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3
votes
0answers
70 views

Embed graph in $\ell_2$ space so that edge and non-edge distances are separated by a constant factor

Suppose I have an undirected unweighted graph $G = (V,E)$. Is there a way to compute points $x_v \in \mathbb{R}^d$ for each vertex $v \in V$ such that $||x_v - x_u|| = 1$ whenever $(u,v) \in E$ and $ |...
22
votes
2answers
531 views

Question about two matrices: Hadamard v. “the magical one” in the proof of the sensitivity conjecture

The recent and incredibly slick proof of the sensitivity conjecture relies on the explicit* construction of a matrix $A_n\in\{-1,0,1\}^{2^n\times 2^n}$, defined recursively as follows: $$A_1 = \begin{...
-1
votes
0answers
33 views

Inclusion problem for strictly non-deterministic CFGs

We proved in a section class the undecidability of the inclusion problem for $\mathsf{CFGs}$––that is, given two $\mathsf{CFGs}$ $G_1$ and $G_2$, is $L(G_1) \subseteq L(G_2)$––by proving that the ...
-2
votes
0answers
59 views

Is there an algorithm similar to Stoer–Wagner mincut algorithm?

Suppose I want to find the minimum weight cut in a weighted undirected graph. I see Stoer–Wagner algorithm mincut algorithm does help to find the minimum cut. Is there an algorithm similar to Stoer–...
8
votes
0answers
119 views

Descriptive Complexity characterzation of BPP

We know of descriptive complexity characterizations of classes such as P, and NP, which use First Order logic, and operators. Does BPP have a characterization under descriptive complexity, too(any ...
-3
votes
0answers
90 views

$SAT$ solvability via strongly polynomial time simplex algorithm?

Are there classes of $SAT$ problems cast as integer programming solvable by parametrized strongly polynomial time simplex algorithm versions such as https://www.tandfonline.com/doi/full/10.1080/...
2
votes
0answers
81 views

Is equivalence of uniform AC0 decidable?

Is there a representation of the functions from $\mathsf{DLogTime}$-uniform $\mathsf{AC}^0$ for which equivalence is decidable? Since the languages defined by nondeterministic pushdown automata ...
11
votes
0answers
301 views

A dynamic data structure to list triangles

Consider an undirected graph with $n$ nodes. Is there an efficient data structure that supports the following operations? Insert an edge into the graph Delete an edge from the graph Given a query ...
5
votes
1answer
129 views

Is sorting pairwise distances as hard as sorting arbitrary points?

If we have $n$ points in $\mathbb{R^d}$, what is the complexity of sorting the $O(n^2)$ pairwise distances? Clearly the complexity is $\Omega(n^2)$ but is there a reduction to show it is as hard as ...
1
vote
0answers
77 views

Switching lemma for polynomials over $\mathbb{F}_2$

Suppose $f$ is in $\mathbb{F}_2[x_1,...,x_n]$ with total degree $d$. Q. Is there any kind of switching lemma or restriction lemma in which by applying the lemma on $f$ we can reduce the total ...
0
votes
1answer
80 views

Chain rule for KL divergence

Is there an inequality to relate the KL divergence of two joint distribution and the sum of the KL divergence of their marginals? Or in particular, is there a proof or a counter example for the ...
2
votes
0answers
61 views

reduction from SAT to approximate set cover

I read this neat result proved in the early 90s: For any $c>1$, There's a poly time map from boolean formulas $\varphi$ to pairs $K, \mathcal S$ where $K$ is a positive integer and $\mathcal S$ ...
6
votes
1answer
162 views

Big-O bounds on the k-th largest element of iid Gaussians

I'm interested in the following problem. Let $X_1, \dots, X_n$ be iid samples with a $N(0,1)$ distribution. Let $X_{[k]}$ be the $k$-th largest element of $\{X_1, \dots, X_n\}$, so e.g. $X_{[1]} = \...
0
votes
1answer
61 views

Grover's algorithm, M out of N, when M is large

The more general version of Grover's algorithm searches for one of $M$ entries that match a criterion, out of $N$ total entries. I have seen it written that this takes $O(\sqrt{N/M})$ iterations, to ...
2
votes
0answers
81 views

Which computational framework lies behind the Chinese “Social Credit System”?

BACKGROUND The Social Credit System is a data-driven reputation system which draws on several sources to label various entities, namely businesses and individual citizens, with a trustworthiness ...
6
votes
1answer
147 views

complexity of deciding whether there's a small polynomial with a given root

Let $f\in (\mathbb{Z}/p\mathbb{Z})^\ast$ be a nonzero element of a prime finite field. For $d, r\in \mathbb{N}$ consider the problem of deciding whether there is a nonzero polynomial $$P(x) = a_0 +...
2
votes
0answers
66 views

What is the communication complexity of approximating addition?

