# Questions tagged [reference-request]

Reference-request is used when the author needs to know about work related to the question.

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### Is this kind of "multi-reduction" interesting?

Given a decision problem $P$, the usual way to show that it is NP-hard is to find a known NP-hard problem $Q$, and show a polynomial-time algorithm that transforms every instance $I_Q$ of $Q$ to an ...
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### The empty tree-word for regular tree languages

Are there references that consider the "empty tree-word" as an allowable element of regular languages of trees? Are there situations where it is more sensible to allow an empty tree-word? ...
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### Cuthill - Mckee Guarantees?

I'm interested in the following problem: given $M$, a $p \times p$ symmetric sparse matrix (the number of non-zero elements in each row is at most $s \ll p$), find a matrix $B = PMP^T$ where $P$ is a ...
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### Böhm tree with pairs (product types)

I am looking for references for a notion of Böhm tree for the λ-calculus with pairs and projections (and the reduction rule $\pi_i\, \langle t_1, t_2\rangle \longrightarrow t_i$). I'm only aware of ...
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### What is the probability a random language in $\mathsf{PSPACE}$ is in $\mathsf{P}$?

Suppose I am drawing a venn diagram of complexity classes, and I don't want one that is the most visually pleasing, but the most accurate. How much of PSPACE should P take up? Let $L$ be chosen ...
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### Straight-line program for sets

Let $\mathcal{S}$ be a collection of sets. A set straight-line program that enumerates $\mathcal{S}$ is a sequence of sets $B_1,\ldots,B_m$, such that $\mathcal{S}\subseteq \{B_1,\ldots,B_m\}$. For ...
• 4,479
1 vote
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### Induced subgraphs with interface

I am interested in hypergraphs with interfaces, I'll call them simply "graphs" in the following. Formally, a graph of sort $k$ is a tuple $(V,E,i)$ with $E\subseteq V^+$ is the set of edges, ...
• 8,903
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### Hardness of Computing Tribes-DNF by Decision Trees

In this paper on "The Polynomial Hierarchy, Random Oracles, and Boolean Circuits", Fact (3.2) states that it is impossible for a polylogarithmic depth decision tree to compute the Tribes-DNF ...
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### What is best lower bound for comparison sort?

Sorting $n$ numbers requires $\lceil \log_2(n!)\rceil$ comparisons and this is asymptotically optimal, but there is an $O(n)$ error term. What is the best known lower bound for large $n$? I couldn't ...
• 14k
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### What infinite sums cannot be approximated in polynomial time?

The following is from the book Geometric algorithms and combinatorial optimization: It shows an infinite sum that has an FPTAS (= an $\epsilon$-approximation can be computed using poly($1/\epsilon$) ...
• 2,262
1 vote
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### Sparsification and critical clauses in SAT

I have been studying the Exponential-Time Hypothesis, ETH, from this paper, there defined $s_k = \inf\{\delta \geq 0 | k\text{-}SAT \in RTIME[2^{\delta n}]\}$. But in that paper's page number 10, I ...
• 109
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### Operation on Sub-exponential Reduction

I have a question concerning the SERF-reducibility of Impagliazzo, Paturi and Zane and subexponential algorithms. My question is: is a composition of two SERF reductions a SERF reduction? Are there a ...
• 109
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### Tree decompositions with unique witness for each edge

In this question I am concerned with tree decompositions of undirected graphs. Recall that a tree decomposition of a graph $G = (V, E)$ is a tree $T$ whose nodes are subsets of $V$ (called bags) ...
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### Rearrange vectors so partial sums are all non-negative

Consider we are given a collection of $n$ vectors in $d$ dimensions, we want to decide if they can be rearranged into $v_1,\ldots,v_n$ such that $\sum_{i=1}^j v_i\geq \textbf{0}$ for all $j\in [n]$. ...
• 4,479
1 vote
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### NP-hardness of partitioning into k sets [closed]

Given a multiset $S$ of positive integers and a fixed positive integer $k$, can $S$ be partitioned into $k$ parts with equal sum? For $k = 2$, this is the well-known Partition problem. For general $k$,...
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### What are pertinent references to cite on Scott domains?

Scott domains are often presented as having been introduced in 1969. However, the first (but numerous!) papers are from the 1970s, so it is not easy to know what the pertinent references are. My two ...
• 509
1 vote
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### A variant of the generalised assignment problem

I am trying to solve this problem: There are $N$ workers and $T$ tasks. Each task can be assigned to at most one worker. Each worker can be assigned any number of tasks. The profit obtained by ...
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### Is there a text that contains all 4 Büchi-Elgot-Trakhtenbrot-style theorems?

There are several natural Büchi-Elgot-Trakhtenbrot-style theorems: The equivalence of various finite automata on finite words and the weak monadic second order theory of 1 successor The equivalence ...
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### What is $\mathrm{NC}^0$-uniform reduction

I am interesting in strict and right'' formulations of results about $\mathrm{NC}^1$-completeness of some languages. Consider for example Barrington's theorem about $\mathrm{NC}^1$-completeness of ...
• 2,013
1 vote
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### Is the Category of $(* \to)^n *$-kinded types freely generated from the discrete graph with $n$ nodes?

In Introduction to Higher Order Categorical Logic part 1, section 4, Lambek defines an adjunction between $\mathbf{Graph}$, the category of graphs and graph homomorphisms, and the category of ...
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### Finding deepest intersection

There must be a name for this problem, but I can't find it: Given $n$ rectangles in the plane, what is the most number of rectangles that a point in the plane belongs to? In other words, thinking of ...
1 vote
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### First-order linear mu-calculus?

There is linear $\mu$-calculus (see e.g. [1]) and first-order $\mu$-calculus (see e.g. here). Has anybody studied first-order linear $\mu$-calculus? [1]: Christian Dax, Martin Hofmann, Martin Lange: A ...
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### Amplifying success probability for PTMs with $poly(n) / \exp(n)$ gap?

The following is a well-known result of BPP in complexity theory, e.g., Theorem 1 and its proof from here: Consider a probabilistic Turing Machine (PTM) $M$, and a language $L \in BPP$: If $x \in L$ (...
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### Approximately sampling from a discrete unimodal distribution with large support

I have an algorithmic problem and I am curious if a solution is known in the literature, because I cannot find it. I came up with an algorithm of my own, but would be curious if something is known. I ...
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