# Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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### The complexity of checking whether two DAG have the same number of topological sorts

This problem is highly related to the CNF one. Here is the problem: given two DAG (directed acyclic graphs), if they have the same counting of topological sorts, answer "Yes", otherwise, answer "No". ...
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### Is Hankelability NP-hard?

I asked this question on SO on April 7 and added a bounty which has now expired but no poly time solution has been found yet. I am trying to write code to detect if a matrix is a permutation of a ...
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### What are consequences of the collapse of CH?

I don't grasp the full complexity of the counting hierarchy $CH$. I understand $CH$ is in $PSPACE$, and contains $PH$ within its second level, due to the Toda's theorem. But, what would be important ...
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### Counting Isomorphism Types of Graphs

Polya's counting theorem leads to an algorithm for counting (precisely) the number of isomorphism types of graphs with $n$ vertices in $\exp (\sqrt n )$ steps. From Polya theorem you get a formula ...
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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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### What is the power of general poly-size permutation branching programs?

Call $\mathsf{PPBP}$ the class of languages decided by poly-size families of permutation branching programs, which are layered branching programs (i.e., the ones defined here) whose transitions ...
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### $RL=L$ Progress Since 2006

Reingold, Trevisan, and Vadhan's breakthrough 2006 paper (http://dl.acm.org/citation.cfm?id=1132583) reduced the problem of showing $RL=L$ to producing pseudorandom walks on regular digraphs that are ...
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### Descriptive complexity characterization of TimeSpace classes

Are there descriptive complexity characterizations for TimeSpace complexity classes like $\mathsf{SC^i}= \mathsf{DTimeSpace}(n^{O(1)},O(\lg^i n))$?
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### Interesting PCP characterization of classes smaller than P?

The PCP theorem, $\mathsf{NP} = \mathsf{PCP}(\mathsf{log}\, n, 1)$, involves probabilistically checkable proofs with polynomial time verifiers, so the smallest class that can be characterized in this ...
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### Complexity of finding the smallest well-covered completion

This is related to an earlier question on which graphs have the property that all maximal independent sets are maximum — such graphs turn out to be known as the well-covered graphs. Any graph $G$ is ...
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### Complexity of interval cover problem

Consider the following problem $Q$: We are given an integer $n$, and $k$ intervals $[l_i,r_i]$ with $1\leq l_i\leq r_i\leq 2n$. We are also given $2n$ integers $d_1,…,d_{2n}\geq 0$. The task is to ...
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### An algebra of complexity classes

A key feature of unrelativized computation is its composability out of smaller fragments, and to partially capture the composability, I came up with an algebra of fine-grained complexity classes. For ...
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### Intermediate problems between PSPACE and EXPTIME

Intermediate problems between P and NP are quite famous, and are sometimes considered as complexity classes by themselves. Do you know of any problem that is known to be PSPACE-hard and in EXPTIME, ...
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### Deeper look at Algorithmica?

Russell Impagliazzo published "A Personal View of Average-Case Complexity" (preprint) back in 1995. He presented five possible worlds we could be living in, depending on how P and NP were related. The ...
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### Complexity of approximating the range of a matrix

Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define: $$S_M = |\{Mx : x \in \{-1,1\}^n\}|.$$ I believe that it is NP-hard to compute $S_M$ exactly, by applying ...
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### What is the evidence for average case separation between EXP and NEXP?

There is significant evidence from cryptography that there exist NP-complete problems that are hard in the average case (meaning that e.g. $AvgP \nsupseteq DistNP$). Namely, we have candidate one-way ...
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### Can short-distance connectivity be harder than connectivity?

Has anybody seen the following (or similar) question being considered: Can it be easier to determine the presence/absence of $s$-$t$ paths than to determine the presence/absence of short $s$-$t$ ...
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### Is graph coloring complete for poly-APX?

Is the graph coloring problem complete for poly-APX under C-reductions (alternatively, under AP-reductions)? For the graph coloring problem, speaking of a feasible solution means a proper coloring for ...
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### Is Node Multiway Cut NP-complete on planar graphs when all terminals lie on the outer face?

I am interested in the following problem. Node Multiway Cut on Planar Graphs with terminals on the outer face Instance: A plane graph G, and integer k, and a set $S \subseteq V(G)$ of terminals ...
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### Williams' Method, Natural Proofs and Constructivity

I have some questions on the previous question which is written bellow. Natural Proof and Constructivity : The topic of the previous question Recently, Ryan Williams proved that Constructivity in ...
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### First use of sans serif for complexity classes

(Apologies for the triviality of this question; nevertheless, it's been bugging me and presumably people here will be able to answer it...) It seems that it has become popular in recent years to ...
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### The best known upper bound for two-way probabilistic finite automata with one-counter

It is known that the class of languages recognized by two-way deterministic finite automata with one-counter (2D1CAs) is a proper subset of $\mathsf{L}$ (deterministic log-space): A 2D1CA can run at ...
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### AC0 many-one reduction of Mod_3 to PRIMES?

Let Mod$_3$ be the language of binary strings with the sum of the bits divisible by 3, and PRIMES be the set of prime integers. In a 2001 paper A Lower Bound for Primality, Allender, Saks, and ...
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### DPLL and Lovász Local Lemma

Let $\varphi$ be a CNF formula. Suppose that each of $\varphi$'s clauses consist of exactly $t$ literals (and, moreover, all literals within one particular clause correspond to different variables). ...
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### Question on Products of Graphs

Let $S_{n}(G)$, $C_{n}(G)$, $T_{n}(G)$ be the $n$-fold Strong Product, Cartesian Product and Tensor Product of a graph $G$ on $|V|$ vertices. Let the chromatic number ($\chi(G)$) and the independence ...
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### Phase Transitions in NP Hard Problems

SAT Problems have a phase transition that depends on the ratio $r$ of variables to clauses. Below $r$, SAT problems are solvable quickly; above, they become difficult. The same is true of NP ...
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### Linear PRAM vs Arithmetic Linear PRAM

A linear PRAM model is a PRAM model without bit operations and at least one operand of the $\times$ instruction is a constant. If in addition we require that the running time does not depend on the ...
Assume we want to refute an unsatisfiable CNF. We can interpret it as an integer program, thus a refutation can be done by applying Lovasz-Schrijver semidefinite cuts ($LS_{+}$ cuts) to its linear ...