arXiv subjects

Jeffery Li

Publications and source records attributed to Jeffery Li.

5 recordsLinked to original sources

arXiv2026

The Impact of Approximation on Algorithmic Progress

In nearly every discipline, scientific computations are limited by the cost and speed of computation. For example, the best-known exact algorithms for the canonical Traveling Salesman Problem would take centuries to run on an instance of size 1 million. A natural response to such limits is to try to find new algorithms or to parallelize existing ones, but many algorithms are already at their theoretically-optimal level and parallelization is often impossible or prohibitively expensive. Starting in the 1960's, computer scientists pursued another solution: allowing solutions to have a small amount of error (i.e. approximating them). In this paper, we survey 118 of the most important algorithm problems in computer science, quantifying the gains and tradeoffs from approximation that have been discovered over the history of the field. Overall, only $\approx$20\% of problems have benefited from approximation. However, those with good approximate algorithms can be dramatically faster to compute with little cost to accuracy. For example, a quarter of computationally intractable problems (e.g. those that take exponential time to compute) have polynomial time approximate algorithms. Approximation also increases the number of algorithms that can run in linear time by 23\%, opening up new computational opportunities for those working in the big data regime. This work also sheds light on what should be expected from progress in AI, where approximation is at the heart of how deep learning works.

cs.DS↗

arXiv2026

Tetris is Hard with Just One Piece Type

We analyze the computational complexity of Tetris clearing (determining whether the player can clear an initial board using a given sequence of pieces) and survival (determining whether the player can avoid losing before placing all the given pieces in an initial board) when restricted to a single polyomino piece type. We prove, for any tetromino piece type $P$ except for O, the NP-hardness of Tetris clearing and survival under the standard Super Rotation System (SRS), even when the input sequence consists of only a specified number of $P$ pieces. These surprising results disprove a 23-year-old conjecture on the computational complexity of Tetris with only I pieces (although our result is only for a specific rotation system). As a corollary, we prove the NP-hardness of Tetris clearing when the sequence of pieces has to be able to be generated from a $7k$-bag randomizer for any positive integer $k\geq 1$. On the positive side, we give polynomial-time algorithms for Tetris clearing and survival when the input sequence consists of only dominoes, assuming a particular rotation model, solving a version of a 9-year-old open problem. Along the way, we give polynomial-time algorithms for Tetris clearing and survival with $1\times k$ pieces (for any fixed $k$), provided the top $k-1$ rows are initially empty, showing that our I NP-hardness result needs to have filled cells in the top three rows.

cs.CC↗

arXiv2026

Planar Graph Orientation Frameworks, Applied to KPlumber and Polyomino Tiling

Given a graph, when can we orient the edges to satisfy local constraints at the vertices, where each vertex specifies which local orientations of its incident edges are allowed? This family of graph orientation problems is a special kind of SAT problem, where each variable (edge orientation) appears in exactly two clauses (vertex constraints) -- once positively and once negatively. We analyze the complexity of many natural vertex types (patterns of allowed vertex neighborhoods), most notably all sets of symmetric vertex types which depend on only the number of incoming edges. In many scenarios, including Planar and Non-Planar Symmetric Graph Orientation with constants, we give a full dichotomy characterizing P vs. NP-complete problem classes. We apply our results to obtain new polynomial-time algorithms, resolving a 20-year-old open problem about KPlumber; to simplify existing NP-hardness proofs for tiling with trominoes; and to prove new NP-completeness results for tiling with tetrominoes.

cs.CC↗

arXiv2025

How fast are algorithms reducing the demands on memory? A survey of progress in space complexity

Algorithm research focuses primarily on how many operations processors need to do (time complexity). But for many problems, both the runtime and energy used are dominated by memory accesses. In this paper, we present the first broad survey of how algorithmic progress has improved memory usage (space complexity). We analyze 118 of the most important algorithm problems in computer science, reviewing the 800+ algorithms used to solve them. Our results show that space complexity has become much more important in recent years as worries have arisen about memory access bottle-necking performance (the ``memory wall''). In 20% of cases we find that space complexity improvements for large problems (n=1 billion) outpaced improvements in DRAM access speed, suggesting that for these problems algorithmic progress played a larger role than hardware progress in minimizing memory access delays. Increasingly, we also see the emergence of algorithmic Pareto frontiers, where getting better asymptotic time complexity for a problem requires getting worse asymptotic space complexity, and vice-versa. This tension implies that programmers will increasingly need to consider multiple algorithmic options to understand which is best for their particular problem. To help theorists and practitioners alike consider these trade-offs, we have created a reference for them at https://algorithm-wiki.csail.mit.edu.

cs.DS↗

arXiv2024

Tetris with Few Piece Types

We prove NP-hardness and #P-hardness of Tetris clearing (clearing an initial board using a given sequence of pieces) with the Super Rotation System (SRS), even when the pieces are limited to any two of the seven Tetris piece types. This result is the first advance on a question posed twenty years ago: which piece sets are easy vs. hard? All previous Tetris NP-hardness proofs used five of the seven piece types. We also prove ASP-completeness of Tetris clearing, using three piece types, as well as versions of 3-Partition and Numerical 3-Dimensional Matching where all input integers are distinct. Finally, we prove NP-hardness of Tetris survival and clearing under the "hard drops only" and "20G" modes, using two piece types, improving on a previous "hard drops only" result that used five piece types.

cs.CC↗