arXiv subjects

Karol Węgrzycki

Publications and source records attributed to Karol Węgrzycki.

44 records · Page 3Linked to original sources

arXiv2021

Improving Schroeppel and Shamir's Algorithm for Subset Sum via Orthogonal Vectors

We present an $\mathcal{O}^\star(2^{0.5n})$ time and $\mathcal{O}^\star(2^{0.249999n})$ space randomized algorithm for solving worst-case Subset Sum instances with $n$ integers. This is the first improvement over the long-standing $\mathcal{O}^\star(2^{n/2})$ time and $\mathcal{O}^\star(2^{n/4})$ space algorithm due to Schroeppel and Shamir (FOCS 1979). We breach this gap in two steps: (1) We present a space efficient reduction to the Orthogonal Vectors Problem (OV), one of the most central problem in Fine-Grained Complexity. The reduction is established via an intricate combination of the method of Schroeppel and Shamir, and the representation technique introduced by Howgrave-Graham and Joux (EUROCRYPT 2010) for designing Subset Sum algorithms for the average case regime. (2) We provide an algorithm for OV that detects an orthogonal pair among $N$ given vectors in $\{0,1\}^d$ with support size $d/4$ in time $\tilde{O}(N\cdot2^d/\binom{d}{d/4})$. Our algorithm for OV is based on and refines the representative families framework developed by Fomin, Lokshtanov, Panolan and Saurabh (J. ACM 2016). Our reduction uncovers a curious tight relation between Subset Sum and OV, because any improvement of our algorithm for OV would imply an improvement over the runtime of Schroeppel and Shamir, which is also a long standing open problem.

cs.DS↗

arXiv2020

Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

For many algorithmic problems on graphs of treewidth $t$, a standard dynamic programming approach gives an algorithm with time and space complexity $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$. It turns out that when one considers the more restrictive parameter treedepth, it is often the case that a variation of this technique can be used to reduce the space complexity to polynomial, while retaining time complexity of the form $2^{\mathcal{O}(d)}\cdot n^{\mathcal{O}(1)}$, where $d$ is the treedepth. This transfer of methodology is, however, far from automatic. For instance, for problems with connectivity constraints, standard dynamic programming techniques give algorithms with time and space complexity $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$ on graphs of treewidth $t$, but it is not clear how to convert them into time-efficient polynomial space algorithms for graphs of low treedepth. Cygan et al. (FOCS'11) introduced the Cut&Count technique and showed that a certain class of problems with connectivity constraints can be solved in time and space complexity $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$. Recently, Hegerfeld and Kratsch (STACS'20) showed that, for some of those problems, the Cut&Count technique can be also applied in the setting of treedepth, and it gives algorithms with running time $2^{\mathcal{O}(d)}\cdot n^{\mathcal{O}(1)}$ and polynomial space usage. However, a number of important problems eluded such a treatment, with the most prominent examples being Hamiltonian Cycle and Longest Path. In this paper we clarify the situation by showing that Hamiltonian Cycle, Hamiltonian Path, Long Cycle, Long Path, and Min Cycle Cover all admit $5^d\cdot n^{\mathcal{O}(1)}$-time and polynomial space algorithms on graphs of treedepth $d$. The algorithms are randomized Monte Carlo with only false negatives.

cs.DS↗

arXiv2019

Approximating APSP without Scaling: Equivalence of Approximate Min-Plus and Exact Min-Max

Zwick's $(1+\varepsilon)$-approximation algorithm for the All Pairs Shortest Path (APSP) problem runs in time $\widetilde{O}(\frac{n^ω}{\varepsilon} \log{W})$, where $ω\le 2.373$ is the exponent of matrix multiplication and $W$ denotes the largest weight. This can be used to approximate several graph characteristics including the diameter, radius, median, minimum-weight triangle, and minimum-weight cycle in the same time bound. Since Zwick's algorithm uses the scaling technique, it has a factor $\log W$ in the running time. In this paper, we study whether APSP and related problems admit approximation schemes avoiding the scaling technique. That is, the number of arithmetic operations should be independent of $W$; this is called strongly polynomial. Our main results are as follows. - We design approximation schemes in strongly polynomial time $O(\frac{n^ω}{\varepsilon} \text{polylog}(\frac{n}{\varepsilon}))$ for APSP on undirected graphs as well as for the graph characteristics diameter, radius, median, minimum-weight triangle, and minimum-weight cycle on directed or undirected graphs. - For APSP on directed graphs we design an approximation scheme in strongly polynomial time $O(n^{\frac{ω+ 3}{2}} \varepsilon^{-1} \text{polylog}(\frac{n}{\varepsilon}))$. This is significantly faster than the best exact algorithm. - We explain why our approximation scheme for APSP on directed graphs has a worse exponent than $ω$: Any improvement over our exponent $\frac{ω+ 3}{2}$ would improve the best known algorithm for Min-Max Product In fact, we prove that approximating directed APSP and exactly computing the Min-Max Product are equivalent.