In my circuit complexity research, I came across the need to find the communication complexity of approximating addition. Specifically, the class of problems I am interested in is parametrized by four ...
2
votes
1answer
84 views

Placing color boxes on a colored image such that color consistency is maximized

I have encountered the following challenging problem that I think to be a non straightforward generalization of the Knapsack problem. Given an image with black background that contains blobs whose ...
-1
votes
0answers
46 views

What are constraints for two k-regular graphs to be isomorphic?

As discussed in the answer for this question, it is not necessary that two k-regular graphs are always isomorphic to each other. However, 2-regular graphs are isomorphic. So my question is if there ...
0
votes
0answers
62 views

Unknown gaps in computation models

I'm looking for computatuon models where it is known that there are problems that we can solve in time T1 and T2. where T1 is smaller then T2 and it is unknown if there are problems where their ...
-4
votes
0answers
145 views

Quadratic Time Algorithim for Sudoku Latin Squares? Any special case of interest involved if this is some form of matrix multiplication?

I've just started taking a serious interest in reading ground material lately, and I'm probably asking this question at the wrong time. So let me know and I'll delete it. I just can't for this ...
0
votes
1answer
71 views

finding maximum weight subgraph

My graph is as follows: I need to find a maximum weight subgraph. The problem is as follows: There are n Vectex clusters, and in every Vextex cluster, there are some vertexes. For two vertexes in ...
0
votes
1answer
79 views

How to build comparison operator (comparator) in an arithmetic circuit

I am trying to convert a basic program into an arithmetic circuit. I am stuck on the step of converting the greater than operator into an arithmetic circuit. To be specific, I do not know how to ...
2
votes
1answer
80 views

Upper bounds on the circut depth

Suppose $f:\{0,1\}^n \to \{0,1\}$ is a function such that it can be computed by a circuit of size $n^c$ for some constant $c>0$. Q. Is there any nontrivial upper bound on the depth of a circuit ...
-1
votes
0answers
37 views

Maximization under constraints

I have a set of $m$ sets, each one has $n$ different items. I have a function $f: 2^n \to \mathbb{R}$. ($f$ could be submodular if it helps). I am trying to maximize the function $f$ under the ...
33
votes
10answers
5k views

Most important new papers in computational complexity

We often hear about classic research and publications in the field of computational complexity (Turing, Cook, Karp, Hartmanis, Razborov etc). I was wondering if there are recently published papers ...
2
votes
0answers
30 views

“Planar graph coloring is not self-reducible” is this about all $p$-relations encoding that problem?

I have a question about the paper "Planar graph coloring is not self-reducible" by Samir Khuller and Vijay Vazirani. The final theorem in that paper states that "Planar Graph k-coloring is not self ...
3
votes
0answers
50 views

Algorithms for Maximum weight connected subgraph in planar graphs

I wonder what is known about the two following maximisation problems. Maximum weight connected subgraph : Input : A graph $G$, with weights $w_v\in \mathbb{R}$ for each vertex $v \in V(G)$ Output :...
0
votes
1answer
148 views

Language in $PSPACE$ and not necessarily in $P$ if $P=PP$?

If $P=PP$ then the counting hierarchy collapses to $CH=P$. Because so many complexity classes are contained in $CH$, this causes most classes to now be contained in $P$. My question is whether this is ...
-1
votes
0answers
23 views

Compression/Minimum Description Length of a Size $n$ List of Rational Numbers

Suppose we have some list of rational numbers where the order in which the numbers are listed is arbitrary. What is the representation of this list which is most compact (i.e. "smallest amount of ...
3
votes
0answers
58 views

Example of a hardness-of-approximation proof which improves the approximation factor?

Are there any examples of hardness-of-approximation reductions where we get a better hardness bound for the problem we've reduced to than the problem we've reduced from? In the examples I've seen so ...
3
votes
0answers
104 views

Examples of ineffective PH collapse theorems?

