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64 views

A Simpler Solution for a special case of the Set Cover problem

We decomposed a simple polygon into many small regions. Then we estimated a visibility polygon of a point by a subset of the small regions. Now I need the minimum set of visibility polygons that can ...
1
vote
1answer
61 views

How do we evalute the difference between a predicted value $\hat{v}$ and the true nash equlibrium value $v$

Consider a bimatrix game problem with matrix $A$ and $B$. The definition of the value $v = [v_1, v_2]$ of the Nash equilibrium $(x, y)$ are as follows, $$v_1 = x^TAy,$$ $$v_2 = x^TBy.$$ The situation ...
2
votes
2answers
99 views

Can finite difference methods approximate the space/time complexity of given programs?

While benchmarking a language prototype, I realized that I had a superlinear implementation of a test program, but wasn't sure if it was quadratic or cubic. I stayed up too late and wrote half a page ...
1
vote
1answer
210 views

Is this a novel technique for determining whether or not two rotated rectangles collide?

I was trying to determine whether or not two rectangles rotated around their centers were colliding and randomly thought to try the following algorithm: Rotate both rectangles by the negative rotation ...
1
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0answers
143 views

I think I may have discovered a new algorithm for a common problem. Now what?

I was trying to solve a common problem the other week and upon checking Stackoverflow there was a solution, but it just seemed like there should be a better way to do it. I thought about it for a ...
9
votes
1answer
241 views

Does Horn SAT (Horn formula in CNF) have an integral polytope?

In some ways, my question is related to this: Is the matching polytope integral? Matching and Horn-SAT are both polynomial time solvable.. So I wonder if there is a Horn polytope, similar to the ...
0
votes
1answer
107 views

Complexity for universal Counter Machine with {0,1}-valued registers

Consider a universal $\{0,1\}$-$k$-counter machine where each of the $k$ registers has a value in $\{0,1\}$ (as opposed to any non-negative integer in the usual formulation), and there are states $q_1,...
1
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0answers
30 views

Reference showing global optimality of local minima for matrix factorization

Consider the following matrix factorization problem: Given an $n\times m$ matrix M, find $n\times r$ and $m\times r$ matrices $U$ and $V$ such that $||UV^T - M||_F^2$ is minimized. I have heard it ...
0
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0answers
79 views

Useful primitives CPUs could provide, from TC0 (or NC1)

I just listened to XYZ talking about "How Universal Is the Idea of Numbers?", and bashing the concept as an accidental historical artifact. He suggested that totally different computational ...
1
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0answers
41 views

Is there a measure of string distance which remains small between two strings after encryption?

I have a theoretical problem inspired by a real world problem I have come across. I am writing code for some spreadsheets, and part of the process is error correcting names with typos. One ...
-1
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2answers
281 views

Partition a graph into two clusters

Suppose given a complete weighted graph $G=(V,E)$, with positive weight. Are there an algorithm that partition $G$ into two clusters $C_1,C_2$ such that sum of heaviest edges in $C_1,C_2$ minimized? ...
2
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1answer
137 views

Does a finite, polynomially-bounded CFG translate into a polynomially-bounded DFA?

We are given a family of context-free grammars $\{ G_1, G_2, G_3, ..., G_n, ...\}$ where each $G_n$ generates strings only of length $n$ and obeys other constraints specified below. We want to study ...
5
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1answer
222 views

Computational complexity of Turán-type problems

According to Turán's theorem (with $r=n/2$), any graph $G$ with $n$ vertices and at least $n(n-2)/2$ edges must contain a clique of size $n/2+1$. My question is: how hard is it to find this clique?$^*$...
4
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0answers
50 views

Coinduction principle for smash products of pointed cpos

In "Relational Properties of Domains", Pitts gives a coinduction principle for pointed cpos (cppos). In corollary 6.13 (below), he specializes it to cppos constructed as fixed points of cppo-...
2
votes
1answer
214 views

3SAT to 1-in-3SAT reduction with additonal constraints

The simplest Reduction for 3-SAT to 1-in-3-SAT reduction is as follows: For each 3SAT clause: $x+y+z=1$ Introduce 4 new variables $\{a, b, c, d\}$ and replace original clause with below 3 clauses: $R(...
2
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0answers
92 views

How typical are odd-H-minor free graphs?

Can anything be said about how typical are odd-H-minor free graphs? (definition of odd-minor-free is in Section 2.2 of notes, page 20 of slides). For instance, for a random graph with $n$ vertices, $...
0
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0answers
18 views

Fast filter for concurrent version vectors (vector clocks, lamport clocks) from a set

I have a set of version vectors (or vector clocks) and need to implement a filter function that takes this set and returns a new set where all elements are either "equal", "happened ...
1
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0answers
63 views

Smallest nonzero eigenvalue of a sum of +1/-1 rank-one matrices?

