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Graph encoding algorithms that you know of ?

Is there any compilation of graph encoding algorithms? I know about Prufer and Huffman encoding. But papers say, prufer is not good enough to represent Minimum Spanning Trees in the sense it may ...
15
votes
1answer
677 views

LogDCFL-complete problems

LogCFL is the set of all languages that are logspace reducible to a context-free language. Similarly, LogDCFL is the set of all languages that are logspace reducible to a deterministic context-free ...
58
votes
4answers
2k views

Problems that can be used to show polynomial-time hardness results

When designing an algorithm for a new problem, if I can't find a polynomial time algorithm after a while, I might try to prove it is NP-hard instead. If I succeed, I've explained why I couldn't find ...
1
vote
1answer
411 views

Complexity of two perfect matchings with minimum shared edges?

Perfect Matching problem is polynomial time solvable in general graphs. Given undirected simple graph, Is the problem of finding two perfect matching with minimum shared edges between them ...
2
votes
2answers
1k views

polygonal triangulation and 3-colorability

Lets define polygonal triangulation a triangulation which has a hamiltonian cycle. It's easy to see that any polygonal triangulation is 3-colorable since any triangulation of a polygon is 3-colorable....
22
votes
2answers
1k views

Polynomial time approximation algorithms for machine scheduling: how many open problems are left?

In 1999, Petra Schuurman and Gerhard J. Woeginger published the paper "Polynomial time approximation algorithms for machine scheduling: Ten open problems". Since then, to the best of my knowledge, ...
39
votes
7answers
4k views

Truly random number generator: Turing computable?

I am seeking a definitive answer to whether or not generation of "truly random" numbers is Turing computable. I don't know how to phrase this precisely. This StackExchange question on "efficient ...
4
votes
1answer
287 views

On Defining Probabilistic/Nondeterministic Circuits

Assume that we are interested in deterministic circuits of size $f(n)$. Here, $n$ represents the number of inputs to the circuit. The standard way of defining probabilistic/nondeterministic circuits ...
17
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3answers
1k views

Properties of Random Directed Graphs with Fixed Out-Degree

I am interested in properties of random directed graphs with fixed out-degree $d$. I am imagining a random graph model where each vertex chooses d neighbors (say, with replacement) u.a.r. ...
11
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1answer
399 views

Do people look at loop nestness in boolean circuits?

While an EE undergrad I attended some lectures that presented a nice characterization of boolean circuits in terms of how many nested loops they have. In complexity, boolean circuits are often thought ...
8
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1answer
1k views

Reducing #SAT to #MONOTONE-2SAT

The problem #MONOTONE-2SAT is known to be #P-complete. This means that #SAT can be reduced to it. My question is: given a #SAT instance $F$, which is the transformation that converts $F$ to its ...
1
vote
1answer
1k views

Removing all but a few cycles in a graph

Let problem $S$ be defined as Given undirected graph $G$ and a set of cycles $C_1,C_2, \ldots, C_n$ in G, find minimum number of vertices that need to be deleted to remove all cycles in the ...
19
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1answer
413 views

Visualizing Unique Games

How would you design a picture to illustrate the unique games conjecture? This is for a "Current Events" presentation on unique games at the next AMS Joint Meeting and for the booklet that will be ...
11
votes
1answer
583 views

Using Kolmogorov complexity to establish proof complexity lower bounds?

The motivation for this question is the fact that most n-bit strings are incompressible. Intuitively, we can propose by analogy that most proofs for Tautologies are incompressible to polynomial size. ...
34
votes
2answers
2k views

Semantic vs. Syntactic Complexity Classes

In his "Computational Complexity" book, Papadimitriou writes: RP is in some sense a new and unusual kind of complexity class. Not any polynomially bounded nondeterministic Turing machine can be the ...
38
votes
9answers
3k views

Optimal greedy algorithms for NP-hard problems

Greed, for lack of a better word, is good. One of the first algorithmic paradigms taught in introductory algorithms course is the greedy approach. Greedy approach results in simple and intuitive ...
46
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20answers
7k views

NP-hard problems on trees

Several optimization problems that are known to be NP-hard on general graphs are trivially solvable in polynomial time (some even in linear time) when the input graph is a tree. Examples include ...
10
votes
2answers
1k views

NP-complete variants of undecidable problems?

