All Questions

38
votes
0answers
1k views

Is there a regular tree language in which the average height of a tree of size $n$ is neither $\Theta(n)$ nor $\Theta(\sqrt{n})$?

We define a regular tree language as in the book TATA: It is the set of trees accepted by a non-deterministic finite tree automaton (Chapter 1) or, equivalently, the set of trees generated by a ...
31
votes
0answers
412 views

Finding an $st$-path in a planar graph which is adjacent to the fewest number of faces

I am curious whether the following problems has been studied before, but wasn't able to find any papers about it: Given a planar graph $G$, and two vertices $s$ and $t$, find an $s$-$t$ path $P$ ...
26
votes
0answers
729 views

Graph problem known to be $NP$-complete only under Cook reduction

The theory of NP-completeness was initially built on Cook (polynomial-time Turing) reductions. Later, Karp introduced polynomial-time many-to-one reductions. A Cook reduction is more powerful than a ...
21
votes
0answers
448 views

Approximate minimum-weighted tree decomposition on complete graphs

Say I have a weighted undirected complete graph $G = (V, E)$. Each edge $e = (u, v, w)$ is assigned with a positive weight $w$. I want to calculate the minimum-weighted $(d, h)$-tree-decomposition. By ...
20
votes
0answers
546 views

Can a calculus have incremental copying and closed scopes?

A few days ago, I proposed the Abstract Calculus, a minimal untyped language that is very similar to the Lambda Calculus, except for the main difference that substitutions are ...
20
votes
0answers
476 views

Largest set of cocircular points

Given $n$ points with integer coordinates in the plane, determine the maximum number of points that lie on the same circle (on its circumference, not its interior). This can be done in $O(n^3)$ ...
17
votes
0answers
514 views

Complexity of deciding whether there is a winning strategy in the following game

The sum divider game for $n$ starts with the set $M_0 = \{1,\dots,n\}$. Player A chooses a number $m_1$ from $M_0 \setminus \{1\}$ and B has to choose a divider $m_2$ of $m_1$ from $M_1 = M_0 \...
16
votes
0answers
310 views

Is finding a weight-balanced tree NP-hard?

In the following, we consider binary trees where only the leaves have weights. Let $T$ be a binary tree and $W(T)$ be the sum of the weights of its leaves. Let $T.l$ and $T.r$ be the left child and ...
14
votes
0answers
186 views

Using logic to prove non-regularity of a language

A language $L$ is regular if and only if it is definiable by a sentence in monadic second order logic (MSO) over strings (J.R. Buchi, Weak second-order arithmetic and Finite automata; Z. Math. Logik ...
13
votes
0answers
295 views

Steps that guarantee exiting a maze

Given a 2-dimensional maze where you can give 4 commands "move up/down/right/left". Knowing the maze but not where the person is, how to find the minimum sequence of commands that guarantees exiting ...
13
votes
0answers
179 views

Why do we have to forbid non-conforming lower and upper type bounds?

(it's a repost of my unanswered question from scala-user@googlegroups.com about Scala) In the Scala Language Specification, §4.4 Type Parameters, there is a requirement: The most general form of a ...
12
votes
0answers
210 views

Flow graph that requires pushing back flow in Ford Fulkerson

Does there exist a flow graph that always requires flow to be pushed back no matter what ordering of augmenting paths is chosen in Ford Fulkerson? Let's assume we use the standard procedure of ...
12
votes
0answers
363 views

Choosing a subset of binary variables to maximize the sum of the highest $K$

Consider the following problem: Input: integers $n > m > k$; $n$ numbers $0 \leq p_1, \ldots, p_n \leq 1$; $n$ numbers $r_1, \ldots, r_n$ where ($r_i \geq 0$). Let $X_1,\dots,X_n$ be $n$ ...
12
votes
0answers
501 views

Test whether two languages are equal, when give in algebraic form

This sub-problem is motivated by Algorithm to test whether a language is regular. Suppose we have two languages $L_1,L_2$ that are expressed in "algebraic" form, as formalized below. I want to ...
11
votes
0answers
475 views

Optimal meeting point in directed graph

Let $G(V, E)$ be a edge-weighted directed connected graph and $v_1, \dots, v_n \in V$ be some vertices. Let $d(a, b)$ denote the length of the shortest path from $a$ to $b$, for $a,b \in V$. I need ...
11
votes
0answers
850 views

Alternative to Bloom filter for extreme parameters

A Bloom filter is a space-efficient probabilistic data structure to perform membership-tests on a set (see Wikipedia's page for a definition; I use the same notations below). I am interested in a ...
11
votes
0answers
282 views

How fast can we compute the size of maximum matching in an unweighted bipartite graph?

