Our Paper on Realizing a Graph on Random Points

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A minimum spanning tree. (image source)

 

Our paper, titled How to Realize a Graph on Random Points, has gone up in the Arxiv. I’ll write more about it in future posts.

Abstract:

Continue reading “Our Paper on Realizing a Graph on Random Points”

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Notes on the PCP Theorem and the Hardness of Approximation: Part 1

In this note, we are going to state the PCP theorem and its relation to the hardness of approximating some NP-hard problem.

PCP Theorem: the Interactive Proof View

Intuitively, a PCP (Probabilistically Checkable Proof) system is an interactive proof system where the verifier is given r(n) random bits and he is allowed to look into the proof \pi in q(n) many locations. If the string x \in \{0,1\}^n is indeed in the language, then there exists a proof so that the verifier always accepts. However, if x is not in the language, no prover can convince this verier with probability more than 1/2. The proof has to be short i.e., of size at most q(n) 2^{r(n)}. This class of language is designated as PCP[r(n), q(n)].

Theorem A (PCP theorem). Every NP language has a highly efficient PCP verifier. In particular,

NP = PCP[O(\log n), O(1)].

Continue reading “Notes on the PCP Theorem and the Hardness of Approximation: Part 1”

In a Hermitian Matrix, the Eigenvectors of Different Eigenvalues are Orthogonal

This is an elementary (yet important) fact in matrix analysis.

Statement

Let M be an n\times n complex Hermitian matrix which means M=M^* where * denotes the conjugate transpose operation. Let \lambda_1, \neq \lambda_2 be two different  eigenvalues of M. Let x, y be the two eigenvectors of M corresponding to the two eigenvalues \lambda_1 and \lambda_2, respectively.

Then the following is true:

\boxed{\lambda_1 \neq \lambda_2 \iff \langle x, y \rangle = 0.}

Here \langle x,y \rangle := y^*x denotes the usual inner product of two vectors x,y.

Continue reading “In a Hermitian Matrix, the Eigenvectors of Different Eigenvalues are Orthogonal”

Eigenvalues of the Laplacian Matrix of the Complete Graph

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A complete graph (source: David Benbennick)

We will show that the eigenvalues of the n\times n Laplacian matrix L of the complete graph K_n are \{0,1\} where  the eigenvalue 0 has (algebraic) multiplicity 1 and the eigenvalue n has multiplicity n-1.

Continue reading “Eigenvalues of the Laplacian Matrix of the Complete Graph”

Kirchhoff’s Matrix Tree Theorem for Counting Spanning Trees — A Proof for Beginners

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Rectilinear minimum spanning tree (source: Rocchini)

In this post, we provide a proof of Kirchhoff’s Matrix Tree theorem [1] which is quite beautiful in our biased opinion. This is a 160-year-old theorem which connects several fundamental concepts of matrix analysis and graph theory (e.g., eigenvalues, determinants, cofactor/minor, Laplacian matrix, incidence matrix, connectedness, graphs, and trees etc.). There are already many expository articles/proof sketches/proofs of this theorem (for example, see [4-6, 8,9]). Here we compile yet-another-expository-article and present arguments in a way which should be accessible to the beginners.

Statement

Suppose  L is the Laplacian of a connected simple graph G with n vertices. Then the number of spanning trees in G, denoted by t(G), is the following:

t(G)=\frac{1}{n} \times \{ the product of all non-zero eigenvalues of L\ \}.

Continue reading “Kirchhoff’s Matrix Tree Theorem for Counting Spanning Trees — A Proof for Beginners”

Spectral Sparsification by Spielman and Srivastava: How They Connected the Dots

In this post, I will discuss my own understanding of the spectral sparsification paper by Daniel Spielman and Nikhil Srivastava (STOC 2008). I will assume the following:

  1. The reader is a beginner, like me, and have already glanced through the Spielman-Srivastava paper (from now on, the SS paper).
  2. The reader has, like me, a basic understanding of spectral sparsification and associated concepts of matrix analysis. I will assume that she has read and understood the Section 2 (Preliminaries) of the SS paper.
  3. The reader holds a copy of the SS paper while reading my post.

First, I will mention the main theorems (actually, I will mention only what they “roughly” say).

Continue reading “Spectral Sparsification by Spielman and Srivastava: How They Connected the Dots”