Examples of bounded $NP$-complete variants of undecidable sets: Bounded Halting problem={ $(M, x, 1^t)$| NTM machine $M$ halts and accepts $x$ within $t$ steps} Bounded Tiling={ $(T, 1^t)$| there is ...
13
votes
3answers
549 views

Hardness Guarantees for AES

Many public-key cryptosystems have some kind of provable security. For example, the Rabin cryptosystem is provably as hard as factoring. I wonder whether such kind of provable security exists for ...
140
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36answers
39k views

What videos should everybody watch?

Stanford University now has a Youtube channel, with free access to HD video of full courses on everything from dynamical systems to quantum entanglement. More conferences and workshops are ...
0
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2answers
10k views

What is the k-SAT problem? [closed]

First of all I am of course aware of the wikipedia article: http://en.wikipedia.org/wiki/Boolean_satisfiability_problem However I still do not understand exactly what the problem is. To demonstrate ...
7
votes
1answer
940 views

Graph Theory Fun Problem

Show that in any graph $G$ with min-degree $k$ ($k \geq 1$ duh!) you can find as its subgraph any tree on $k+1$ vertices. I have not been able to solve the question so far. However, I would like if ...
449
votes
70answers
160k views

What papers should everyone read?

This question is (inspired by)/(shamefully stolen from) a similar question at MathOverflow, but I expect the answers here will be quite different. We all have favorite papers in our own respective ...
7
votes
2answers
794 views

Universal Turing Machines in “Computational Complexity” by Papadimitriou

The first part of this question has been solved (see comments). In the book "computational complexity" by Papadimitriou, a Universal Turing Machine is given. But this machine is not concrete, in the ...
13
votes
2answers
999 views

Projective Plane of Order 12

Objective: Settle the conjecture that there is no projective plane of order 12. In 1989, using computer search on a Cray, Lam proved that no projective plane of order 10 exists. Now that God's ...
4
votes
3answers
347 views

How can I model this usage scenario mathematically?

I want to create a fairly simple mathematical model that describes usage patterns and performance trade-offs in a system. The system behaves as follows: clients periodically issue multi-cast packets ...
-2
votes
1answer
2k views

How do I formally describe a rooted, directed, acyclic graph?

I need a formalism to describe the following requirements: I have a graph comprised of nodes and transitions between nodes Nodes maybe one of three types, all are sub-classes of a base abstract node ...
22
votes
6answers
2k views

Introduction to spectral graph theory

What are the basic references? Are there any good, high-level surveys of SGT and its applications to CS in general and machine learning more specifically?
4
votes
1answer
423 views

Is this problem mappable to 3SAT or is it weaker than 3SAT?

Consider a variant of a satisifiability problem. Given n dimensions (n >= 3, n < 10,000 think of n as large but finite) The range of each dimension is either an interval over the integers or an ...
3
votes
2answers
505 views

Counting complexity of a scheduling problem. [closed]

Let $T={1,…,n}$ be a set of tasks. Each task $i$ has associated a non negative processing time $p_i$ and a deadline $d_i$. A feasible schedule of the tasks consists of a permutation of $n$ elements $\...
5
votes
1answer
2k views

Turing Machines and Subroutine Simulation

I have been reading Wikipedia as an introduction to Turing machines. I found a reference to John Hopcroft and Jeffrey Ullman, (1979). Introduction to Automata Theory, Languages and Computation (1st ed....
11
votes
2answers
463 views

Average distortion embeddings

Consider two metric spaces $(X, d)$ and $(Y, f)$, and an embedding $\mu : X \rightarrow Y$. Traditional metric space embeddings measure the quality of $\mu$ as the worst-case ratio of original to ...
30
votes
4answers
1k views

Are there “small” machines which can efficiently match regular expressions?

It's well-known that a regular expression can be recognized by a nondeterministic finite automaton of size proportional to the regular expression, or by a deterministic FA which is potentially ...
29
votes
3answers
1k views

What does one mean by heuristic statistical physics arguments?

