP-complete problems to some other? If no, can you together make such list?
-1
votes
1answer
26 views

Poly-time reduction from HAMPATH to HAMPATH-E

I need to prove that HAMPATH-e = { < G,s,t,e > | G is directed graph, s, t are vertices and e a edge } there is hamiltonian path between s to t that cross the edge e is an NP complete. i've ...
-2
votes
1answer
31 views

Knapsack: there is a polynomial solution in bit terms?

I'm reading about Knapsack problem. The approaches to solve that I found: Branch and bound Brute force Dynamic programming Memory functions Greedy All solutions have exponential time in terms of ...
1
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0answers
38 views

prove that the satisfiability problem with each clause containing at most 3 literals, denoted by ≤3SAT, is NP-complete

I've tried to prove it for several days but I can't make sure if it is equivalent to max-3-SAT problem? This problem seems similar to the proof of SAT ∝ 3-SAT except the case where there are more than ...
2
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2answers
73 views

What are the hardest problems that are in P if and only if P=NP?

I used to think that NP complete problems are the "hardest" problems of all problems that would still be in P if P=NP. Now I think otherwise. What I'm asking is if there are any problems that are ...
3
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1answer
62 views

Why is Knapsack and ILP NP-complete

I have a question concerning several NP-hard problems and why they are (or are not NP-complete). I understand the concepts behind NP-hard and NP-complete: Problem lies in NPC if it is NP-hard and ...
-1
votes
2answers
66 views

What is the simplest known NP-Complete problem for testing P=NP solutions? [closed]

About a year and a half ago I ask this question regarding $P=NP$. The answers have helped me understand the problem tremendously and since then I've dabbled further into the topic. With that stated, ...
4
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3answers
2k views

Does a polynomial solution for an NP-complete problem that can only be implemented for small N *still* imply P=NP?

Basic sanity check on NP-complete solutions: Suppose there was a polynomial time solution for an NP-complete problem, but the size of NP-complete problems that could be solved is still relatively ...
0
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1answer
20 views

How do I show this variant of the longest path problem is NP-hard?

The problem is as follows: "Given a weighted graph G and a path p, show that p is the longest simple path in G." I'm thinking a reduction from HAMPATH would work, but after 3 hours of racking my ...
5
votes
2answers
302 views

Is this an instance of a well-known problem?

Context I am developing an application and came across a problem that seemed difficult to solve. Before attempting to reinvent the wheel (and trying to solve an NP complete problem on my own), I ...
2
votes
2answers
97 views

How are games like chess provably harder than NP?

From this question, I had the debate about how problems harder than NP are proved. I said that intuitively I understand it as (from this video explaining that some problems are provably harder than ...
0
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0answers
25 views

If problems P1 and P2 are known to be NP-hard, then we can conclude that P1∝P2 and P2∝P1? [duplicate]

I know the definition of NP-hard is that “a problem(P1 or P2) is NP-hard if every NP problem could be polynomially reduce to (P1 or P2)”. However, P1∝P2 means P1 could be polynomially reduced to P2, ...
3
votes
1answer
41 views

Can we reduce an NP complete item to an NP item which is $\bf{non}$ P?

I'm curious if we can reduce an $NP$-complete problem to an $NP$ problem which is not a part of the $P$ set. Meaning, can we take an algorithm for this kind of $NP$ problem and use it to solve a ...
6
votes
1answer
117 views

How to use an old SAT solver to discover a new one, as is done in The Golden Ticket?

In Lance Fortnow's book The Golden Ticket, he mentions that once you have a polynomial-time algorithm for an NP-complete problem, you can use it to find a faster algorithm. Can you tell me how that is ...
5
votes
1answer
181 views

How do we know for sure that EXPTIME ≠ P?

I'm a beginner in learning about computational complexity and this has stumped me. I've read that by the time hierarchy theorem, it's known that EXP-complete problems are not in P. (Wikipedia) It ...
2
votes
1answer
31 views

Flaw in linear programming solution for multi-commodity flow problem?

The multi-commodity flow problem problem statement - wiki According to constraints of multi-commodity flow problem a given material must start at source s with demand d and end up at its target t. ...
6
votes
2answers
470 views

Reconciling NP and the decision problem

So I've seen that most NP-Complete problems seem to take the form of decision problems - problems which require only a yes/no answer. However, how can this be reconciled with the requirement that the ...
1
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0answers
44 views

Reduction of 3-SAT to Vertex Cover?

