On Avoiding Sufficiently Long Abelian Squares

A finite word $w$ is an abelian square if $w = xx^\prime$ with $x^\prime$ a permutation of $x$. In 1972, Entringer, Jackson, and Schatz proved that every binary word of length $k^2 + 6k$ contains an abelian square of length $\geq 2k$. We use Cartesian lattice paths to characterize abelian squares in binary sequences, and construct a binary word of length $q(q+1)$ avoiding abelian squares of length $\geq 2\sqrt{2q(q+1)}$ or greater. We thus prove that the length of the longest binary word avoiding abelian squares of length $2k$ is $\Theta(k^2)$.Comment: 5 page

A Geometric Approach to Combinatorial Fixed-Point Theorems

We develop a geometric framework that unifies several different combinatorial fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing them to be different geometric manifestations of the same topological phenomena. In doing so, we obtain (1) new Tucker-like and Sperner-like fixed-point theorems involving an exponential-sized label set; (2) a generalization of Fan's parity proof of Tucker's Lemma to a much broader class of label sets; and (3) direct proofs of several Sperner-like lemmas from Tucker's lemma via explicit geometric embeddings, without the need for topological fixed-point theorems. Our work naturally suggests several interesting open questions for future research.Comment: 10 pages; an extended abstract appeared at Eurocomb 201

Covering Problems via Structural Approaches

The minimum set cover problem is, without question, among the most ubiquitous and well-studied problems in computer science. Its theoretical hardness has been fully characterized--logarithmic approximability has been established, and no sublogarithmic approximation exists unless P=NP. However, the gap between real-world instances and the theoretical worst case is often immense--many covering problems of practical relevance admit much better approximations, or even solvability in polynomial time. Simple combinatorial or geometric structure can often be exploited to obtain improved algorithms on a problem-by-problem basis, but there is no general method of determining the extent to which this is possible. In this thesis, we aim to shed light on the relationship between the structure and the hardness of covering problems. We discuss several measures of structural complexity of set cover instances and prove new algorithmic and hardness results linking the approximability of a set cover problem to its underlying structure. In particular, we provide: - An APX-hardness proof for a wide family of problems that encode a simple covering problem known as Special-3SC. - A class of polynomial dynamic programming algorithms for a group of weighted geometric set cover problems having simple structure. - A simplified quasi-uniform sampling algorithm that yields improved approximations for weighted covering problems having low cell complexity or geometric union complexity. - Applications of the above to various capacitated covering problems via linear programming strengthening and rounding. In total, we obtain new results for dozens of covering problems exhibiting geometric or combinatorial structure. We tabulate these problems and classify them according to their approximability

Dimension reduction algorithms for near-optimal low-dimensional embeddings and compressive sensing

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 41-42).In this thesis, we establish theoretical guarantees for several dimension reduction algorithms developed for applications in compressive sensing and signal processing. In each instance, the input is a point or set of points in d-dimensional Euclidean space, and the goal is to find a linear function from Rd into Rk , where k << d, such that the resulting embedding of the input pointset into k-dimensional Euclidean space has various desirable properties. We focus on two classes of theoretical results: -- First, we examine linear embeddings of arbitrary pointsets with the aim of minimizing distortion. We present an exhaustive-search-based algorithm that yields a k-dimensional linear embedding with distortion at most ... is the smallest possible distortion over all orthonormal embeddings into k dimensions. This PTAS-like result transcends lower bounds for well-known embedding teclhniques such as the Johnson-Lindenstrauss transform. -- Next, motivated by compressive sensing of images, we examine linear embeddings of datasets containing points that are sparse in the pixel basis, with the goal of recoving a nearly-optimal sparse approximation to the original data. We present several algorithms that achieve strong recovery guarantees using the near-optimal bound of measurements, while also being highly "local" so that they can be implemented more easily in physical devices. We also present some impossibility results concerning the existence of such embeddings with stronger locality properties.by Elyot Grant.S.M

The School Bus Problem on Trees

The School Bus Problem is an NP-hard vehicle routing problem in which the goal is to route buses that transport children to a school such that for each child, the distance travelled on the bus does not exceed the shortest distance from the child's home to the school by more than a given regret threshold. Subject to this constraint and bus capacity limit, the goal is to minimize the number of buses required. In this paper, we give a polynomial time 4-approximation algorithm when the children and school are located at vertices of a fixed tree. As a byproduct of our analysis, we show that the integrality gap of the natural set-cover formulation for this problem is also bounded by 4. We also present a constant factor approximation for the variant where we have a fixed number of buses to use, and the goal is to minimize the maximum regre

The School Bus Problem on Trees

The School Bus Problem is an NP-hard vehicle routing problem in which the goal is to route buses that transport children to a school such that for each child, the distance travelled on the bus does not exceed the shortest distance from the child's home to the school by more than a given regret threshold. Subject to this constraint and bus capacity limit, the goal is to minimize the number of buses required. In this paper, we give a polynomial time 4-approximation algorithm when the children and school are located at vertices of a fixed tree. As a byproduct of our analysis, we show that the integrality gap of the natural set-cover formulation for this problem is also bounded by 4. We also present a constant factor approximation for the variant where we have a fixed number of buses to use, and the goal is to minimize the maximum regret

Compressive sensing using locality-preserving matrices

Compressive sensing is a method for acquiring high-dimensional signals (e.g., images) using a small number of linear measurements. Consider an n-pixel image x ∈ R n, where each pixel p has value xp. The image is acquired by computing the measurement vector Ax, where A is an m×n measurement matrix, for some m &lt;&lt; n. The goal is to design the matrix A and the recovery algorithm which, given Ax, returns an approximation to x. It is known that m = O(k log(n/k)) measurements suffices to recover the k-sparse approximation of x. Unfortunately, this result uses matrices A that are random. Such matrices are difficult to implement in physical devices. In this paper we propose compressive sensing schemes that use matrices A that achieve the near-optimal bound of m = O(k log n), while being highly “local”. We also show impossibility results for stronger notions of locality

Nearly optimal linear embeddings into very low dimensions

We propose algorithms for constructing linear embeddings of a finite dataset V ⊂ ℝ[superscript d] into a k-dimensional subspace with provable, nearly optimal distortions. First, we propose an exhaustive-search-based algorithm that yields a k-dimensional linear embedding with distortion at most ε[subscript opt](k)+δ, for any δ > 0 where ε[subscript opt](k) is the smallest achievable distortion over all possible orthonormal embeddings. This algorithm is space-efficient and can be achieved by a single pass over the data V. However, the runtime of this algorithm is exponential in k. Second, we propose a convex-programming-based algorithm that yields an O(k/δ)-dimensional orthonormal embedding with distortion at most (1 + δ)ε[subscript opt](k). The runtime of this algorithm is polynomial in d and independent of k. Several experiments demonstrate the benefits of our approach over conventional linear embedding techniques, such as principal components analysis (PCA) or random projections.National Science Foundation (U.S.)Natural Sciences and Engineering Research Council of CanadaCenter for Massive Data Algorithmics (MADALGO)David & Lucile Packard FoundationMITEI-Shell Progra