I have heard that there are heuristic arguments in statistical physics that yield results in probability theory for which rigorous proofs are either unknown or very difficult to arrive at. What is a ...
7
votes
2answers
398 views

On the class of the FNP version of the Hamiltonian Cycle problem

This post is linked to: FNP complexity class Many places say that the decision version of Hamiltonian Cycle is NP-Complete, and NP-Complete problems are those whose solution can be verified in ...
9
votes
1answer
307 views

Metric graph theory database search algorithms

I am (slowly) writing a review of the Handbook of Chemoinformatics Algorithms for SIGACT News. One chapter discusses current software implementations, and the database searches (and other ...
48
votes
2answers
7k views

Realizability theory: difference in power between Lambda calculus and Turing Machines

I have three related subquestions, which are highlighted by bullet points below (no, they could not be split, if you are wondering). Andrej Bauer wrote, here, that some functions are realizable ...
7
votes
2answers
551 views

SAT Solution Space - Definition of Cluster of Solutions

I'm looking for a formal definition of Cluster of Solutions. My current understanding is the following. Let $x$ be a boolean assignment on $n$ variables. Let $f: \{ 0,1 \} ^n \to \mathbb{N}$ be a ...
4
votes
1answer
216 views

Optimizing multiplication in a partly commutative semigroup

Let us say I have a semigroup M and its basis B. I know which elements of B commute. What is the most efficient way to do multiplication in such a semigroup? Essentially, this is a question of how ...
13
votes
1answer
451 views

Fast sparse boolean matrix chain product

So, I've got about 100-200 very sparse square boolean matrices with side length ~several dozens, and I need to compute their product. I know that if I multiply them serially, the product will usually ...
11
votes
7answers
1k views

Quantum Computation - Postulates of QM

I have just started (independent) learning about quantum computation in general from Nielsen-Chuang book. I wanted to ask if anyone could try finding time to help me with whats going on with the ...
10
votes
1answer
417 views

Hardness of constrained star system problem?

A star system is a family $F$ of n subsets of n-elements set $S$. A star system is graphical if there is some graph $G(V,E)$ such that $F$ is the family of vertex neighborhoods in $G$. It is $NP$-...
1
vote
1answer
249 views

Constraint Satisfaction Problem: Choosing real numbers with certain characteristics

I have a set of n real numbers. I also have a set of functions, f_1, f_2, ..., f_m. Each of these functions takes a list of numbers as its argument. I also have ...
17
votes
2answers
1k views

Beginner's Guide to Derandomization

I found the book Pairwise Independence and Derandomization on the subject, but it's more research-oriented than tutorial oriented. I'm new to the subject of "Derandomization," and as such, I wanted ...
6
votes
0answers
149 views

Constraint Satisfaction Problem: Choosing real numbers with variance in a certain range

I have a set of n real numbers. I want to repeatedly choose subsets of k elements such that the variance of these k numbers falls within some specified range, r = [l, u]. Moreover I want to do this ...
4
votes
3answers
503 views

FNP complexity class

Where can I find more information about the FNP complexity class? The only place I did find anything on FNP was http://en.wikipedia.org/wiki/FNP_(complexity) However, that isn't sufficient for me to ...
3
votes
2answers
3k views

Using decision version of TSP to solve optimization version

Given an oracle for solving the decision version of TSP, how would I use this to solve the optimization version of TSP. This is not a homework assignment, but of general interest. I have been trying ...
33
votes
2answers
2k views

NTIME(n^k) ≠ DTIME(n^k) ?

In "On determinism versus nondeterminism and related problems" (Proc. IEEE FOCS, pages 429–438, 1983), Paul, Pippenger, Szemerédi and Trotter proved that $\mathsf{NTIME}(n)\neq\mathsf{DTIME}(n)$. ...
0
votes
3answers
3k views

Is it possible to have a 4-coloring for a non-planar graph ? [closed]

I have been working on this thread Grid $k$-coloring without monochromatic rectangles, and I am aware that the four color theorem implies that all planar graphs are four colorable. The question is ...
14
votes
1answer
882 views

Integer relation detection for Subset Sum or NPP?

Is there a way to encode an instance of Subset Sum or the Number Partition Problem so that a (small) solution to an integer relation yields an answer? If not definitely, then in some probabilistic ...

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