Can someone explain to me in the most simplest possible way, how to reduce $3-SAT$ to $Vertex\:Cover$ ? I am following the explanation here(scroll to page 4 bottom). I understand the basic setup of ...
0
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0answers
18 views

On subset sum instance length

Suppose we want to have a $NP$-complete subset sum instance of $m$ distinct integers from $\{-2^{n^\frac1\beta},\dots,0,\dots,2^{n^\frac1\beta}\}$ should $m=O({n^{\frac1\alpha}})$ hold where $\alpha,\...
2
votes
2answers
134 views

Time complexity of a problem inspired by palindromes

This was inspired by Bradshaw's question originally posted on Math.SatckExchange. EVEN PALINDROME: Input: Given a list of strings $[v_i, v_2, ... ,v_n]$ where $\Sigma |v_i| $ is even number. ...
0
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0answers
49 views

How do you reduce graph edge colouring problem to the graph node colouring problem

I know that if the nodes of a graph can be coloured by $n$ colours such that no two nodes sharing an edge have the same colour, I can also colour its edges with $n$ colours such that no two different ...
1
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1answer
53 views

Turing NP complete but not Karp NP complete?

Is there some examples of candidate problems that have Turing reduction from SAT but no known Karp reduction? Conversely is there some examples of candidate problems that have Turing reduction to SAT ...
0
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1answer
30 views

Finding top k which are the most different from each other

Assume I have a set of items $A$ and each item $a \in A$ has a score $s(a)$. Also, each two items $a_1,a_2 \in A$ have variety score $var(a_1,a_2)$ which tells how different they are. I want to ...
5
votes
1answer
57 views

Why is determining if there is a solution to a Battleship puzzle NP-Complete?

This paper http://www.mountainvistasoft.com/docs/BattleshipsAsDecidabilityProblem.pdf says that the decision problem, "Given a particular puzzle, is there a solution?" is NP-Complete. I don't ...
4
votes
0answers
40 views

find a minimum-cost pair of arc-disjoint paths, both within a given restricted distance

Is there a polynomial algorithm that can find a pair of arc-disjoint paths in a directed graph that has a minimum total cost, subject to the condition that both paths are within the same distance. ...
1
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3answers
145 views

What is wrong with this reasoning that finding the genus of a degree 3 bipartite graph is NP-complete?

Finding genus of a biparite graph is $NP$-complete and finding genus of a degree $3$ graph is $NP$-complete and so finding genus of a degree $3$ bipartite graph is $NP$-complete. Though implication ...
11
votes
3answers
846 views

Can any finite problem be in NP-Complete?

My lecturer made the statement Any finite problem cannot be NP-Complete He was talking about Sudoku's at the time saying something along the lines that for a 8x8 Sudoku there is a finite set of ...
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2answers
72 views

Is the “modular subset product” problem NP-complete?

While examining some $NP$-complete problems relating to sets of integers, a question flashed through my mind: whether the $NP$-completeness of these problems is retained when integer arithmetic is ...
1
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1answer
40 views

How can ships in Battleship be partitioned into B subsets, each with a capacity C?

I have been doing some research on the reduction of Battleship to bin-packing, but I do not completely understand the input to the problem from this academic paper: http://www.mountainvistasoft.com/...
3
votes
1answer
47 views

What is the time complexity of Summing Triples with duplicates?

Summing Triples problem is strongly $NP$-complete as shown by McDiarmid. Summing Triples problem: Input: list of 3N distinct positive integers Question: Is there a partition of the list into N ...
0
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1answer
51 views

show that special case of NP-complete problem is also NP-complete?

I want to show that a problem is NP-hard by reducing a known NP-complete problem to it. However, I will have to use a special case of the NP-complete problem for the reduction to work. I'm pretty sure ...
4
votes
1answer
41 views

Maximize function over a set with a transitive and antisymmetric relation

Let $\mathcal{R}$ be a transitive and antisymmetric relation defined over a finite set $X$. For any set $S\subseteq X$ define $\Gamma(S)=\left\{y\in S \mid \not \exists x\in S . (x,y)\in\mathcal{R}\...
6
votes
1answer
42 views

What does Cellular Automata Pre-image problem actually means?

