P versus NP and other resource-bounded computation.

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Properties of “second-order” NP (complete) languages

Reading the question Natural NP-Complete Problems with Large Witnesses, I was interested in this language: $L = \{ \varphi ~~:~~ \varphi \text{ is SAT formula with more than } |\varphi|^2 \text{ ...
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Reconstructing labeled poset from linear extensions

Let $(P, <, \mu)$ be a labeled poset, that is, a partial order $(P, <)$ with a labeling function $\mu$ that maps the elements of $P$ to labels in an alphabet $\Sigma$. A label list (or word) is ...
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How many sets of vectors can be represented as the solutions of a Horn-SAT instance?

Let the solution space of a SAT instance be the set of Boolean vectors of satisfying assignments of $\{0,1\}$ to the variables (that result in the formula evaluating to TRUE). In other words, a ...
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Show there exists a turing machine for the following language with these properties [on hold]

I'm struggling to understand a question I've been given. The question asks: Let $\psi$ be a boolean formula in $n$ variables. There are $2^n$ different combinations of assigning values to the ...
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1answer
158 views

What is the complexity of decision tree complexity?

Given a boolean function $f$ on $n$ bits, how hard is it to determine its decision tree complexity? (I assume the decision tree is simple, i.e., the allowed questions are the bits of the input.) If ...
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Understanding MA protocol as a variant of TM for small space setting

MA protocol is one of the most basic models of interactive proofs. Merlin is a prover sending a witness $w$ for given input string $x$, and Arthur is a verifier who verifies if $w$ is a positive ...
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Straight line complexity of monomials

Let $k$ be some field. As usual, for an $f\in k[x_{1},x_{2},\ldots,x_{n}]$ we define $L(f)$ to be the straight-line complexity of $f$ over $k$. Let $F$ be the set of monomials of $f$, namely the ...
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How to judge the definition of computational complexity of reals is natural or suitable?

As we know, definition of computational complexity of algorithm is almost without controversy, but the definition of computational complexity of reals or the computation models over reals is not in ...
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How to reduce the computational complexity max algorithm in this specific case

We work over $\mathbb{R}_+^L$. Let $V$ be the set of column vectors whose coordinates take values $0$ or $1$. Thus, $V$ contains $2^L$ vectors. Let $\mathbf{w}(t)$ (in $\mathbb{R}_+^L$) a vector that ...
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Why is HAMILTONIAN CYCLE so different from PERMANENT?

A polynomial $f(x_1,\ldots,x_n)$ is a monotone projection of a polynomial $g(y_1,\ldots,y_m)$ if $m$ = poly$(n)$, and there is an assignment $\pi:\{y_1,\ldots,y_m\}\to\{x_1,\ldots,x_n, 0,1\}$ such ...
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Reducing computational complexity of sorting algorithm in this specific case [duplicate]

We work over $\mathbb{R}^L$. Let $V$ be the set of column vectors whose coordinates take values $0$ or $1$. Thus, $V$ contains $2^L$ vectors. Let $\mathbf{w}(t)$ (in $\mathbb{R}^L$) a vector that ...
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Hitting set of very restricted linear forms

We say that $f\in\mathbb{Z}[x_{1},\dots,x_{n}]$ is a {-1,0,1}-linear form if $f=\sum_{i\in S}x_{i}-\sum_{i\in T}x_{i}$ where $S,T\subseteq[n]$. A hitting set $H\subseteq\mathbb{Z}^{n}$ for ...
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Trying to find polynomial-time algorithms for knapsack-like problems

Consider two related problems: You have n cannisters that must go into m trucks that can each carry k cannisters. You require that no truck becomes overloaded, and for each cannister, there is a ...
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best known space lower bound for SAT?

Following on from a previous question, what are the best current space lower bounds for SAT? With a space lower bound I here mean the number of worktape cells used by a Turing machine which uses ...
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145 views

Partition problem

We know that Partition problem is NP-complete: "Given a multi-set of positive integers like X = < a_1, a_2, ... a_n >, is there any bi-partition for X such that the summation of the numbers in S ...
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Implication of lower bounds in Boolean circuit to other models of computation

Suppose that one can prove that some hard function $f$ with $n$ bit input does not admit any Boolean circuit of size at most $n^t$. Then, how strong can we say about how hard $f$ is in other models? ...
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197 views

Is Kolmogorov complexity a surjective function?