Is there a way to compute the size of a maximum matching in an unweighted bipartite graph more efficiently (e.g. faster) than computing a maximum matching? It is a long shot but it is often an ...
11
votes
0answers
241 views

Change in the distances in a graph after removal of a node

Given an undirected unweighted graph $G=(V,E)$ and a node $s \in V$, we are looking for a vector $\operatorname{diff}[]$, such that, $$\operatorname{diff}[v] = \sum_{u \in V \setminus \{v\}}{(d^{G \...
11
votes
0answers
306 views

Proof of PCP theorem

I am reading the proof of PCP theorem in Proof Verication and Hardness of Approximation Problems. The following paragraph appears in section 3 (page 4), "Outline of the Proof of the Main Theorem". ...
11
votes
0answers
661 views

Fast algorithm for max-convolution with concave functions?

I'm interested in a discrete max-convolution problem, which is to compute $$r(c) = \max_{x | x \ge 0, \sum_k x_k = c} \left[ \sum_{k=1} f_k(x_k) \right] $$ for all values $c=0, \ldots, C$, where $x=(...
10
votes
0answers
158 views

How would a CPU designed purely for functional programming be different?

CPU's are to an extent designed with in mind the software that people will write for it, implicitly or explicitly. It seems to me that if you look at the design of instruction set architectures, ...
10
votes
0answers
271 views

Is Agda sound as a proof system?

I was browsing Agda's stdlib source code, since I was trying to get into it seriously and therefore wanted to know more. I was amazed at that Agda is way more developed than I thought and it's ...
10
votes
0answers
123 views

When can you “invert” an equation in the lambda calculus

Suppose that $M$ is a full model of the simply typed lambda calculus. Suppose each base type is infinite. Now suppose that $f$ and $g$ are two functions in $M$ (not necessarily in the same domain) ...
10
votes
0answers
205 views

Minimum edge deletion partitioning of a planar graph

I'm interested in the time complexity of the following problem: Given an undirected planar graph $G=(V,E)$ and a weight function $w:E \rightarrow \mathbb{Z}$ (so weights can be negative, too), color ...
10
votes
0answers
132 views

Complexity class for probabilistic approximation algorithms with bounded error

What's the name of a complexity class of optimization problems that have "bounded error probabilistic approximation algorithms"? Bounded error probabilistic version of APX (as BPP is bounded error ...
10
votes
0answers
158 views

(Slightly) faster simulation of quantum Fourier transform

Suppose I want to write a classical software simulator of a quantum circuit with $N$ qubits. When it comes time to simulate the quantum Fourier transform I can evaluate all $2^N$ states to determine ...
10
votes
0answers
291 views

Shift-resolve parsing - questions

I've recently came across a paper describing the parsing technique mentioned in the title. Unfortunately, the terminology used in said paper is somewhat beyond my comprehension, so I've been ...
10
votes
0answers
747 views

Universal Turing Machine simulation with bounded time overhead

Is it possible to design a Universal Turing Machine in which the simulation time of a given Turing Machine $M$ is bounded by a factor of $\mathcal{O}(\log|\Gamma|+\log|Q|)$ of the original running-...
10
votes
0answers
1k views

Alternatives to SVD for rank factorization

I have rank-deficient matrix $M \in \mathbb{R}^{n\times m}$ with $\text{rank}(M) = k$ and I want to find a rank factorization $M = PQ$ with $P \in \mathbb{R}^{n \times k}$ and $Q \in \mathbb{R}^{k \...
10
votes
0answers
196 views

“Essential” problem for MA

I am trying to understand different interactive proof systems, in particular AM and MA. Is there a typical problem for the complexity class MA as Graph-NonIsomorphism problem is for AM? Is there ...
10
votes
0answers
127 views

Applying the graph mining algorithm Leap Search in an unlabeled setting

I am reading Mining Significant Graph Patterns by Leap Search (Yan et al., 2008), and I am unclear on how their technique translates to the unlabeled setting, since $p$ and $q$ (the frequency ...
9
votes
0answers
95 views

Can we say McCarthy and Hoare had the same objective in the 60s regarding a mathematical theory of computation?

