arXiv subjects

Jesper Nederlof

Publications and source records attributed to Jesper Nederlof.

At least 37 records · Page 2Linked to original sources

arXiv2022

Tight Lower Bounds for Problems Parameterized by Rank-width

We show that there is no $2^{o(k^2)} n^{O(1)}$ time algorithm for Independent Set on $n$-vertex graphs with rank-width $k$, unless the Exponential Time Hypothesis (ETH) fails. Our lower bound matches the $2^{O(k^2)} n^{O(1)}$ time algorithm given by Bui-Xuan, Telle, and Vatshelle [Discret. Appl. Math., 2010] and it answers the open question of Bergougnoux and Kanté [SIAM J. Discret. Math., 2021]. We also show that the known $2^{O(k^2)} n^{O(1)}$ time algorithms for Weighted Dominating Set, Maximum Induced Matching and Feedback Vertex Set parameterized by rank-width $k$ are optimal assuming ETH. Our results are the first tight ETH lower bounds parameterized by rank-width that do not follow directly from lower bounds for $n$-vertex graphs.

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arXiv2022

Makespan Scheduling of Unit Jobs with Precedence Constraints in $O(1.995^n)$ time

In a classical scheduling problem, we are given a set of $n$ jobs of unit length along with precedence constraints and the goal is to find a schedule of these jobs on $m$ identical machines that minimizes the makespan. This problem is well-known to be NP-hard for an unbounded number of machines. Using standard 3-field notation, it is known as $P|\text{prec}, p_j=1|C_{\max}$. We present an algorithm for this problem that runs in $O(1.995^n)$ time. Before our work, even for $m=3$ machines the best known algorithms ran in $O^\ast(2^n)$ time. In contrast, our algorithm works when the number of machines $m$ is unbounded. A crucial ingredient of our approach is an algorithm with a runtime that is only single-exponential in the vertex cover of the comparability graph of the precedence constraint graph. This heavily relies on insights from a classical result by Dolev and Warmuth (Journal of Algorithms 1984) for precedence graphs without long chains.

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arXiv2022

Bounding generalized coloring numbers of planar graphs using coin models

We study Koebe orderings of planar graphs: vertex orderings obtained by modelling the graph as the intersection graph of pairwise internally-disjoint discs in the plane, and ordering the vertices by non-increasing radii of the associated discs. We prove that for every $d\in \mathbb{N}$, any such ordering has $d$-admissibility bounded by $O(d/\ln d)$ and weak $d$-coloring number bounded by $O(d^4 \ln d)$. This in particular shows that the $d$-admissibility of planar graphs is bounded by $O(d/\ln d)$, which asymptotically matches a known lower bound due to Dvořák and Siebertz.

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arXiv2021

On the Parameterized Complexity of the Connected Flow and Many Visits TSP Problem

We study a variant of Min Cost Flow in which the flow needs to be connected. Specifically, in the Connected Flow problem one is given a directed graph $G$, along with a set of demand vertices $D \subseteq V(G)$ with demands $\mathsf{dem}: D \rightarrow \mathbb{N}$, and costs and capacities for each edge. The goal is to find a minimum cost flow that satisfies the demands, respects the capacities and induces a (strongly) connected subgraph. This generalizes previously studied problems like the (Many Visits) TSP. We study the parameterized complexity of Connected Flow parameterized by $|D|$, the treewidth $tw$ and by vertex cover size $k$ of $G$ and provide: (i) $\mathsf{NP}$-completeness already for the case $|D|=2$ with only unit demands and capacities and no edge costs, and fixed-parameter tractability if there are no capacities, (ii) a fixed-parameter tractable $\mathcal{O}^{\star}(k^{\mathcal{O}(k)})$ time algorithm for the general case, and a kernel of size polynomial in $k$ for the special case of Many Visits TSP, (iii) a $|V(G)|^{\mathcal{O}(tw)}$ time algorithm and a matching $|V(G)|^{o(tw)}$ time conditional lower bound conditioned on the Exponential Time Hypothesis. To achieve some of our results, we significantly extend an approach by Kowalik et al.~[ESA'20].