What are some examples of theorems of the form "If $X$, then $PH$ collapses", but where the proof does not say which level $PH$ collapses to? The canonical example would be "If $PH = PSPACE$, then $...
0
votes
0answers
57 views

Lower bound for permutation generator

I'm interested in a problem akin to combinatorial circuits, but in terms of complexity. Apologies for missing the correct terminology, I'll appreciate any corrections. Given $n$ inputs numbered $1 ......
0
votes
0answers
39 views

Intersection of two deterministic parity automata

Given two deterministic parity automata $A_1=(Q_1,\Sigma,\delta,q_{01},c_1)$ and $A_2=(Q_2,\Sigma,\delta,q_{02},c_2)$ with the finite set of states $Q_i$, the finite alphabet $\Sigma_i$, the ...
0
votes
0answers
19 views

Algorithmic gap for greedy algorithm for (metric) uncapacitated facility location

In Jain et. al (2003), at the bottom of page 801, they construct an instance of (metric) uncapacitated facility location for which they claim the greedy (Hochbaum's) algorithm has gap $\Omega(\frac{\...
0
votes
1answer
128 views

Is it possible to prove that a general purpose integer factorization algorithm must contain a loop?

1) Let $A$ be a (general purpose) algorithm that factors $n$. Suppose $A$ contains a loop (which is hard to imagine if not impossible that it does not.) If $A$ contains nested loops then these loops ...
5
votes
1answer
112 views

Complexity of Finding Largest Set of Intersecting Convex Polytopes

I have a set of $n$ convex polytopes, and I wish to find the largest subset of those polytopes that shares at least one point in common. I think that this problem should be NP-hard, but I am ...
4
votes
0answers
46 views

$\ell_\infty$ partially enclosing ball problem

Suppose I have $n$ points in $\mathbb{R}^d$ endowed with the $\ell_\infty$ metric, and I wish to find a minimum-diameter ball that contains some $k$ of these points. What is known about this problem? ...
5
votes
0answers
43 views

Hard family for degree-$D$ MAX-3LIN

Max 3LIN is the problem where one is given a linear system of equations $C$ over $\mathbb{F}_2$ with $3$ variables per equation, and needs to determine the maximum number of equations that can be ...
0
votes
0answers
26 views

Two question regarding coreset construictions

I have two questions regarding coreset construction of clustering problem In A Unified Framework for Approximating and Clustering Data, a very general framework is given to construct coresets for ...
0
votes
0answers
55 views

Linear time algorithm for projective clustering

There is a lot of work in clustering of high dimensional data. In case of k-means, it is shown here that one can get an $(1+\epsilon)$-approximation in linear time, yielding a PTAS, by random sampling....
0
votes
0answers
67 views

What is sequence unification?

And why is it interesting? Please provide some examples. This text is here to circumvent the anti-spam filter which thinks my question is bad.
0
votes
0answers
109 views

Oracle separation between PH and PSPACE

I am having difficulty understanding the concept and intuition behind this proof. The proof deals with constructing an oracle $A$ relative to which $PH$ is separated from $PSPACE$. I have several ...
0
votes
0answers
83 views

Is it possible to sort by only knowing the sign of pairwise sums?

I am currently thinking of how much structure one actually needs in order to be able to sort things at all. All comparison-based algorithms need a direct comparability, but are we able to remove this ...
4
votes
0answers
85 views

Where in $PH$ are these problems?

Is 'Given two codes with alphabet in $\mathbb F_2$ with Generator matrices $G_1$ and $G_2$ do they have the same minimum distance?' in $NP$ or is it in $coNP$ (I can see it in $P^{NP}$)? If $G_1$ is ...
3
votes
2answers
187 views

How do continuations represent negations (under the Curry–Howard correspondence)?

Under the Curry–Howard correspondence, types can be thought of as propositions, and values inhabiting a type can be thought of as proofs that the corresponding proposition is true. (E.g., the ...
2
votes
0answers
81 views

Is $MSB$ of permanent and certifying half number of witnesses easy?

Can there be a $P$ algorithm to decide if number of perfect matchings is at least $(n!/2)+1$ for a bipartite graph on $n+n$ vertices? Can there be a $P$ algorithm to decide if number of witnesses ...
12
votes
1answer
370 views

Deterministic error reduction, state-of-the-art?

Assume one has a randomized (BPP) algorithm $A$ using $r$ bits of randomness. Natural ways to amplify its probability of success to $1-\delta$, for any chosen $\delta>0$, are Independent runs + ...
2
votes
0answers
67 views

Looking for Research in Cryptographic Computing

I recall reading in college about a nascent research area regarding cryptographic techniques for secure computing, relating to zero-knowledge proofs, but I am having trouble remembering the exact term....
0
votes
1answer
81 views

Partially persistent linked list data structure: would lookup of the first element at a specific version be O(|versions|) and not O(1)?

I'm following course material from the course Advanced Data Structures. The result by Driscoll et al 1989 states the following (wording of the following theorem taken from lec notes, page 4, which ...

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