Use for algorithms, algorithm-analysis and complexity-theory questions that aim for polynomial running time resp. time complexity. Such questions often are are reference-requests or about runtime-analysis or time-complexity.

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2
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1answer
81 views

Polynomial hierarchy: inclusion between spaces

Using the definition for the polynomial hierarchy: $$ \Sigma_{i+1}^P = NP^{\Sigma_i^P} $$ $$ \Pi_{i+1}^P = coNP^{\Sigma_i^P} $$ I have been asked to to show that: $$ P^{\Pi_k^P } \subseteq ...
4
votes
0answers
39 views

What's the complexity of solving a packing LP?

As we know, we can solve general linear programs in weakly polynomial time and it remains open if it is possible to solve them in strongly polynomial time as well. But what is the situation in the ...
1
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1answer
33 views

If a CSP (over a finite domain) has only linear inequalities as constraints, is it solvable in linear time?

I have an optimization problem in fuzzy logic that I want to model and solve as a CSP. If I could use only linear inequalities in my encoding, is the resulting CSP solvable in linear time? Problem ...
-1
votes
1answer
38 views

A program for polytime languages

Does their exist a program P[m,s] which always halts and for any polytime language exists an m; possibly incomputable; such that P[m,s] accepts only those strings s which are in the language.
4
votes
1answer
43 views

Can we use reductions to design approximation algorithms for NP-hard problems?

Let us say that I have a problem $P(n)$ that I need to solve (where $n$ is the size of the input of problem $P$). I used a polynomial-time reduction from a known NP-hard problem $Q(m)$ (where $m$ is ...
7
votes
3answers
169 views

Why are most (or all?) polynomial time algorithms practical?

I read an interesting comment in a paper recently about how weirdly useful maths turns out to be. It mentions how polynomial time doesn't have to mean efficient in reality (e.g., ...
8
votes
3answers
268 views

Isn't polynomial identity testing over arithmetic *expressions* trivial?

Polynomial identity testing is the standard example of a problem known to be in co-RP but not known to be in P. Over arithmetic circuits, it does indeed seem hard, since the degree of the polynomial ...
10
votes
2answers
224 views

Why not to take the unary representation of numbers in numeric algorithms?

A pseudo-polynomial time algorithm is an algorithm that has polynomial running time on input value (magnitude) but exponential running time on input size(number of bits). For example testing whether ...
-1
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1answer
30 views

P is closed under power of integer

I'm new in this area of complexity and I'm trying to get into it by understanding basic proofs. I want to prove that if $L\in P$, so $L^k\in P$, where $k$ is non-negative integer. How to prove it in ...
0
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0answers
21 views

Pedagogic reference on cut generating functions

Can you recommend an introduction to the topic of cut generating functions? I am looking for introductory or review-like material. I did find the following survey paper, but it seems to be addressed ...
4
votes
2answers
336 views

Can the isomorphic graph problem be solved in deterministic polynomial time?

Here is a recent homework problem of mine: Call graphs G and H isomorphic if the nodes of G may be reordered so that it is identical to H. Let ISO = {⟨G,H⟩| G and H are isomorphic graphs}. ...
5
votes
1answer
544 views

Can we show that non-determinism adds no power, for some specific running time?

$NP = \cup_{k \in \mathbb{N}} NTIME(n^k)$ $P = \cup_{k \in \mathbb{N}} TIME(n^k)$ Can we show that $NTIME(n^k) = TIME(n^k)$ for a specific $k$? For how large of a $k$ can we show the above ...
1
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0answers
25 views

Besides practical computing applications, is there a reason polynomial is “good” and exponential is “bad”? [duplicate]

The overarching theme of computer science seems to be that polynomial time or space or what-have-you for an algorithm is a success, and exponential is a failure. The definitions of P and NP revolve ...
9
votes
1answer
110 views

If $n^{\log n}$ is not polynomial or exponential, then what this function is called?