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arXiv2021

Isolation schemes for problems on decomposable graphs

The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a self-reduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its introduction, much effort has been dedicated towards derandomization of the Isolation Lemma for specific classes of problems. So far, the focus was mainly on problems solvable in polynomial time. In this paper, we study a setting that is more typical for $\mathsf{NP}$-complete problems, and obtain partial derandomizations in the form of significantly decreasing the number of required random bits. In particular, motivated by the advances in parameterized algorithms, we focus on problems on decomposable graphs. For example, for the problem of detecting a Hamiltonian cycle, we build upon the rank-based approach from [Bodlaender et al., Inf. Comput.'15] and design isolation schemes that use - $O(t\log n + \log^2{n})$ random bits on graphs of treewidth at most $t$; - $O(\sqrt{n})$ random bits on planar or $H$-minor free graphs; and - $O(n)$-random bits on general graphs. In all these schemes, the weights are bounded exponentially in the number of random bits used. As a corollary, for every fixed $H$ we obtain an algorithm for detecting a Hamiltonian cycle in an $H$-minor-free graph that runs in deterministic time $2^{O(\sqrt{n})}$ and uses polynomial space; this is the first algorithm to achieve such complexity guarantees. For problems of more local nature, such as finding an independent set of maximum size, we obtain isolation schemes on graphs of treedepth at most $d$ that use $O(d)$ random bits and assign polynomially-bounded weights. We also complement our findings with several unconditional and conditional lower bounds, which show that many of the results cannot be significantly improved.

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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.

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arXiv2020

Parameterized Complexity of Partial Scheduling

We study a natural variant of scheduling that we call \emph{partial scheduling}: In this variant an instance of a scheduling problem along with an integer $k$ is given and one seeks an optimal schedule where not all, but only $k$ jobs, have to be processed. Specifically, we aim to determine the fine-grained parameterized complexity of partial scheduling problems parameterized by $k$ for all variants of scheduling problems that minimize the makespan and involve unit/arbitrary processing times, identical/unrelated parallel machines, release/due dates, and precedence constraints. That is, we investigate whether algorithms with runtimes of the type $f(k)n^{\mathcal{O}(1)}$ or $n^{\mathcal{O}(f(k))}$ exist for a function $f$ that is as small as possible. Our contribution is two-fold: First, we categorize each variant to be either in $\mathsf{P}$, $\mathsf{NP}$-complete and fixed-parameter tractable by $k$, or $\mathsf{W}[1]$-hard parameterized by $k$. Second, for many interesting cases we further investigate the run time on a finer scale and obtain run times that are (almost) optimal assuming the Exponential Time Hypothesis. As one of our main technical contributions, we give an $\mathcal{O}(8^kk(|V|+|E|))$ time algorithm to solve instances of partial scheduling problems minimizing the makespan with unit length jobs, precedence constraints and release dates, where $G=(V,E)$ is the graph with precedence constraints.

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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.

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arXiv2019

Detecting Feedback Vertex Sets of Size $k$ in $O^\star(2.7^k)$ Time

In the Feedback Vertex Set problem, one is given an undirected graph $G$ and an integer $k$, and one needs to determine whether there exists a set of $k$ vertices that intersects all cycles of $G$ (a so-called feedback vertex set). Feedback Vertex Set is one of the most central problems in parameterized complexity: It served as an excellent test bed for many important algorithmic techniques in the field such as Iterative Compression~[Guo et al. (JCSS'06)], Randomized Branching~[Becker et al. (J. Artif. Intell. Res'00)] and Cut\&Count~[Cygan et al. (FOCS'11)]. In particular, there has been a long race for the smallest dependence $f(k)$ in run times of the type $O^\star(f(k))$, where the $O^\star$ notation omits factors polynomial in $n$. This race seemed to be run in 2011, when a randomized algorithm $O^\star(3^k)$ time algorithm based on Cut\&Count was introduced. In this work, we show the contrary and give a $O^\star(2.7^k)$ time randomized algorithm. Our algorithm combines all mentioned techniques with substantial new ideas: First, we show that, given a feedback vertex set of size $k$ of bounded average degree, a tree decomposition of width $(1-Ω(1))k$ can be found in polynomial time. Second, we give a randomized branching strategy inspired by the one from~[Becker et al. (J. Artif. Intell. Res'00)] to reduce to the aforementioned bounded average degree setting. Third, we obtain significant run time improvements by employing fast matrix multiplication.

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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.