I am reading about Cellular Automata and Computational Complexity and i found a related paper by F. Green, NP-Complete Problems in Cellular Automata. In the 2nd page he lists three NP-Complete ...
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0answers
26 views

Reducing partition to a partition where sum(partition1) = 3 times sum(partition2)

Given the following NP-complete problem: PARTITION Input: A list of positive integers a1,a2...,an Question: Can the list be partitioned into 2 parts, A1 & A2 such that the sum of each part is ...
1
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1answer
21 views

About the interpretation of the SOS hardness results of the planted Max-Clique problem

One can look at these two papers http://arxiv.org/abs/1502.06590 and http://arxiv.org/abs/1507.05136 and see their main theorems. If I understand right then both these papers are talking of the ...
0
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3answers
82 views

Size of instance after reduction

A decision problem $C$ is $NP$-complete if $C$ is in $NP$, and every problem in $NP$ is reducible to $C$ in polynomial time. Reduction means transforming an instance of one problem $A$ to an instance ...
5
votes
1answer
101 views

Implication of Berman and Hartmanis conjecture

I am reading "Complexity and Cryptography" by Talbolt and Welsh. The book mentions the Berman and Hartmanis conjecture : All $NP$-Complete languages are $p$-isomorphic. Then the book says that ...
3
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0answers
20 views

A particular type of SOS hardness proof

Is there an example of a sum of squares (SOS) hardness proof where the constraint is something non-trivial (like with some polynomial constraint) rather than just imposing the the typical $x_i^2 =1$ ...
1
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1answer
48 views

If a language is X-complete, is its complement is X-complete as well?

I'm looking for an information about closure of complexity complete classes. Is it true that any language, if the language is X-complete, then its complement is X-complete? Why? I was thinking ...
1
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1answer
95 views

Is this a well-known NP-hard problem?

Let $R = \{1, \ldots, n\}$ and $S = \{S_1, \ldots, S_m\}$ a collection of subsets of $R$ such that $R = \bigcup_{i = 1}^m S_i$ and, for $n > 3$, $$3 \leq \vert S_i \vert \leq 4 \, , \enspace i \in \...
2
votes
1answer
48 views

Is the unweighted vertex cover problem equivalent to its weighted version?

Consider the unweighted and weighted versions of the vertex cover problem (UVC and WVC for short, respectively). As UVC is a special case of WVC, is it true that $$\text{UVC} \leq_\mathrm{m} \text{WVC}...
3
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0answers
70 views

Typical NP-complete/hard problems in machine learning

I know little about machine Learning, but I work on optimization (solving NP-hard problems with SAT solvers or MIP). Examples of this would be solving TSP, Steiner tree problems, path finding with ...
3
votes
2answers
33 views

How can we use the FPTAS for problem B to solve problem A

Given an optimization problem A which is NP-complete, and can be polynomially reduced to another optimization problem B which is also NP-complete. If we use an FPTAS to solve the reduced problem B' (A ...
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0answers
19 views

A question about SOS duality

Let us start with the optimization question, \begin{eqnarray*} min \{ c \vert c - f \in SOS_d \} \end{eqnarray*} for some function $f : \{0,1\}^n \rightarrow \mathbb{R}$ and $SOS_d $ being the cone ...
4
votes
1answer
113 views

Complexity of $n \times \log n$ tiling problem

Is there a polynomial time algorithm for an $n \times \log n$ tiling problem? For instance: Suppose $A$ is a finite alphabet. A tile is a $2 \times 2$ matrix of elements from $A$. A tiling is a ...
2
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1answer
31 views

Is a degree-$d$ pseudo distribution always a relaxation?

The optimization problem we are generally concerned with looks like the following, \begin{eqnarray*} &\inf \{ p(x) \vert x \in K\} \\ &K = \{ x \in \mathbb{R}^n \vert q_i(x) \geq 0, i = 1,..,m ...
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1answer
88 views

Reducing co3SAT to UNIQUE-SAT

I am having trouble with this problem: Let N3SAT denote the non-satisfiability problem for 3CNF’s. Show that $N3SAT\leq_p UNQ$ where in UNQ, given a CNF φ we want to know whether there is a unique ...