I don't think there's any way to ask a very precise question here, so this might be considered opinion based. Nevertheless, it seems the question is clear enough because I'm asking whether these two ...
9
votes
0answers
255 views

Advantages of algorithm W over algorithm J for type inference in Hindley-Milner type system

According to A modern eye on ML type inference Furthermore, for some unknown reason, W appears to have become more popular than J, even though the latter is viewed—with reason!—by Milner as ...
9
votes
0answers
119 views

In the beginning, computable functions where always total, but when where the partial functions included

The modern definition of computable functions $f : \mathbb N \to \mathbb N$ as given on wikipedia quite naturally describes partial functions, and not just total functions. Now I am reading some ...
9
votes
0answers
116 views

Using naturality to prove $f: \forall\alpha. \alpha\times\alpha\to\alpha$ must be a projection

Suppose we have a System F term $f : \forall \alpha. \alpha\times\alpha\to\alpha$, interpreted in a parametric model which is a bicartesian closed category. I was wondering if in such context it is ...
9
votes
0answers
111 views

Denotational semantics of object-oriented languages

I am interested in denotational semantics of object oriented languages. Namely, what are the common/typical denotations of objects used in the literature? Is this an interesting topic these days? The ...
9
votes
0answers
263 views

P vs NP and the Time Hierarchy

Assuming $P\neq NP$, is it possible that there exists a $k$ such that $P\subseteq\textsf{NTIME}(t^k)$? There reason I ask this is that I assume the following: $$P=NP \implies \forall k\ \exists j.\ \...
9
votes
0answers
81 views

Interval density of time bounded Kolmogorov complexity

The Kolmogorov complexity of a string $x$ is the size of the smallest Turing machine $M$ that started on empty tape produces $x$. To make it computable, we can add a bound on the time used by $M$ to ...
8
votes
0answers
188 views

Are there consequences for P ≠ NP that are unintuitive?

It's often regarded that the most intuitive answer to the question of $P$ vs $NP$ is that $P ≠ NP$. This is often illustrated with some consequences that would follow if $P = NP$ were true. Things ...
8
votes
0answers
127 views

Covering a complete graph with n copies of an arbitrary graph: NP-complete?

Given a complete graph $G$, an arbitrary graph $H$, and a positive integer $n$, are there subgraphs $A_1,\dots,A_n$ of $G$ (not necessarily disjoint) such that their union is $G$, and each of them ...
8
votes
0answers
142 views

Can you multiply complex 2x2 matrices in fewer than 21 real multiplies?

It is well known that 2x2 matrices can be multiplied using just 7 (instead of the obvious 8) multiplications in the ground field (Strassen-Winograd, etc.). It is also well known that complex numbers ...
8
votes
0answers
123 views

Problem with manipulation of colored graph

Consider a finite set of colors and a given an unweighted graph with the following properties: 1) Graph is connected. 2) All vertices of the graph has a color in the given set of colors. 3) No two ...
8
votes
0answers
125 views

Extensional constructs in minimal extensional type theory without eta equality

The extensional version of Intuitionistic Type Theory is usually formulated in a way that makes extensional concepts like functional extensionality derivable. In particular, equality reflection, ...
8
votes
0answers
231 views

Type-classes for type inference

I'm creating a semantic analyzer with type inference. For the basics I've got a type variable and a type construct with name and a list of types. I want to support overloading and I know that Haskell ...
8
votes
0answers
85 views

Data Structures for Non-Orientable Manifolds

I am looking for a data structure to represent non-orientable manifolds (i.e. meshes like Moebius Strip, but without self-intersection). I will then implement other algorithms using this DS such as, ...
8
votes
0answers
139 views

What is the best algorithm to compute ALL homomorphisms between two rooted labeled trees?

Lets consider two node-labeled rooted trees Q and D. According to wikipedia definition ( https://en.wikipedia.org/wiki/Tree_homomorphism ) a mapping m from the nodes of Q to the nodes of D is a tree ...
8
votes
0answers
75 views

Connections between circuit complexity and Unique Games Conjecture?

Circuit complexity has connections to many questions in complexity theory. For a couple examples, Ryan Williams shared some in a recent talk and Section 3 of these notes gives simple relations to $\...
8
votes
0answers
137 views

Is extensionality for coinductive datatypes consistent with Coq's logic?

Given a coinductive datatype, one can usually (always?) define a bisimulation as the smallest equivalence relation over it. I would like to add an axiom stating that if two members of the type are ...
8
votes
0answers
174 views

How to solve the loan graph problem

The problem A loan graph is a directed weighted graph $\mathcal{G} = (V, A),$ where $A \subseteq V \times V.$ If we have a directed arc $(u, v)$, we interpret it as the node $u$ gave a loan of $w(u, ...
8
votes
0answers
224 views

Can any PEG grammar be parsed in linear time?

On the Wikipedia for PEG it is claimed: Any PEG can be parsed in linear time by using a packrat parser, as described above. However, packrat parsers can't handle left recursion. You can eliminate ...

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