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arXiv2019

Detecting and Counting Small Patterns in Planar Graphs in Subexponential Parameterized Time

We present an algorithm that takes as input an $n$-vertex planar graph $G$ and a $k$-vertex pattern graph $P$, and computes the number of (induced) copies of $P$ in $G$ in $2^{O(k/\log k)}n^{O(1)}$ time. If $P$ is a matching, independent set, or connected bounded maximum degree graph, the runtime reduces to $2^{\tilde{O}(\sqrt{k})}n^{O(1)}$. While our algorithm counts all copies of $P$, it also improves the fastest algorithms that only detect copies of $P$. Before our work, no $2^{O(k/\log k)}n^{O(1)}$ time algorithms for detecting unrestricted patterns $P$ were known, and by a result of Bodlaender et al. [ICALP 2016] a $2^{o(k/\log k)}n^{O(1)}$ time algorithm would violate the Exponential Time Hypothesis (ETH). Furthermore, it was only known how to detect copies of a fixed connected bounded maximum degree pattern $P$ in $2^{\tilde{O}(\sqrt{k})}n^{O(1)}$ time probabilistically. For counting problems, it was a repeatedly asked open question whether $2^{o(k)}n^{O(1)}$ time algorithms exist that count even special patterns such as independent sets, matchings and paths in planar graphs. The above results resolve this question in a strong sense by giving algorithms for counting versions of problems with running times equal to the ETH lower bounds for their decision versions. Generally speaking, our algorithm counts copies of $P$ in time proportional to its number of non-isomorphic separations of order $\tilde{O}(\sqrt{k})$. The algorithm introduces a new recursive approach to construct families of balanced cycle separators in planar graphs that have limited overlap inspired by methods from Fomin et al. [FOCS 2016], a new `efficient' inclusion-exclusion based argument and uses methods from Bodlaender et al. [ICALP 2016].

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arXiv2019

Hamiltonicity below Dirac's condition

Dirac's theorem (1952) is a classical result of graph theory, stating that an $n$-vertex graph ($n \geq 3$) is Hamiltonian if every vertex has degree at least $n/2$. Both the value $n/2$ and the requirement for every vertex to have high degree are necessary for the theorem to hold. In this work we give efficient algorithms for determining Hamiltonicity when either of the two conditions are relaxed. More precisely, we show that the Hamiltonian cycle problem can be solved in time $c^k \cdot n^{O(1)}$, for some fixed constant $c$, if at least $n-k$ vertices have degree at least $n/2$, or if all vertices have degree at least $n/2-k$. The running time is, in both cases, asymptotically optimal, under the exponential-time hypothesis (ETH). The results extend the range of tractability of the Hamiltonian cycle problem, showing that it is fixed-parameter tractable when parameterized below a natural bound. In addition, for the first parameterization we show that a kernel with $O(k)$ vertices can be found in polynomial time.

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arXiv2018

Nearly ETH-Tight Algorithms for Planar Steiner Tree with Terminals on Few Faces

The Planar Steiner Tree problem is one of the most fundamental NP-complete problems as it models many network design problems. Recall that an instance of this problem consists of a graph with edge weights, and a subset of vertices (often called terminals); the goal is to find a subtree of the graph of minimum total weight that connects all terminals. A seminal paper by Erickson et al. [Math. Oper. Res., 1987] considers instances where the underlying graph is planar and all terminals can be covered by the boundary of $k$ faces. Erickson et al. show that the problem can be solved by an algorithm using $n^{O(k)}$ time and $n^{O(k)}$ space, where $n$ denotes the number of vertices of the input graph. In the past 30 years there has been no significant improvement of this algorithm, despite several efforts. In this work, we give an algorithm for Planar Steiner Tree with running time $2^{O(k)} n^{O(\sqrt{k})}$ using only polynomial space. Furthermore, we show that the running time of our algorithm is almost tight: we prove that there is no $f(k)n^{o(\sqrt{k})}$ algorithm for Planar Steiner Tree for any computable function $f$, unless the Exponential Time Hypothesis fails.

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arXiv2018

Computing the Chromatic Number Using Graph Decompositions via Matrix Rank

Computing the smallest number $q$ such that the vertices of a given graph can be properly $q$-colored is one of the oldest and most fundamental problems in combinatorial optimization. The $q$-Coloring problem has been studied intensively using the framework of parameterized algorithmics, resulting in a very good understanding of the best-possible algorithms for several parameterizations based on the structure of the graph. While there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators, almost nothing is known about parameterizations based on edge separators. We fill this gap by studying $q$-Coloring parameterized by cutwidth, and parameterized by pathwidth in bounded-degree graphs. Our research uncovers interesting new ways to exploit small edge separators. We present two algorithms for $q$-Coloring parameterized by cutwidth $cutw$: a deterministic one that runs in time $O^*(2^{ω\cdot cutw})$, where $ω$ is the matrix multiplication constant, and a randomized one with runtime $O^*(2^{cutw})$. In sharp contrast to earlier work, the running time is independent of $q$. The dependence on cutwidth is optimal: we prove that even 3-Coloring cannot be solved in $O^*((2-\varepsilon)^{cutw})$ time assuming the Strong Exponential Time Hypothesis (SETH). Our algorithms rely on a new rank bound for a matrix that describes compatible colorings. Combined with a simple communication protocol for evaluating a product of two polynomials, this also yields an $O^*((\lfloor d/2\rfloor+1)^{pw})$ time randomized algorithm for $q$-Coloring on graphs of pathwidth $pw$ and maximum degree $d$. Such a runtime was first obtained by Björklund, but only for graphs with few proper colorings. We also prove that this result is optimal in the sense that no $O^*((\lfloor d/2\rfloor+1-\varepsilon)^{pw})$-time algorithm exists assuming SETH.

