Computer Science

We study the fixed-support Wasserstein barycenter problem (FS-WBP), which consists in computing the Wasserstein barycenter of m discrete probability measures supported on a finite metric space of size n . We show first that the constraint matrix arising from the standard linear programming (LP) representation of the FS-WBP is \textit{not totally unimodular} when m≥3 and n≥3 . This result resolves an open question pertaining to the relationship between the FS-WBP and the minimum-cost flow (MCF) problem since it proves that the FS-WBP in the standard LP form is not an MCF problem when m≥3 and n≥3 . We also develop a provably fast \textit{deterministic} variant of the celebrated iterative Bregman projection (IBP) algorithm, named \textsc{FastIBP}, with a complexity bound of O ~ (m n 7/3 ε −4/3 ) , where ε∈(0,1) is the desired tolerance. This complexity bound is better than the best known complexity bound of O ~ (m n 2 ε −2 ) for the IBP algorithm in terms of ε , and that of O ~ (m n 5/2 ε −1 ) from accelerated alternating minimization algorithm or accelerated primal-dual adaptive gradient algorithm in terms of n . Finally, we conduct extensive experiments with both synthetic data and real images and demonstrate the favorable performance of the \textsc{FastIBP} algorithm in practice.

The parameterized complexity of counting minimum cuts stands as a natural question because Ball and Provan showed its #P-completeness. For any undirected graph G=(V,E) and two disjoint sets of its vertices S,T , we design a fixed-parameter tractable algorithm which counts minimum edge (S,T) -cuts parameterized by their size p . Our algorithm operates on a transformed graph instance. This transformation, called drainage, reveals a collection of at most n=|V| successive minimum (S,T) -cuts Z i . We prove that any minimum (S,T) -cut X contains edges of at least one cut Z i . This observation, together with Menger's theorem, allows us to build the algorithm counting all minimum (S,T) -cuts with running time 2 O( p 2 ) n O(1) . Initially dedicated to counting minimum cuts, it can be modified to obtain an FPT sampling of minimum edge (S,T) -cuts.

We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay {et al.} [CHHL19,CHLT19] that exploit L 1 Fourier tail bounds for classes of Boolean functions to construct pseudorandom generators (PRGs). We show that given a bound on the k -th level of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with k . This interpolates previous works, which either require Fourier bounds on all levels [CHHL19], or have polynomial dependence on the error parameter in the seed length [CHLT10], and thus answers an open question in [CHLT19]. As an example, we show that for polynomial error, Fourier bounds on the first O(logn) levels is sufficient to recover the seed length in [CHHL19], which requires bounds on the entire tail. We obtain our results by an alternate analysis of fractional PRGs using Taylor's theorem and bounding the degree- k Lagrange remainder term using multilinearity and random restrictions. Interestingly, our analysis relies only on the \emph{level-k unsigned Fourier sum}, which is potentially a much smaller quantity than the L 1 notion in previous works. By generalizing a connection established in [CHH+20], we give a new reduction from constructing PRGs to proving correlation bounds. Finally, using these improvements we show how to obtain a PRG for F 2 polynomials with seed length close to the state-of-the-art construction due to Viola [Vio09], which was not known to be possible using this framework.

The purpose of this paper is to extend the definition of Frechet distance which measures the distance between two curves to a distance (Frechet-Like distance) which measures the similarity between two rooted trees. The definition of Frechet-Like distance is as follows: Tow men start from the roots of two trees. When they reach to a node with the degree of more than 2 , they construct k−1 men which k is the outgoing degree of the node and each man monitor a man in another tree (there is a rope between them). The distance is the minimum length of the ropes between the men and the men whom are monitored and they all go forward (the geodesic distance between them to the root of the tree increases) and reach to the leaves of the trees. Here, I prove that the Frechet-Like distance between two trees is SNP-hard to compute. I modify the definition of Frechet-Like distance to measure the distance between tow merge trees, and I prove the relation between the interleaving distance and the modified Frechet-Like distance.

The question whether P equals NP revolves around the discrepancy between active production and mere verification by Turing machines. In this paper, we examine the analogous problem for finite transducers and automata. Every nondeterministic finite transducer defines a binary relation associating each input word with all output words that the transducer can successfully produce on the given input. Finite-valued transducers are those for which there is a finite upper bound on the number of output words that the relation associates with every input word. We characterize finite-valued, functional, and unambiguous nondeterministic transducers whose relations can be verified by a deterministic two-tape automaton, show how to construct such an automaton if one exists, and prove the undecidability of the criterion.