cs.DS↗

arXiv2019

Equal-Subset-Sum Faster Than the Meet-in-the-Middle

In the Equal-Subset-Sum problem, we are given a set $S$ of $n$ integers and the problem is to decide if there exist two disjoint nonempty subsets $A,B \subseteq S$, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in $O^{*}(3^{n/2}) \le O^{*}(1.7321^n)$ time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give $O^{*}(1.7088^n)$ worst case Monte Carlo algorithm. This answers the open problem from Woeginger's inspirational survey. Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in $O^{*}(3^n)$ time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in $O^{*}(2.6817^n)$ time and polynomial space.

cs.DS↗

arXiv2019

A Subquadratic Approximation Scheme for Partition

The subject of this paper is the time complexity of approximating Knapsack, Subset Sum, Partition, and some other related problems. The main result is an $\widetilde{O}(n+1/\varepsilon^{5/3})$ time randomized FPTAS for Partition, which is derived from a certain relaxed form of a randomized FPTAS for Subset Sum. To the best of our knowledge, this is the first NP-hard problem that has been shown to admit a subquadratic time approximation scheme, i.e., one with time complexity of $O((n+1/\varepsilon)^{2-δ})$ for some $δ>0$. To put these developments in context, note that a quadratic FPTAS for \partition has been known for 40 years. Our main contribution lies in designing a mechanism that reduces an instance of Subset Sum to several simpler instances, each with some special structure, and keeps track of interactions between them. This allows us to combine techniques from approximation algorithms, pseudo-polynomial algorithms, and additive combinatorics. We also prove several related results. Notably, we improve approximation schemes for 3SUM, (min,+)-convolution, and Tree Sparsity. Finally, we argue why breaking the quadratic barrier for approximate Knapsack is unlikely by giving an $Ω((n+1/\varepsilon)^{2-o(1)})$ conditional lower bound.

cs.DS↗

arXiv2019

On problems equivalent to (min,+)-convolution

In recent years, significant progress has been made in explaining the apparent hardness of improving upon the naive solutions for many fundamental polynomially solvable problems. This progress has come in the form of conditional lower bounds -- reductions from a problem assumed to be hard. The hard problems include 3SUM, All-Pairs Shortest Path, SAT, Orthogonal Vectors, and others. In the $(\min,+)$-convolution problem, the goal is to compute a sequence $(c[i])^{n-1}_{i=0}$, where $c[k] = $ $\min_{i=0,\ldots,k} $ $\{a[i] $ $+$ $b[k-i]\}$, given sequences $(a[i])^{n-1}_{i=0}$ and $(b[i])_{i=0}^{n-1}$. This can easily be done in $O(n^2)$ time, but no $O(n^{2-\varepsilon})$ algorithm is known for $\varepsilon > 0$. In this paper, we undertake a systematic study of the $(\min,+)$-convolution problem as a hardness assumption. First, we establish the equivalence of this problem to a group of other problems, including variants of the classic knapsack problem and problems related to subadditive sequences. The $(\min,+)$-convolution problem has been used as a building block in algorithms for many problems, notably problems in stringology. It has also appeared as an ad hoc hardness assumption. Second, we investigate some of these connections and provide new reductions and other results. We also explain why replacing this assumption with the SETH might not be possible for some problems.

cs.DS↗

arXiv2017

Why Do Cascade Sizes Follow a Power-Law?

We introduce random directed acyclic graph and use it to model the information diffusion network. Subsequently, we analyze the cascade generation model (CGM) introduced by Leskovec et al. [19]. Until now only empirical studies of this model were done. In this paper, we present the first theoretical proof that the sizes of cascades generated by the CGM follow the power-law distribution, which is consistent with multiple empirical analysis of the large social networks. We compared the assumptions of our model with the Twitter social network and tested the goodness of approximation.

cs.SI↗

arXiv2016

RecSys Challenge 2016: job recommendations based on preselection of offers and gradient boosting

We present the Mim-Solution's approach to the RecSys Challenge 2016, which ranked 2nd. The goal of the competition was to prepare job recommendations for the users of the website Xing.com. Our two phase algorithm consists of candidate selection followed by the candidate ranking. We ranked the candidates by the predicted probability that the user will positively interact with the job offer. We have used Gradient Boosting Decision Trees as the regression tool.

cs.AI↗