The Frobenius Problem in a Free Monoid

Given positive integers c1,c2,...,ck with gcd(c1,c2,...,ck) = 1, the Frobenius problem (FP) is to compute the largest integer g(c1,c2,...,ck) that cannot be written as a non-negative integer linear combination of c1,c2,...,ck. The Frobenius problem in a free monoid (FPFM) is a non-commutative generalization of the Frobenius problem. Given words x1,x2,...,xk such that there are only finitely many words that cannot be written as concatenations of words in {x1,x2,...,xk}, the FPFM is to find the longest such words. Unlike the FP, where the upper bound g(c1,c2,...,ck)≤max 1≤i≤k ci2 is quadratic, the upper bound on the length of the longest words in the FPFM can be exponential in certain measures and some of the exponential upper bounds are tight. For the 2FPFM, where the given words over Σ are of only two distinct lengths m and n with 1<m<n, the length of the longest omitted words is ≤g(m, m|Σ|n-m + n - m). In Chapter 1, I give the definition of the FP in integers and summarize some of the interesting properties of the FP. In Chapter 2, I give the definition of the FPFM and discuss some general properties of the FPFM. Then I mainly focus on the 2FPFM. I discuss the 2FPFM from different points of view and present two equivalent problems, one of which is about combinatorics on words and the other is about the word graph. In Chapter 3, I discuss some variations on the FPFM and related problems, including input in other forms, bases with constant size, the case of infinite words, the case of concatenation with overlap, and the generalization of the local postage-stamp problem in a free monoid. In Chapter 4, I present the construction of some essential examples to complement the theory of the 2FPFM discussed in Chapter 2. The theory and examples of the 2FPFM are the main contribution of the thesis. In Chapter 5, I discuss the algorithms for and computational complexity of the FPFM and related problems. In the last chapter, I summarize the main results and list some open problems. Part of my work in the thesis has appeared in the papers

Automatic Summarization for Student Reflective Responses

Educational research has demonstrated that asking students to respond to reflection prompts can improve both teaching and learning. However, summarizing student responses to these prompts is an onerous task for humans and poses challenges for existing summarization methods. From the input perspective, there are three challenges. First, there is a lexical variety problem due to the fact that different students tend to use different expressions. Second, there is a length variety problem that student inputs range from single words to multiple sentences. Third, there is a redundancy issue since some content among student responses are not useful. From the output perspective, there are two additional challenges. First, the human summaries consist of a list of important phrases instead of sentences. Second, from an instructor's perspective, the number of students who have a particular problem or are interested in a particular topic is valuable. The goal of this research is to enhance student response summarization at multiple levels of granularity. At the sentence level, we propose a novel summarization algorithm by extending traditional ILP-based framework with a low-rank matrix approximation to address the challenge of lexical variety. At the phrase level, we propose a phrase summarization framework by a combination of phrase extraction, phrase clustering, and phrase ranking. Experimental results show the effectiveness on multiple student response data sets. Also at the phrase level, we propose a quantitative phrase summarization algorithm in order to estimate the number of students who semantically mention the phrases in a summary. We first introduce a new phrase-based highlighting scheme for automatic summarization. It highlights the phrases in the human summaries and also the corresponding semantically-equivalent phrases in student responses. Enabled by the highlighting scheme, we improve the previous phrase-based summarization framework by developing a supervised candidate phrase extraction, learning to estimate the phrase similarities, and experimenting with different clustering algorithms to group phrases into clusters. Experimental results show that our proposed methods not only yield better summarization performance evaluated using ROUGE, but also produce summaries that capture the pressing student needs