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10
votes
1answer
240 views

Identifying useless edges for shortest path

Consider a graph $G$ (the problem makes sense both for directed and undirected graphs). Call $M_G$ the matrix of distances of $G$: $M_G[i, j]$ is the shortest path distance from vertex $i$ to vertex ...
11
votes
2answers
1k views

Hamiltonian cycle on graphs without small cycles

While answering this question on cstheory, I (informally) proved on the fly the following theorem: Theorem: For any fixed $l \geq 3$ the Hamiltonian cycle probem remains NP-complete even if ...
6
votes
0answers
81 views

Hard extendability problems

In extendability problem, we are given part of the solution and we want to decide whether we can extend it to a complete solution. Some extendability problems are efficiently solvable while other ...
5
votes
1answer
234 views

A good exposition of the random restriction method

I'm wondering if there are good references that describe the random restriction method as a lower bound technique ? I'm aware that it's linked to the switching lemma and shows up in many different ...
-1
votes
1answer
84 views

SAT formula specifying that exactly $k$ of $N$ boolean variables are active using less than $N$-choose-$k$ terms

Is there a way to express the condition that exactly $k$ of $N$ boolean variables are active without writing a disjunction of $N$-choose-$k$ terms, i.e., all possible configurations of the $N$ ...
1
vote
1answer
84 views

description of continuous probability distribution

The minimum description length of the realization of a random variable $x$ is given by the entropy of its probability mass function. Entropy of the representation of a (generic) probability ...
1
vote
0answers
85 views

Time-inhomogeneous Markov Chains

I'm trying to find out what is known about time-inhomogeneous ergodic Markov Chains where the transition matrix can vary over time. All textbooks and lecture notes I could find initially introduce ...
13
votes
1answer
272 views

Las Vegas vs Monte Carlo randomized decision tree complexity

Background: Decision tree complexity or query complexity is a simple model of computation defined as follows. Let $f:\{0,1\}^n\to \{0,1\}$ be a Boolean function. The deterministic query complexity of ...
0
votes
1answer
46 views

How to define the _regret_ in multiagent systems? Any good definition please?

I am reading this book. In chapter 7, section 7.5 page 240 (in the pdf), the authors defined (definition 7.5.1) the regret as being the difference between the average per-period reward the agent ...
0
votes
0answers
68 views

planted cliques in continuous graphs?

The question is pretty simple (almost the same as that mentioned in the title). Is there an equivalent definition of the planted clique problem for Continuous graphs ...
2
votes
1answer
100 views

Reducing the bandwidth of non-symmetric matrix

Is there an efficient algorithm to reduce the bandwidth of a directed graph's adjacency matrix? Something like the reverse Cuthill-McKee, but for non-symmetric matrices.
2
votes
1answer
162 views

Literature for Generalized Load Balancing

i am looking for literature on this kind of problem. $$ \begin{align} \min_x \max_k &\quad \sum_{i,j} x_{ij}c_{ijk}\\ \text{subject to}&\\ &\sum_j x_{ij}=1,&& \forall i\in\mathcal ...
5
votes
2answers
333 views

Would a purely topological computational model be useful in decision problems in topology?

If one were to develop a purely topological computational model based upon the equivalence of information in knots and the model would perform transformations of that information. This would be the ...
3
votes
0answers
139 views

Standard reference for efficient computation of non-intersecting Eulerian circuit

A plane graph $G$ defines a cyclic ordering $O(v) = \langle v_1, v_2, \dotsc, n_{\deg(v)}\rangle$ on the neighborhood $N(v)$ of each vertex $v \in V(G)$. A non-intersecting Eulerian circuit $C$ is an ...
0
votes
0answers
61 views

Problems solved by the greedy algorithm for matroids

Given a matroid $\mathcal{M} = (E, \mathcal{I})$ with weights $w : E \rightarrow Q^+$, it is well known that the greedy algorithm finds a maximum weight independent set $S \in \mathcal{I}$ in ...
3
votes
2answers
655 views

if I were to define a new complexity class how should I start?