Holant problems are intimately connected with quantum theory as tensor networks. We first use techniques from Holant theory to derive new and improved results for quantum entanglement theory. We discover two particular entangled states | Ψ 6 ⟩ of 6 qubits and | Ψ 8 ⟩ of 8 qubits respectively, that have extraordinary and unique closure properties in terms of the Bell property. Then we use entanglement properties of constraint functions to derive a new complexity dichotomy for all real-valued Holant problems containing an odd-arity signature. The signatures need not be symmetric, and no auxiliary signatures are assumed.

A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H) . Let H be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom( H ), we are given a graph G , whose every vertex v is assigned with a list L(v) of vertices of H . We ask whether there exists a homomorphism h from G to H , which respects lists L , i.e., for every v∈V(G) it holds that h(v)∈L(v) . The complexity dichotomy for LHom( H ) was proven by Feder, Hell, and Huang [JGT 2003]. We are interested in the complexity of the problem, parameterized by the treewidth of the input graph. This problem was investigated by Egri, Marx, and Rzążewski [STACS 2018], who obtained tight complexity bounds for the special case of reflexive graphs H . In this paper we extend and generalize their results for \emph{all} relevant graphs H , i.e., those, for which the LHom{H} problem is NP-hard. For every such H we find a constant k=k(H) , such that LHom( H ) on instances with n vertices and treewidth t * can be solved in time k t ⋅ n O(1) , provided that the input graph is given along with a tree decomposition of width t , * cannot be solved in time (k−ε ) t ⋅ n O(1) , for any ε>0 , unless the SETH fails. For some graphs H the value of k(H) is much smaller than the trivial upper bound, i.e., |V(H)| . Obtaining matching upper and lower bounds shows that the set of algorithmic tools we have discovered cannot be extended in order to obtain faster algorithms for LHom( H ) in bounded-treewidth graphs. Furthermore, neither the algorithm, nor the proof of the lower bound, is very specific to treewidth. We believe that they can be used for other variants of LHom( H ), e.g. with different parameterizations.

These are notes for the course CS-172 I first taught in the Fall 1986 at UC Berkeley and subsequently at Boston University. The goal was to introduce the undergraduates to basic concepts of Theory of Computation and to provoke their interest in further study. Model-dependent effects were systematically ignored. Concrete computational problems were considered only as illustrations of general principles. The notes are skeletal: they do have (terse) proofs, but exercises, references, intuitive comments, examples are missing or inadequate. The notes can be used for designing a course or by students who want to refresh the known material or are bright and have access to an instructor for questions. Each subsection takes about a week of the course.

Computers are known to solve a wide spectrum of problems, however not all problems are computationally solvable. Further, the solvable problems themselves vary on the amount of computational resources they require for being solved. The rigorous analysis of problems and assigning them to complexity classes what makes up the immense field of complexity theory. Do protein folding and sudoku have something in common? It might not seem so but complexity theory tells us that if we had an algorithm that could solve sudoku efficiently then we could adapt it to predict for protein folding. This same property is held by classic platformer games such as Super Mario Bros, which was proven to be NP-complete by Erik Demaine et. al. This article attempts to review the analysis of classical platformer games. Here, we explore the field of complexity theory through a broad survey of literature and then use it to prove that that solving a generalized level in the game "Celeste" is NP-complete. Later, we also show how a small change in it makes the game presumably harder to compute. Various abstractions and formalisms related to modelling of games in general (namely game theory and constraint logic) and 2D platformer video games, including the generalized meta-theorems originally formulated by Giovanni Viglietta are also presented.

Motivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and hypergraphs. We prove that the geometric rank is an upper bound on the subrank of tensors and the independence number of hypergraphs. We prove that the geometric rank is smaller than the slice rank of Tao, and relate geometric rank to the analytic rank of Gowers and Wolf in an asymptotic fashion. As a first application, we use geometric rank to prove a tight upper bound on the (border) subrank of the matrix multiplication tensors, matching Strassen's well-known lower bound from 1987.