Damian Straszak

Strong Inapproximability of the Shortest Reset Word

The Cerný conjecture states that every n-state synchronizing automaton has a reset word of length at most \((n-1)^2\). We study the hardness of finding short reset words. It is known that the exact version of the problem, i.e., finding the shortest reset word, is \(\mathrm {NP}\)-hard and \(\mathrm {coNP}\)-hard, and complete for the \(\mathrm {DP}\) class, and that approximating the length of the shortest reset word within a factor of \(O(\log n)\) is \(\mathrm {NP}\)-hard [Gerbush and Heeringa, CIAA’10], even for the binary alphabet [Berlinkov, DLT’13]. We significantly improve on these results by showing that, for every \(\varepsilon >0\), it is \(\mathrm {NP}\)-hard to approximate the length of the shortest reset word within a factor of \(n^{1-\varepsilon }\). This is essentially tight since a simple O(n)-approximation algorithm exists.

Real stable polynomials and matroids: optimization and counting

Several fundamental optimization and counting problems arising in computer science, mathematics and physics can be reduced to one of the following computational tasks involving polynomials and set systems: given an oracle access to an m-variate real polynomial g and to a family of (multi-)subsets ℬ of [m], (1) compute the sum of coefficients of monomials in g corresponding to all the sets that appear in B(1), or find S ε ℬ such that the monomial in g corresponding to S has the largest coefficient in g. Special cases of these problems, such as computing permanents and mixed discriminants, sampling from determinantal point processes, and maximizing sub-determinants with combinatorial constraints have been topics of much recent interest in theoretical computer science. In this paper we present a general convex programming framework geared to solve both of these problems. Subsequently, we show that roughly, when g is a real stable polynomial with non-negative coefficients and B is a matroid, the integrality gap of our convex relaxation is finite and depends only on m (and not on the coefficients of g). Prior to this work, such results were known only in important but sporadic cases that relied heavily on the structure of either g or ℬ; it was not even a priori clear if one could formulate a convex relaxation that has a finite integrality gap beyond these special cases. Two notable examples are a result by Gurvits for real stable polynomials g when ℬ contains one element, and a result by Nikolov and Singh for a family of multi-linear real stable polynomials when B is the partition matroid. This work, which encapsulates almost all interesting cases of g and B, benefits from both - it is inspired by the latter in coming up with the right convex programming relaxation and the former in deriving the integrality gap. However, proving our results requires extensions of both; in that process we come up with new notions and connections between real stable polynomials and matroids which might be of independent interest.

On a Natural Dynamics for Linear Programming

In this paper we study dynamics inspired by Physarum polycephalum (a slime mold) for solving linear programs [NTY00, IJNT11, JZ12]. These dynamics are arrived at by a local and mechanistic interpretation of the inner workings of the slime mold and a global optimization perspective has been lacking even in the simplest of instances. Our first result is an interpretation of the dynamics as an optimization process. We show that Physarum dynamics can be seen as a steepest-descent type algorithm on a certain Riemannian manifold. Moreover, we prove that the trajectories of Physarum are in fact paths of optimizers to a parametrized family of convex programs, in which the objective is a linear cost function regularized by an entropy barrier. Subsequently, we rigorously establish several important properties of solution curves of Physarum. We prove global existence of such solutions and show that they have limits, being optimal solutions of the underlying LP. Finally, we show that the discretization of the Physarum dynamics is efficient for a class of linear programs, which include unimodular constraint matrices. Thus, together, our results shed some light on how nature might be solving instances of perhaps the most complex problem in P: linear programming.

On the Complexity of Constrained Determinantal Point Processes

Determinantal Point Processes (DPPs) are probabilistic models that arise in quantum physics and random matrix theory and have recently found numerous applications in computer science. DPPs define distributions over subsets of a given ground set, they exhibit interesting properties such as negative correlation, and, unlike other models, have efficient algorithms for sampling. When applied to kernel methods in machine learning, DPPs favor subsets of the given data with more diverse features. However, many real-world applications require efficient algorithms to sample from DPPs with additional constraints on the subset, e.g., partition or matroid constraints that are important to ensure priors, resource or fairness constraints on the sampled subset. Whether one can efficiently sample from DPPs in such constrained settings is an important problem that was first raised in a survey of DPPs by \cite{KuleszaTaskar12} and studied in some recent works in the machine learning literature. The main contribution of our paper is the first resolution of the complexity of sampling from DPPs with constraints. We give exact efficient algorithms for sampling from constrained DPPs when their description is in unary. Furthermore, we prove that when the constraints are specified in binary, this problem is #P-hard via a reduction from the problem of computing mixed discriminants implying that it may be unlikely that there is an FPRAS. Our results benefit from viewing the constrained sampling problem via the lens of polynomials. Consequently, we obtain a few algorithms of independent interest: 1) to count over the base polytope of regular matroids when there are additional (succinct) budget constraints and, 2) to evaluate and compute the mixed characteristic polynomials, that played a central role in the resolution of the Kadison-Singer problem, for certain special cases.

Beating O(nm) in Approximate LZW-Compressed Pattern Matching

Given an LZW/LZ78 compressed text, we want to find an approximate occurrence of a given pattern of length m. The goal is to achieve time complexity depending on the size n of the compressed representation of the text instead of its length. We consider two specific definitions of approximate matching, namely the Hamming distance and the edit distance, and show how to achieve \(\mathcal{O}(n\sqrt{m}k^{2})\) and \(\mathcal{O}(n\sqrt{m}k^{3})\) running time, respectively, where k is the bound on the distance, both in linear space. Even for very small values of k, the best previously known solutions required Ω(nm) time. Our main contribution is applying a periodicity-based argument in a way that is computationally effective even if we operate on a compressed representation of a string, while the previous solutions were either based on a dynamic programming, or a black-box application of tools developed for uncompressed strings.