If I wanted to define my own complexity class would you first define it as a set of problems with reductions to those problems? How would one go about doing this?
19
votes
1answer
265 views

Languages recognized by polynomial-size DFAs

For a fixed finite alphabet $\Sigma$, a formal language $L$ over $\Sigma$ is regular if there exists a deterministic finite automaton (DFA) over $\Sigma$ which accepts exactly $L$. I am interested in ...
0
votes
0answers
43 views

Competitive throughput model definition

I am reading a paper on optimal node routing and it is mentioning the phrase "competitive throughput model". I have searched for a definition, but I didn't find anything. Could anybody give me a ...
8
votes
1answer
160 views

Multidimensional arithmetic progression variant

For $\vec{d} \in \mathbb{N}^n$, let $Q(\vec{d}) \subset \mathbb{N}^n$ be the set of vertices of the $n$-dimensional cube scaled in the direction of the $i$-th coordinate by $d_i$, i.e. $Q(\vec{d} = ...
0
votes
0answers
102 views

Edges in a graph with girth greater than 4

According to the following paper by Füredi, the maximum number of edges for a graph with $n$ vertices is upper bounded by $O(n^{\frac{3}{2}})$, where the leading co-efficient of the term ...
10
votes
1answer
251 views

Another variant of PARTITION

I've got a reduction of the following partition problem to a certain scheduling problem: Input: A list $a_1\leqslant\cdots\leqslant a_n$ of positive integers in non-decreasing order. Question: Does ...
6
votes
1answer
208 views

Distinguishing between two coins

It is well known that the complexity of distinguishing an $\epsilon$ biased coin from a fair one is $\theta(\epsilon^{-2})$. Are there results for distinguishing a $p$ coin from a $p+\epsilon$ coin? I ...
9
votes
3answers
314 views

Does $\delta^+(G)+\delta^-(G) \geq n$ imply strong connectivity?

Denote by $\delta^+(G)$ the minimal out degree in $G$, and by $\delta^-(G)$ the minimal in-degree. In a related question, I've mentioned the Ghouila-Houri extension of Dirac's theorem on Hamiltonian ...
0
votes
0answers
122 views

An interesting class of colored graphs?

Let $G$ be a complete graph edge-colored with $k$ colors. We say that $G$ is Gallai-colored if no triangle is colored with three distinct colors. Fix a tuple of integers $c = (c_1,\ldots,c_k)$. We may ...
3
votes
0answers
109 views

Generating quadratic optimization problems amenable to quantum annealing

Some context: there is a current debate in adiabatic quantum computing over whether a particular machine, the D-Wave quantum annealer, can outperform a classical algorithm [*]. Earlier this year, a ...
7
votes
3answers
375 views

Exponential gap on neural network layers

I read it here that there are function families which need $\mathcal{O}(2^n)$ nodes on neural network with at most $d - 1$ layers to represent the function while need only $\mathcal{O}(n)$ if the ...
7
votes
1answer
169 views

Independent set size of a large girth graphs

For triangle-free (girth $\geq 4$) graph $G$. The following theorem holds true Theorem (Ajtai et al.): For a triangle-free graph $G$ with maximum degree $\Delta$, $$\alpha(G) \geq ...
3
votes
0answers
148 views

#EXP-Complete problems

Let #EXP be the counting variant of NEXP, in the same way that #P is the counting variant of NP. Are there any known #EXP-complete problems? In particular, has #Succinct Sat (the natural candidate) ...
2
votes
0answers
129 views

Fixed-parameter tractability of SCS: finding the missing proof(s)

I am interested in two "super-objects" problems from computational biology. The first problem, dubbed Shortest Common Supersequence ($SCSy$), takes a family of sequences $s_1,\ldots,s_k$, and seeks a ...
12
votes
1answer
204 views

Testing isomorphism of asymmetric graphs

While reading the question Examples where the uniqueness of the solution makes it easier to find, a new (easier?) question came to my mind: actually we don't know if the Graph Isomorphism ($GI$) ...
3
votes
2answers
129 views

TCS oriented refs/survey on group theoretic word problem

The word problem for groups was shown to be Turing-complete in 1955 but has many decidable subcases. This problem arose more in mathematical group theory than in theoretical computer science, but now ...
2
votes
1answer
70 views

Paper about the upper bound of the number of inequalities to describe the Integer hull of a polyhedron

I am interested in the upper bound of the number of inequalities to describe the integer hull of a polyhedron. That is, given an integer programming problem with n inequalities which construct a ...
4
votes
1answer
102 views

parameterized algorithms for geometric set cover

Are there any parametrized algorithms $W$-hardness results known for the computational problem Geometric Set Cover? It is known that set cover problem is $W[2]$ hard when parametrized by the solution ...
5
votes
2answers
126 views

Learnability of constraint satisfaction problems CSPs?

