The search for max-cliques in graphs is a well established NP-complete problem in graph theory and algorithm design, with many algorithms designed to make use of internal structures of specific types of graphs. We study the extension of the problem of searching for max-cliques in graphs to hypergraphs with constant edge size k, and adapt existing algorithms for graphs to work in k-hypergraphs. In particular, we are interested in the generalization of circulant graphs to circulant k-hypergraphs, and provide a definition of this type of hypergraph. We design and implement a new algorithm to perform max-clique searches on circulant k-hypergraphs. This algorithm combines ideas from a Russian doll algorithm for max-cliques in graphs (Ostergard 2002) with an algorithm based on necklaces for a class of circulant k-hypergraphs (Tzanakis, Moura, Stevens and Panario 2016). We examine the performance of our new algorithm against a set of adapted algorithms (backtracking and Russian doll search for general k-hypergraphs, and necklace-based search for circulant k-hypergraphs) in a set of benchmarking experiments across various densities and edge sizes. This study reveals that the new algorithm outperforms the others when edge density of the hypergraph is high, and that the pure necklace-based algorithm is best in the case of low densities. Finally, we use our new algorithm to perform an exhaustive search on circulant 4-hypergraphs constructed from linear feedback shift register sequences on finite fields of order q that yields covering arrays. The search is completed for 2 <= q <= 5 which solves the open case of q=5 left by Tzanakis et al.

In this thesis, we show results for some well-studied problems from learning theory and combinatorial optimization. Learning Parities under the Uniform Distribution: We study the learnability of parities in the agnostic learning framework of Haussler and Kearns et al. We show that under the uniform distribution, agnostically learning parities reduces to learning parities with random classification noise, commonly referred to as the noisy parity problem. Together with the parity learning algorithm of Blum et al, this gives the first nontrivial algorithm for agnostic learning of parities. We use similar techniques to reduce learning of two other fundamental concept classes under the uniform distribution to learning of noisy parities. Namely, we show that learning of DNF expressions reduces to learning noisy parities of just logarithmic number of variables and learning of k-juntas reduces to learning noisy parities of k variables. Agnostic Learning of Halfspaces: We give an essentially optimal hardness result for agnostic learning of halfspaces over rationals. We show that for any constant ε finding a halfspace that agrees with an unknown function on 1/2+ε fraction of examples is NP-hard even when there exists a halfspace that agrees with the unknown function on 1-ε fraction of examples. This significantly improves on a number of previous hardness results for this problem. We extend the result to ε = 2[superscript-Ω(sqrt{log n})] assuming NP is not contained in DTIME(2[superscript(log n)O(1)]). Majorities of Halfspaces: We show that majorities of halfspaces are hard to PAC-learn using any representation, based on the cryptographic assumption underlying the Ajtai-Dwork cryptosystem. This also implies a hardness result for learning halfspaces with a high rate of adversarial noise even if the learning algorithm can output any efficiently computable hypothesis. Max-Clique, Chromatic Number and Min-3Lin-Deletion: We prove an improved hardness of approximation result for two problems, namely, the problem of finding the size of the largest clique in a graph (also referred to as the Max-Clique problem) and the problem of finding the chromatic number of a graph. We show that for any constant γ > 0, there is no polynomial time algorithm that approximates these problems within factor n/2[superscript(log n)3/4+γ] in an n vertex graph, assuming NP is not contained in BPTIME(2[superscript(log n)O(1)]). This improves the hardness factor of n/2[superscript (log n)1-γ'] for some small (unspecified) constant γ' > 0 shown by Khot. Our main idea is to show an improved hardness result for the Min-3Lin-Deletion problem. An instance of Min-3Lin-Deletion is a system of linear equations modulo 2, where each equation is over three variables. The objective is to find the minimum number of equations that need to be deleted so that the remaining system of equations has a satisfying assignment. We show a hardness factor of 2[superscript sqrt{log n}] for this problem, improving upon the hardness factor of (log n)[superscriptβ] shown by Hastad, for some small (unspecified) constant β > 0. The hardness results for Max-Clique and chromatic number are then obtained using the reduction from Min-3Lin-Deletion as given by Khot. Monotone Multilinear Boolean Circuits for Bipartite Perfect Matching: A monotone Boolean circuit is said to be multilinear if for any AND gate in the circuit, the minimal representation of the two input functions to the gate do not have any variable in common. We show that monotone multilinear Boolean circuits for computing bipartite perfect matching require exponential size. In fact we prove a stronger result by characterizing the structure of the smallest monotone multilinear Boolean circuits for the problem.