I just found this sentence on page 6 of Garey and Johnson's "Computers and Intractability". Any algorithm whose time complexity function cannot be so bounded is called an exponential time ...
4
votes
1answer
56 views

Machine with an oracle for a language that cannot decide another language in polynomial time

We usually see examples of languages contained in $P^A$ for some language $A$, or cases where $P^A=P^B$ (or $P^A\subseteq P^B$) for two languages $P^A$ and $P^B$. However, there is any explicit ...
0
votes
0answers
21 views

How to prove an algorithm with subroutines to belong to the polynomial time category using correct notation? [duplicate]

I'm having some trouble with writing my answer with the correct notation. I suppose I get the basic idea how to solve this and wrote down my answer but my gut tells me that that it's not ...
3
votes
1answer
49 views

Do poly-time algorithms exist whose time complexity is unprovable?

If not, is there a decision procedure that successfully classifies any polynomial time algorithm as poly-time within a time polynomially bounded by the length of the input algorithm?
2
votes
1answer
42 views

Can one find the minima of a convex function efficiently?

Say I have a real valued convex function $f$ on the hypercube $[-1,1]^n$ and let $f'$ be its restriction on the discrete hypercube $\{-1,1\}^n$. Is there any $poly(n)$ algorithm that for any class ...
10
votes
1answer
309 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 ...
9
votes
2answers
692 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
vote
1answer
55 views

The running time of the knapsack problem is $O(n\cdot \min(B,V))$ and is not polynomial, why?

My question is why the dynamic programming of the knapsack problem does run in polynomial time? The question is answered here Why is the O(nW) algorithm for the Knapsack problem not a polynomial one? ...
2
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0answers
27 views

Cobham's characterization of FP

Does anyone know of an accessible introduction to Cobham's model independent characterization of FP and it's equivalence to the standard definition using Turing machines? The best source I could find ...
1
vote
0answers
61 views

Suppose P = NC - what then? [duplicate]

Suppose tomorrow someone discovered a proof that P = NC. What would the consequences for computer science research and practical applications be in this case?
7
votes
0answers
99 views

P vs NP and the Time Hierarchy

Assuming P $\neq$ NP, is it possible that there exists a $k$ such that for all $j$, $\textsf{DTIME}(t^j) \subseteq \textsf{NTIME}(t^k)$? There reason I ask is that I assume P = NP implies that for ...
6
votes
3answers
178 views

When did polynomial-time algorithm become of interest?

I would like to understand why and when polynomial algorithms became of interest. When did people realize the role and importance of efficient versus non-efficient algorithms? Did that happen when ...
1
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0answers
23 views

Mixing time of three particle systems

Is there anything known about mixing time of Markov chains for three particle systems? It is proved here http://www.ams.org/journals/tran/2005-357-08/S0002-9947-05-03610-X/S0002-9947-05-03610-X.pdf ...
3
votes
1answer
103 views

What is the decidable language in $P/poly$ but not in $P$?

Except for the undecidable unaries I have no idea if there is anything in the gap between $P/poly$ and $P$
3
votes
1answer
43 views

Is there a known way to convert any $QBF_2$-formula into an equisatisfiable $QBF_2$-formula in CNF in polynomial time?

It is easy to turn any boolean formula and any quantified boolean formula into an equisatisfiable formula in CNF using Tseitin transformation: $$ Q_1 z_1 Q_2 z_2 \ldots Q_n z_n \Phi \Rightarrow Q_1 ...
4
votes
2answers
213 views

Why is it important to solve a problem in Polynomial time, In cryptography?

I have just started to learn Cryptography. I am trying to learn "Merkle-Hellman Knapsack Cryptosystem". So, right at the beginning of the discussion, a question came in my mind: Why is it important ...
5
votes
1answer
124 views

Polynomial time algorithm for finding two or more vertex-disjoint cycles

The cycle detection problem for a directed graph has well-known polynomial time solutions, graph traversal algorithms such as Dijkstra algorithm can be used to find whether or not a cycle exists in a ...
-1
votes
2answers
363 views

Time Complexity of k-clique problem with fixed k [closed]

My question expands on a related question on the link, Why is the clique problem NP-complete? In that post the author argued that while the $k$-clique problem is NP-complete; for a fixed $k$ the ...
1
vote
1answer
110 views

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 ...
21
votes
4answers
3k views

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 ...
2
votes
3answers
84 views

Graph cycles on 40 vertices

I'm trying to create an algorithm in polynomial time, that detects wether or not a graph is in a language. The language specifies that a graph is only part of this language if it has a cycle on 40 ...
4
votes
1answer
149 views

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? ...
3
votes
1answer
50 views

Methods of turning a decision problem into finding the certificate?