This may sound more like a soft question but I am struggling to find an answer for it. While the learnability of Bayesian Networks and other graphical models are well detailed in the literature of ...
4
votes
1answer
101 views

Dual Barnette's Conjecture

Is every Eulerian triangulated (planar) graph Hamiltonian? On the other hand we have that: Barnette's Conjecture (Open): Every cubic bipartite (3-connected) planar graph is Hamiltonian. ...
25
votes
2answers
668 views

Kolmogorov's conjecture that $P$ has linear-size circuits

In his book, Boolean Function Complexity, Stasys Jukna mentions (page 564) that Kolmogorov believed that every language in P has circuits of linear size. No reference is mentioned and I couldn't find ...
7
votes
1answer
175 views

Tight examples for approximating the feedback vertex set problem

There are several 2-approximation algorithms for the UNWEIGHTED feedback vertex set problem (FVS), which are summarized in [4]. Note that the reduction from vertex cover to FVS is ...
-1
votes
1answer
114 views

is determining an unknown CFL from intersection of two CFLs decidable?

this problem was asked over a week ago on cs.se now with 7v and no answers so far, ie still "open". (there are many somewhat related problems/near variants re CFLs but its not obvious how to reduce it ...
2
votes
0answers
74 views

Assessing the unavoidability of a set kof structures?

Consider a family of structures equipped with a "sub-structure" relation - e.g. subwords, subpermutations or induced subgraphs. Given a set of structures $\cal{K}$, we denote by $Av(\cal{K})$ the ...
4
votes
0answers
63 views

Complexity of equivalence testing of arithmetic circuits

I have a very specific problem that appears to be close to equivalence testing for arithmetic circuits (checking whether the computed functions are the same). Since I'm completely new to the field, ...
2
votes
5answers
428 views

Problems that are hard to prove in $\mathcal{P}$

What is the famous "hard" problems that were shown to be in $\mathcal{P}$ after? I want to know a list of problems that are difficult to prove in the class of "easy" problems? Maybe like matching, ...
2
votes
0answers
73 views

Existence of bijective proofs involving Yamanouchi words?

The following notion comes from algebraic combinatorics and might have some connections to language/permutation theory. Fix an integer $k$. Let us call a $k$-Yamanouchi word a sequence $u$ of integers ...
14
votes
2answers
192 views

On the status of learnability inside $\mathsf{TC}^0$

I'm trying to understand the complexity of functions expressible via threshold gates and this led me to $\mathsf{TC}^0$. In particular, I'm interested what's currently known about learning inside ...
5
votes
1answer
107 views

Probability of random (in)finite graphs being isomorphic

I once skimmed a paper which examined the probability of two (infinite) graphs picked at random being isomorph. The surprising result was that for two random infinite graphs this probability is quite ...
2
votes
0answers
62 views

A number-theoretic bijection in modular arithmetic

Fix an integer $n$. Considered as a multiplicative group, the sets $A = (\mathbb{Z} / n \mathbb{Z})^*$ and $B = \mathbb{Z} / \phi(n) \mathbb{Z}$ have the same cardinality $\phi(n)$, but it does not ...
6
votes
3answers
326 views

Multidimensional knapsack STRONGLY NP-complete

Who can point me to a reference where it is actually shown that multidimensional knapsack is strongly NP-complete? I have found loads of papers where they claim it is, without citation; I have found ...
4
votes
1answer
348 views

How can I prove formally semantic equivalence of programming languages?

I would like to compare two languages which are from different programming paradigms. Both langauges are object oriented languages, but one of them a multiparadigm language because it supports ...
3
votes
1answer
141 views

$NP$ set as a function of combinatorial space

Inspired by this answer given by Noam, which I think implies (if I understood correctly his answer ) that a set $A \in NP$ if and only if there is polynomial-time computable function $f$ from a ...
1
vote
4answers
193 views

Turing-complete computation models on graphs

There are many Turing complete computation models and new ones are devised all the time. I am looking for Turing-complete computation models based on graphs?
2
votes
1answer
134 views

A curious Wilf equivalence class of function compositions

I was enumerating pairs of functions from a size $n$ set into itself, and ran into these three relations which all generate the same integer sequence starting at index zero: 1, 1, 6, 87, 2200, 84245. ...