Suppose we have a $k\times k$ matrix $A = \sum_{i=1}^{n} a_i a_i^T$ where $n \leq \mathrm{poly}(k)$ and each $a_i\in\{-1,1\}^{k}$. It is easy to prove that the largest eigenvalue of $A$ is at most $\...
0
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0answers
104 views

Does this sum-product algorithm make more sense?

Recall the belief propagation "sum-product" algorithm (from wikipedia): $\forall x_v\in Dom(v),\;$ $\mu_{v \to a} (x_v) = \prod_{a^* \in N(v)\setminus\{a\} } \mu_{a^* \to v} (x_v)$ $ \mu_{...
0
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0answers
47 views

Functor Image example in Roman’s CT book

In his book, An Introduction to the Language of Category Theory,Steven Roman provide us with the following diagram: Then, he states: Note that in this example, the object part of F is not injective, ...
1
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0answers
82 views

Is there some n such that lambda calculus with only n variables is Turing-complete?

Typically in lambda calculus you have an infinite stock of variables. Could we get away with a finite set?
0
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0answers
131 views

Is this "choose and switch" game just 2SAT in disguise?

Suppose you have $n$ switches lined up on a wall, labeled left to right from $1$ to $n$. The switches are either on or off. You get a list of $q$ choices that must be made. A choice consist of two ...
2
votes
2answers
304 views

Some issues with proof of Fundamental Theorem of Statistical learning

I am reading the book "Understanding Machine Learning" by Shai Shalev-Shwartz and Shai Ben-David. The theorem 6.7 has several equivalent statements for a class of functions $H$. The first ...
0
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0answers
21 views

Going from one base packing to another using basis exchanges

Suppose I have a matroid $M = (E, \mathcal{I})$. It is a known fact that given any two bases $X_0$ and $X_n$, we can transform $X_0$ into $X_n$ by repeatedly applying the basis exchange axiom. So ...
0
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1answer
122 views

Finding top-K items in a sliding window

Imagine we have a stream of bank transactions. Each transaction has a target account and some amount of money. I'd like to find top K accounts over some period of time (e.g. last 7 days) which ...
2
votes
2answers
126 views

Information and Coding Theory Texts

I am coming from a pure mathematics (in analysis) background and am curious to learn some information and coding theory. I am after some recommendations on texts. Due to my personal background I am ...
3
votes
1answer
81 views

Recursive generic oracles

In Fenner, Stephen; Fortnow, Lance; Kurtz, Stuart A.; Li, Lide, An oracle builder’s toolkit, Inf. Comput. 182, No. 2, 95-136 (2003). ZBL1025.68041, the authors go through a variety of generic oracles. ...
6
votes
5answers
320 views

Relationship between Random Graph Theory and TCS

Sorry for this large and vague question. I am a new grad probability student recently interested in random graph theory(RG). I heard from someone in math department that RG has close relationship to ...
0
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1answer
38 views

Are perceptual hashes connected to audio/video compression?

Without loss of generality, I'll only talk about video, but this should apply to any sort of signal. A perceptual hash function (WP) maps videos to fingerprints such that each fingerprint's preimage ...
0
votes
1answer
127 views

What is a known sequence for which being constant is not provable?

My question concerns the property of being constant for computable functions ${\mathbb N}\to \{0,1\}$, within any common framework $T$ strong enough to include Heyting arithmetic (and of course not ...
1
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0answers
88 views

Input length and calculation time to simulate a quantum measurement

Let us consider $n$ quits $b_i$. Let us start from the state $|0,0,...,0>$ and apply a circuit $C$ composed by $m$ quantum gates, with $m$ polynomial in $n$. The final state is $C|0,0,...,0>$. ...
1
vote
1answer
108 views

Lower bound for proving a random 3-SAT formula is unsat?

For a random 3-CNF formula with n variables and m clauses, assume this formula is unsat, what is the lower bound for proving it to be unsat? Some results posted in Lower bounds for random 3-SAT via ...
1
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0answers
148 views

Is polynomial-time the same in all classical computational models?

There are many models of computability, all giving the same notion of 'computable function'. To pick a few examples: Turing machines (with variants: one-ended, two-ended, multiple tapes...) RAM ...
1
vote
1answer
89 views

Are coproduct types redundant in presence of natural numbers and $\Sigma$-types?