I usually find this in the context of asking about NP-complete problems, but any decision problem works. We start by assuming there's a polynomial time algorithm that gives the yes or no answer. If ...
2
votes
1answer
21 views

Is it Polynomial to decide whether any product of input numbers satisfies a boolean expression?

I have an input number c of n bits and its prime factorization. I want to find a divisor of c with certain fixed bits "f". For example: ...
1
vote
1answer
27 views

Polynomial Identity Testing Evaluating a polynomial on a circuit

Say I have a polynomial over $Q$. Let it be given in the form of arithmetic circuit family ${C_n}$. The randomised poly time algorithm evaluates the polynomial at a random point. What if the number of ...
0
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0answers
18 views

General methods for polynomial reductions? [duplicate]

Let's say you want to show $A \leq_{p} B$ (this is usually in the context of showing $B$ is NP-complete, but I'm just asking about the reductions. We are specifically looking at polynomial (Karp) ...
7
votes
2answers
93 views

Common method for solving satisfiability problems which lie in P

I know from Schaefer's Dichotomy Theorem that only a few types of satisfiability problems are in P and any other problem is NP-complete. However, all of the algorithms I know for them use specific ...
-1
votes
1answer
51 views

NXOR for 2 inputs on a turing machine, in P?

Question: L is the language of $\langle M,x,y\rangle$ s.t TM $M$ accepts both inputs $x$ and $y$ or doesn't accept either. Prove that given some $M$, finding 2 inputs $x$ and $y$ s.t. $\langle ...
3
votes
1answer
124 views

How does this proof show that sequences of $O(1)$ polynomially bounded Kolmogorov complexity are NOT the polynomial computable ones?

I'm trying to understand a proof from the paper: Balcázar, José L.; Gavaldà, Ricard; Hermo, Montserrat: Compressiblity of infinite binary sequences, Published in: Complexity, Logic, and Recursion ...
2
votes
2answers
120 views

Heuristic for Tournament Scheduling

I am holding a bi-yearly tournament in my city, for which I want to write a program that gives me (nearly-)optimal pairings, and waiting time. The setup is as follows: ...
3
votes
1answer
58 views

Characterizing the range of a polytime function

Is it true that an infinite language is in P iff it is the range of a length increasing polytime function? I ask because I know that it is a basic result in computability theory that a set is ...
0
votes
1answer
70 views

If A is polynomial time reducible to B such that B <= A, does it mean B must be a polynomial time algorithm?

I don't understand what it means for A to be polynomial time reducible to B. I'm guessing is that we can revised the algorithm some how such that it becomes B, where B is a polynomial time algorithm. ...
2
votes
1answer
191 views

task scheduling optimization problem

I'm interested in such problem. I have a set of $n$ tasks ${T_i}$ and directed acyclic graph, which nodes correspond to tasks and edges correspond to order of execution two tasks. In other words if I ...
1
vote
3answers
128 views

If A is in P and B is non-trivial, then A ≤p B [duplicate]

On wikipedia's article on Polynomial-time reduction it states: Every nontrivial decision problem in P (the class of polynomial-time decision problems, where nontrivial means that not every input ...
0
votes
1answer
52 views

Vertex Cover of size k in a tree?

What is a polynomial time algorithm for finding a vertex cover of size $k$ in a tree? Would depth first or breadth first search be efficient or is there some other algorithm that finds the vertex ...
3
votes
1answer
62 views

What does the 2 in a 2-approximation algorithm mean?

Does the 2 in a 2-approximation algorithm mean the solution is within 2*OPT or OPT/2?
2
votes
1answer
128 views

Find a subgraph whose edge weights sum to at least the number of nodes

Given a graph G = (V,E) every edge is assigned a real number Xe $\in$ [0,1] The sum of x variables for all edges is equal to the number of edges -1 : $\sum x_V = |V|-1$ For a subset S ...