Let us fix an encoding of Turing-machines and a universal Turing-machine, U, that on input (T,x) outputs whatever T outputs on input x (possibly both running forever). Define the Kolmogorov complexity ...
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272 views

Decide the existence of a string homomorphism

Consider the following problem: Given two strings x,y, decide whether there exists a string homomorphism f such that f(x)=y. It is easy to show that this problem is in $NP$. Are there other ...
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intuition that VP=?VNP is (not?) connected to P=?NP

recently there has been major progress into the VP=?VNP problem for algebraic circuits originated by Valiant, inspiring some optimistic outlook on its eventual or imminent resolution.[1] what is ...
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173 views

A curious asymmetry in good characterizations

There are quite a few theorems, mostly in graph theory and combinatorial optimization, that are often referred to as good characterizations. They typically put a property in $NP\cap co-NP$, by showing ...
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How to find a CSP-hard group?

Suppose $G$ is a finite non-abelian group. The conjugator search problem (CSP) is defined as: Find $z\in G$ for given a pair $(x,y=zxz^{-1})\in G^2$. Then, $G$ is said to be CSP-hard if there is no ...
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Which factors make the problem of inferring the grammar difficult?

Scott Aaronson said in the paper entitled "Why Philosophers Should Care About Computational Complexity" (Please see ECCC Report: TR11-108): Following the work of Kearns and Valiant, we now know that ...
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Partial circulant matrices: Is there a non-zero vector $v\in \{-1,0,1\}^n$ such that $Mv=0$?

The following question arose as a side product of some work I have been part of recently. An $m$ by $n$ $(0,1)$-matrix $M$ is partial circulant if it can be formed by taking the first $m$ rows of a ...
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Special case of Bin-packing problem

We know that the decision version of Bin-packing problem is NP-complete: Given an integer B, an integer k, and a list of integers X = (x1, x2, . . . , xn) where xi ∈ [0, B], is there any partition of ...
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Practically Good Algorithms of a Very Low Computational Complexity Class

I am looking for one (or more) examples of a parametric class of algorithms $P_t$ for approximately solving a class $\cal A$ of algorithmic questions with the following properties: 1) Solving the ...
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$NP$-complete problem with quasi-polynomial bound on the number of solutions

FewP is the class of $NP$-problems with polynomial bound on the number of solutions (in the input size). There is no known $NP$-complete problem in $fewP$. I am interested in how far we can stretch ...
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132 views

SAT variant with almost disjoint clauses

I'm wondering whether the following SAT variant is NP-complete or polynomial. The formula given in input has n*m variables, and it has two parts. A part with the n positive clauses: $$(x_{1,1} \lor ...
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91 views

What's the complexity of recognizing equivalence for the following relation?

Consider the set $\mathcal{M}_{m,n}(\mathbb{Z})$ of $m$-by-$n$ matrices over, e.g., integers. We say that two matrices $A$, $B \in \mathcal{M}_{m,n}(\mathbb{Z})$ are equivalent if $A$ can be obtained ...
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118 views

“Largest” class properly contained in PSPACE for a random oracle

Green [1] showed that $PP^{PH}$ is properly contained in $PSPACE$ relative to some oracle. Around the same time, in the famous "voting polynomials" paper [2], it was shown that $PP$ is properly ...
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Pseudorandom generators indistinguishable by uniform deterministic adversaries

I've seen pseudorandom generators defined for nonuniform efficient adversaries, or uniform probabilistic efficient adversaries. (For example, a monograph Pseudorandomness by Vadhan (here's its draft ...
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What are some natural problems that we can quickly find a solution to using massive parallelism but not a canonical solution?

For many problems, more than one output is acceptable. For instance, the problem of finding an assignment that satisfies a boolean formula. If randomness buys us something then it could be that it ...
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Natural NP-complete problems with “large” witnesses

The question on cstheory "What is NP restricted to linear size witnesses?" asks about the class NP restricted to linear size $O(n)$ witnesses, but Are there natural NP-complete problems in which ...
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What is this variant of set cover problem known as?