In the homotopy type theory book section A.2.5 defines $\Sigma$-types, A.2.6 coproduct types and A.2.9 the natural numbers type. If we already have $\Sigma$-types and the natural numbers type can we ...
6
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0answers
167 views

Fastest exact algorithm for MAXCUT

Is the algorithm introduced in the following paper still the fastest exact algorithm for general MAXCUT problems? TIA Ryan Williams, A new algorithm for optimal $2$-constraint satisfaction and its ...
7
votes
1answer
195 views

What is the complexity class of higher-order primitive recursion?

Short and sweet: The complexity class of primitive recursive functions (WP) is PR (WP, Zoo). What's the complexity class of higher-order primitive recursion (WP)? The motivating context is simply that ...
5
votes
2answers
256 views

Complexity of "can we get a cycle by stacking directed bipartite graphs?"

Preliminaries We consider directed bipartite graphs of the form $G = (V,V',E)$, in which the nodes are partitioned into $V = \{1,\ldots,n\}$ and $V'=\{1',\ldots,n'\}$, with $|V|=|V'|=n$, and $E\...
0
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0answers
54 views

Number of composite factors as a function of the number of bits of an integer

Is there a standard formula to calculate the number of composite factors using the number of bits of an integer?
0
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0answers
45 views

Using bin-packing algorithms to approximate maximum-makespan

Bin-packing (BP) and maximum-makespan (MM) are dual problems. In both problems, the input can be defined as a set $S$ of positive integers, and the output is a partition of $S$. In BP, there is a ...
15
votes
6answers
5k views

Can theoretical computer science be applied in social sciences?

I'm very new to this field - technically not in it but want to be. I'm very early in my academic career (sophomore at a community college) but decided that I want to add a math major along with my ...
1
vote
1answer
136 views

Is QMA known to contain Co-NP?

Is QMA known to contain Co-NP? If not, would Co-NP being contained in QMA have any implications for other complexity classes. (e.g. Causing the polynomial heirachy to collapse.)
2
votes
1answer
170 views

A variant of two-counter machine

I would like to show that the halting problem for some variant of two counter machine (Minsky machine) is undecidable: instead of "if c=0 goto i else goto j", there are "if c>d goto ...
0
votes
1answer
168 views

Lexicographic Boolean satisfiability

Maximal Satisfying Assignment (Lexicographic Boolean satisfiability / LexMaxSAT), the problem of finding the lexicographically maximum x_1, . . . , x_n ∈ {0, 1}^n that satisfies Boolean formula φ, or ...
0
votes
0answers
34 views

Bin Packing And Preemptive Multi-Core Scheduling

I am trying to solve a preemptive multi-core scheduling problem, where the input is the tasks. The number of cores M can be decided after seeing the input tasks. I ...
1
vote
1answer
98 views

Partition the edges of a bipartite graph into perfect $b$-matchings

Any $r$-regular bipartite graph can be partitioned into $r$ disjoint perfect matchings. I want to know whether a version of this extends to perfect $b$-matchings. Suppose we have a bipartite graph $G =...
0
votes
0answers
104 views

Questions about the equivalence between PH and depth-d circuits with respect to an oracle

Consider an oracle $A$ and the language \begin{equation} P(A) = \{x\in \{0, 1\}^{n}: \text{the number of strings in }~A~\text{of length}~|x|~\text{is odd}\}. \end{equation} I am trying to make sense ...
6
votes
1answer
231 views

What is the impact of encodings of sparse structures on the complexity of the model checking problem?

Some preliminaries first. Consider a purely-relational structure (a.k.a. database) $\mathfrak{A} = (A, R_1^{\mathfrak{A}}, \ldots, R_{|\tau|}^{\mathfrak{A}})$ over some finite signature $\tau = \{ R_1,...
7
votes
0answers
165 views

$\lambda$-definability and structure preserved by homomorphisms

I imagine there are some standard results that bear on this, but I'm having trouble finding a proof or refutation of it. Some prelimary definitions. A Henkin structure $A = (A^\cdot, ⟦\cdot⟧_A)$ for ...
4
votes
1answer
114 views

What is tightest known (VC-style) sample complexity bound for uniform convergence of empirical means?

The following result is adapted from Anthony and Bartlett, 1999 (Theorem 4.9). Theorem There exist positive constants $m_0 \le 400$, $c_1 \le 8$, $c_2 \le 41$, $c_3 \ge 1/576$ such that, if $(\Omega,\...
2
votes
0answers
84 views

Resource on phase transition in MAXCUT problems

Could anyone suggest reading materials on phase transition in MAXCUT problems other than [1]? Thanks. Ref: Coppersmith, Don, David Gamarnik, Mohammad Taghi Hajiaghayi, and Gregory B. Sorkin. "...

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