Jason Li

LOWER CENTRAL SERIES OF A FREE ASSOCIATIVE ALGEBRA OVER THE INTEGERS AND FINITE FIELDS

Abstract Consider the free algebra A n generated over Q by n generators x 1 , … , x n . Interesting objects attached to A = A n are members of its lower central series, L i = L i ( A ) , defined inductively by L 1 = A , L i + 1 = [ A , L i ] , and their associated graded components B i = B i ( A ) defined as B i = L i / L i + 1 . These quotients B i for i ⩾ 2 , as well as the reduced quotient B ¯ 1 = A / ( L 2 + A L 3 ) , exhibit a rich geometric structure, as shown by Feigin and Shoikhet (2007) [FS] and later authors (Dobrovolska et al., 1997 [DKM] , Dobrovolska and Etingof, 2008 [DE] , Arbesfeld and Jordan, 2010 [AJ] , Bapat and Jordan, 2010 [BJ] ). We study the same problem over the integers Z and finite fields F p . New phenomena arise, namely, torsion in B i over Z , and jumps in dimension over F p . We describe the torsion in the reduced quotient B ¯ 1 and B 2 geometrically in terms of the De Rham cohomology of Z n . As a corollary we obtain a complete description of B ¯ 1 ( A n ( Z ) ) and B ¯ 1 ( A n ( F p ) ) , as well as of B 2 ( A n ( Z [ 1 / 2 ] ) ) and B 2 ( A n ( F p ) ) , p > 2 . We also give theoretical and experimental results for B i with i > 2 , formulating a number of conjectures and questions on their basis. Finally, we discuss the supercase, when some of the generators are odd and some are even, and provide some theoretical results and experimental data in this case.

Improved distributed algorithms for exact shortest paths

Computing shortest paths is one of the central problems in the theory of distributed computing. For the last few years, substantial progress has been made on the approximate single source shortest paths problem, culminating in an algorithm of Henzinger, Krinninger, and Nanongkai [STOC’16] which deterministically computes (1+o(1))-approximate shortest paths in Õ(D+√n) time, where D is the hop-diameter of the graph. Up to logarithmic factors, this time complexity is optimal, matching the lower bound of Elkin [STOC’04]. The question of exact shortest paths however saw no algorithmic progress for decades, until the recent breakthrough of Elkin [STOC’17], which established a sublinear-time algorithm for exact single source shortest paths on undirected graphs. Shortly after, Huang et al. [FOCS’17] provided improved algorithms for exact all pairs shortest paths problem on directed graphs. In this paper, we provide an alternative single-source shortest path algorithm with complexity Õ(n3/4D1/4). For polylogarithmic D, this improves on Elkin’s Õ(n5/6) bound and gets closer to the Ω(n1/2) lower bound of Elkin [STOC’04]. For larger values of D, we present an improved variant of our algorithm which achieves complexity Õ(max{ n3/4+o(1) , n3/4D1/6} + D ), and thus compares favorably with Elkin’s bound of Õ(max{ n5/6, n2/3D1/3} + D ) in essentially the entire range of parameters. This algorithm provides also a qualitative improvement, because it works for the more challenging case of directed graph (i.e., graphs where the two directions of an edge can have different weights), constituting the first sublinear-time algorithm for directed graphs. Our algorithm also extends to the case of exact r-source shortest paths, in which we provide the fastest algorithm for moderately small r and D, improving on those of Huang et al.

Minor Excluded Network Families Admit Fast Distributed Algorithms

Distributed network optimization problems, such as minimum spanning tree, minimum cut, and shortest path, are an active research area in distributed computing. This paper presents a fast distributed algorithm for such problems in the CONGEST model, on networks that exclude a fixed minor. On general graphs, many optimization problems, including the ones mentioned above, require Ω(√ n) rounds of communication in the CONGEST model, even if the network graph has a much smaller diameter. Naturally, the next step in algorithm design is to design efficient algorithms which bypass this lower bound on a restricted class of graphs. Currently, the only known method of doing so uses the low-congestion shortcut framework of Ghaffari and Haeupler [SODA16]. Building off of their work, this paper proves that excluded minor graphs admit high-quality shortcuts, leading to an Õ(D^2) round algorithm for the aforementioned problems, where D is the diameter of the network graph. To work with excluded minor graph families, we utilize the Graph Structure Theorem of Robertson and Seymour. To the best of our knowledge, this is the first time the Graph Structure Theorem has been used for an algorithmic result in the distributed setting. Even though the proof is involved, merely showing the existence of good shortcuts is sufficient to obtain simple, efficient distributed algorithms. In particular, the shortcut framework can efficiently construct near-optimal shortcuts and then use them to solve the optimization problems. This, combined with the very general family of excluded minor graphs, which includes most other important graph classes, makes this result of significant interest.

Bounding Laconic Proof Systems by Solving CSPs in Parallel

We show that the basic semidefinite programming relaxation value of any constraint satisfaction problem can be computed in NC; that is, in parallel polylogarithmic time and polynomial work. As a complexity-theoretic consequence we get that MIPone[k,c,s] subseteq PSPACE provided s/c leq (.62-o(1))k/2^k, resolving a question of Austrin, Haa stad, and Pass. Here MIPone[k,c,s] is the class of languages decidable with completeness c and soundness s by an interactive proof system with k provers, each constrained to communicate just 1 bit.