Input is a universe $U$ and a family of subsets of $U$, say, ${\cal F} \subseteq 2^U$. We assume that the subsets in ${\cal F}$ can cover $U$, i.e., $\bigcup_{E\in {\cal F}}E=U$. An incremental ...
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EXPSPACE-complete problems

I am currently trying to find EXPSPACE-complete problems (mainly to find inspiration for a reduction), and I am surprised by the small number of results coming up. So far, I found these, and I have ...
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Ackermann Function Time Complexity

Are there any known problems that have an Ackermann function time complexity lower bound?
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Are there problems for which divide-and-conquer / recursion is provably useless?

When we try to construct an algorithm for a new problem, divide-and-conquer (using recursion) is one of the first approaches that we try. But in some cases, this approach seems fruitless as the ...
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Can we decide Red-blue cut problem in polynomial time?

Given a directed graph whose arcs are coloured red and blue and integers $r$ and $b$, can we decide in polynomial time whether the digraph has a cut with at most $r$ red arcs and at most $b$ blue ...
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Does ${\bf AC^0PAD} = {\bf PPAD}$?

What happens if we define ${\bf PPAD}$ such that instead of a polytime Turing-machine/polysize circuit, a logspace Turing-machine or an ${\bf AC^0}$ circuit encodes the problem? Recently giving ...
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What is the minimum size of a circuit that computes PARITY?

It is a classic result that every fan-in 2 AND-OR-NOT circuit that computes PARITY from the input variables has size at least $3(n-1)$ and this is sharp. (We define size as the number of AND and OR ...
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Determine the complexity class of a language [closed]

Let $L'$ be the language containing all the pairs $(G,v)$ where $G$ is a directed graph and $v$ is a vertex in $G$ such that $G$ contains a cycle (i.e. closed walk) that contains $v$ and the number of ...
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Statements that imply $\mathbf{P}\neq \mathbf{NP}$

This is sort of an open-ended question - for which I apologize in advance. Are there examples of statements that (seemingly) don't have anything to do with complexity or Turing machines but the ...
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Recognition problem of cycle permutation graphs

A cycle permutation graph is a cubic graph composed from two $n$-cycles each labeled $1, 2,..., n$, with additional edges connecting vertex $i$ in the first cycle to vertex $\pi(i)$ in the second, ...
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Is it possible to solve perfect matching in linear time

As we know matching can be solve in polynomial time. One classical and famous algorithm is designed by Karp and Hopcroft. Is it possible to solve perfect matching problem in linear time for given ...
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What if an $\mathsf L$-complete problem has $\mathsf{NC}^1$ circuits? More generally, what evidence is there against $\mathsf{NC}^1=\mathsf{L}$?

Edit: let me reformulate the question in a more specific way (and change the title accordingly). A slightly edited version of the original question follows. Is there a result comparable to the ...
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Variation on partial Set Cover with penalties

I'm looking at the following problem, which I'm hoping is perhaps a known variant of the Set Cover problem: Input: We're given a universe $U$ and two disjoint subsets $A$ and $B$ such that $A \cup ...
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Is DAG isomorphism NP-C

Given two directed acyclic graphs $G_1$ and $G_2$, is it NP-Complete to find a one-to-one mapping $f:V(G_1) \rightarrow V(G_2)$ such that $(v_i,v_j) \in D(G_1)$ if and only if $(f(v_i),f(v_j)) \in ...
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Is this graph polynomial known? Can it be efficiently computed?

Consider a connected simple graph $G$ with $n$ vertices and $m$ edges. View each edge $\ell$ as a transposition $t_{\ell}$ acting on the set of vertices. [To be more explicit, given an edge $\ell$ ...
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2answers
336 views

Complexity of finding even cuts for a graph

Given a graph $G=(V,E)$, what is known about the classical computational complexity of finding a non-trivial cut which partitions the vertices into two sets $V_a$ and $V_b$ such that every vertex in ...
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Problems in dynamic algorithms in computational geometry

The publication of Chiang and Tamassia's paper on dynamic algorithms in computational geometry included several algorithms used in solving dynamic computational geometry problems such as: Dynamic ...
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Complexity of the coset intersection problem

Given the symmetry group $S_n$ and two subgroups $G, H\leq S_n$, and $\pi\in S_n$, does $G\pi\cap H=\emptyset$ hold? As far as I know, the problem is known as the coset intersection problem. I am ...