Alan Guo

Trivariate monomial complete intersections and plane partitions

We consider the homogeneous components U_r of the map on R = k[x,y,z]/(x^A, y^B, z^C) that multiplies by x + y + z. We prove a relationship between the Smith normal forms of submatrices of an arbitrary Toeplitz matrix using Schur polynomials, and use this to give a relationship between Smith normal form entries of U_r. We also give a bijective proof of an identity proven by J. Li and F. Zanello equating the determinant of the middle homogeneous component U_r when (A, B, C) = (a + b, a + c, b + c) to the number of plane partitions in an a by b by c box. Finally, we prove that, for certain vector subspaces of R, similar identities hold relating determinants to symmetry classes of plane partitions, in particular classes 3, 6, and 8.

Classic Nintendo Games Are (Computationally) Hard

We prove NP-hardness results for five of Nintendo’s largest video game franchises: Mario, Donkey Kong, Legend of Zelda, Metroid, and Pokemon. Our results apply to generalized versions of Super Mario Bros. 1, 3, Lost Levels, and Super Mario World; Donkey Kong Country 1–3; all Legend of Zelda games; all Metroid games; and all Pokemon role-playing games. In addition, we prove PSPACE-completeness of the Donkey Kong Country games and several Legend of Zelda games.

Classic Nintendo games are (computationally) hard

We prove NP-hardness results for five of Nintendos largest video game franchises: Mario, Donkey Kong, Legend of Zelda, Metroid, and Pokemon. Our results apply to generalized versions of Super Mario Bros.?1-3, The Lost Levels, and Super Mario World; Donkey Kong Country 1-3; all Legend of Zelda games; all Metroid games; and all Pokemon role-playing games. In addition, we prove PSPACE-completeness of the Donkey Kong Country games and several Legend of Zelda games.

Local Reconstructors and Tolerant Testers for Connectivity and Diameter

A local property reconstructor for a graph property is an algorithm which, given oracle access to the adjacency list of a graph that is “close” to having the property, provides oracle access to the adjacency matrix of a “correction” of the graph, i.e. a graph which has the property and is close to the given graph. For this model, we achieve local property reconstructors for the properties of connectivity and k-connectivity in undirected graphs, and the property of strong connectivity in directed graphs. Along the way, we present a method of transforming a local reconstructor (which acts as a “adjacency matrix oracle” for the corrected graph) into an “adjacency list oracle”. This allows us to recursively use our local reconstructor for (k − 1)-connectivity to obtain a local reconstructor for k-connectivity.

High-Rate Locally Correctable Codes via Lifting

We present a general framework for constructing high-rate error correcting codes that are locally correctable (and hence locally decodable if linear) with a sublinear number of queries, based on lifting codes with respect to functions on the coordinates. Our approach generalizes the lifting of affine-invariant codes (of Guo, Kopparty, and Sudan) and its generalization automorphic lifting (alluded to in the work of Ben-Sasson et al., but distinct from their degree lifting), which lifts algebraic geometry codes with respect to a group of automorphisms of the code. Our notion of lifting is a natural alternative to the degree lifting of Ben-Sasson et al. and it carries two advantages. First, it overcomes the rate barrier inherent in degree lifting. Second, it requires no special properties (e.g. linearity and invariance) of the base code, and requires a very little structure on the set of functions on the coordinates of the code. As an application, we construct new explicit families of locally correctable codes by lifting algebraic geometry codes. Like the multiplicity codes of Kopparty, Saraf, Yekhanin, and the affine-lifted codes of Guo, Kopparty, and Sudan, our codes of block length N can achieve N∈ query complexity and 1 - α rate for any given ∈, α > 0, while correcting a constant fraction of errors, in contrast to the Reed-Muller codes and the degree-lifted AG codes of Ben-Sasson et al., which face a rate barrier of ∈O(1/∈). However, like the degree-lifted AG codes, our codes are over an alphabet significantly smaller than that obtained by Reed-Muller codes, affine-lifted codes, and multiplicity codes.

Winning strategies for aperiodic subtraction games

We provide a winning strategy for sums of games of MARK-t, an impartial game played on the nonnegative integers where each move consists of subtraction by an integer between 1 and t-1 inclusive, or division by t, rounding down when necessary. Our algorithm computes the Sprague-Grundy values for arbitrary n in quadratic time. This solves a problem posed by Aviezri Fraenkel. In addition, we characterize the P-positions and N-positions for the game in mis\`ere play.

List-Decoding Algorithms for Lifted Codes

Lifted Reed-Solomon codes are a natural affine-invariant family of error-correcting codes, which generalize Reed-Muller codes. They were known to have efficient local-testing and local-decoding algorithms (comparable with the known algorithms for Reed-Muller codes), but with significantly better rate. We give efficient algorithms for list decoding and local list decoding of lifted codes. Our algorithms are based on a new technical lemma, which says that the codewords of lifted codes are low degree polynomials when viewed as univariate polynomials over a big field (even though they may be very high degree when viewed as multivariate polynomials over a small field).

List Decoding Group Homomorphisms Between Supersolvable Groups

We show that the set of homomorphisms between two supersolvable groups can be locally list decoded up to the minimum distance of the code, extending the results of Dinur et al who studied the case where the groups are abelian. Moreover, when specialized to the abelian case, our proof is more streamlined and gives a better constant in the exponent of the list size. The constant is improved from about 3.5 million to 105.