Unanswered Questions
738 questions with no upvoted or accepted answers
29
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Does $EXP\neq ZPP$ imply sub-exponential simulation of BPP or NP?
By simulation I mean in the Impaglazzio-Widgerson [IW98] sense, i.e. sub-exponential deterministic simulation which appears correct i.o to every efficient adversary.
I think this is a proof: if $EXP\...
- 21.8k
27
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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 ...
- 5,803
24
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523
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Can we do integer addition in linear time?
Why, yes, of course. But I'm actually interested in the cost of computing the sum of multiple integers:
Input: A sequence of nonnegative integers $\langle X_i:i<k\rangle$ written in binary.
Output: ...
- 18.6k
20
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591
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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 ...
CommunityBot
- 1
19
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Lower bounds on single-source shortest paths in directed graphs
Are there any non-trivial lower bounds on the complexity of single-source shortest paths (SSSP) in a directed graph, where all edges have non-negative edge weights? Can we rule out the possibility of ...
CommunityBot
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19
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1
answer
317
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Is there a geometrical picture for adiabatic quantum computation?
In adiabatic quantum computation (AQC), one encodes the solution to an optimization problem in the ground state of a [problem] Hamiltonian $H_p$. To get to this ground state, you start in an easily ...
- 11.1k
18
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442
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Complexity of the homomorphism problem parameterized by treewidth
The homomorphism problem $\text{Hom}(\mathcal{G}, \mathcal{H})$ for two
classes $\mathcal{G}$ and $\mathcal{H}$ of graphs is defined as follows:
Input: a graph $G$ in $\mathcal{G}$, a graph $H$ in $...
- 9,838
18
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550
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Complexity of the densest $k$-subgraph problem on planar graphs
In the densest $k$-subgraph problem, one is given an undirected graph $G$ and wants to find a set of vertices $N$ with $|N| = k$ such that the number of edges in the subgraph of $G$ induced by $N$ is ...
- 99
17
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610
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Linear-time algorithm to test if clique number equals degeneracy bound?
Given a connected simple graph $G=(V,E)$, let $d$ denote its degeneracy and let $\omega$ denote the size of a maximum clique.
A well-known bound on the clique number is $\omega\le d+1$, which is ...
- 1,166
17
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0
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984
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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 ...
CommunityBot
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17
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494
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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 $s$-$t$ ...
CommunityBot
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16
votes
0
answers
2k
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What is the fastest deterministic algorithm for incremental DAG reachability?
As the title. The incremental algorithm maintains the reachability information of a DAG when it undergoes a series of edge insertions (but no deletions). And the algorithm supports constant query (if ...
- 519
15
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0
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515
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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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15
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346
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Intersecting Complexity Classes with Advice
In on hiding information from an oracle, the authors (Abadi, Feigenbaum, and Kilian) wrote:
$(\mathsf{NP/poly} \cap \mathsf{co\text-NP}{/poly})$ ... is not known to be equal to $(\mathsf{NP}...
- 9,508
15
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1
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615
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Exact Algorithm for edge labeling problem in DAG
I am implementing some system part of which requires some help. I am therefore framing it as a graph problem to make it domain independent.
Problem: We are given directed acyclic graph $G=(V,E)$. ...
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