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arXiv2018

More Consequences of Falsifying SETH and the Orthogonal Vectors Conjecture

The Strong Exponential Time Hypothesis and the OV-conjecture are two popular hardness assumptions used to prove a plethora of lower bounds, especially in the realm of polynomial-time algorithms. The OV-conjecture in moderate dimension states there is no $ε>0$ for which an $O(N^{2-ε})\mathrm{poly}(D)$ time algorithm can decide whether there is a pair of orthogonal vectors in a given set of size $N$ that contains $D$-dimensional binary vectors. We strengthen the evidence for these hardness assumptions. In particular, we show that if the OV-conjecture fails, then two problems for which we are far from obtaining even tiny improvements over exhaustive search would have surprisingly fast algorithms. If the OV conjecture is false, then there is a fixed $ε>0$ such that: (1) For all $d$ and all large enough $k$, there is a randomized algorithm that takes $O(n^{(1-ε)k})$ time to solve the Zero-Weight-$k$-Clique and Min-Weight-$k$-Clique problems on $d$-hypergraphs with $n$ vertices. As a consequence, the OV-conjecture is implied by the Weighted Clique conjecture. (2) For all $c$, the satisfiability of sparse TC1 circuits on $n$ inputs (that is, circuits with $cn$ wires, depth $c\log n$, and negation, AND, OR, and threshold gates) can be computed in time ${O((2-ε)^n)}$.

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arXiv2017

A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

For even $k$, the matchings connectivity matrix $\mathbf{M}_k$ encodes which pairs of perfect matchings on $k$ vertices form a single cycle. Cygan et al. (STOC 2013) showed that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_2$ is $Θ(\sqrt 2^k)$ and used this to give an $O^*((2+\sqrt{2})^{\mathsf{pw}})$ time algorithm for counting Hamiltonian cycles modulo $2$ on graphs of pathwidth $\mathsf{pw}$. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within $\mathbf{M}_k$, which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011). We present a new technique for a similar pattern propagation when only a black-box lower bound on the asymptotic rank of $\mathbf{M}_k$ is given; no stronger structural insights such as the existence of large permutation submatrices in $\mathbf{M}_k$ are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes $p$) parameterized by pathwidth. To apply this technique, we prove that the rank of $\mathbf{M}_k$ over the rationals is $4^k / \mathrm{poly}(k)$. We also show that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_p$ is $Ω(1.97^k)$ for any prime $p\neq 2$ and even $Ω(2.15^k)$ for some primes. As a consequence, we obtain that Hamiltonian cycles cannot be counted in time $O^*((6-ε)^{\mathsf{pw}})$ for any $ε>0$ unless SETH fails. This bound is tight due to a $O^*(6^{\mathsf{pw}})$ time algorithm by Bodlaender et al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be counted modulo primes $p\neq 2$ in time $O^*(3.97^\mathsf{pw})$, indicating that the modulus can affect the complexity in intricate ways.

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arXiv2017

On Directed Feedback Vertex Set parameterized by treewidth

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$.

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arXiv2017

New Tools and Connections for Exponential-time Approximation

In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and a parameter $α>1$, and the goal is to design an $α$-approximation algorithm with the fastest possible running time. We show the following results: - An $r$-approximation for maximum independent set in $O^*(\exp(\tilde O(n/r \log^2 r+r\log^2r)))$ time, - An $r$-approximation for chromatic number in $O^*(\exp(\tilde{O}(n/r \log r+r\log^2r)))$ time, - A $(2-1/r)$-approximation for minimum vertex cover in $O^*(\exp(n/r^{Ω(r)}))$ time, and - A $(k-1/r)$-approximation for minimum $k$-hypergraph vertex cover in $O^*(\exp(n/(kr)^{Ω(kr)}))$ time. (Throughout, $\tilde O$ and $O^*$ omit $\mathrm{polyloglog}(r)$ and factors polynomial in the input size, respectively.) The best known time bounds for all problems were $O^*(2^{n/r})$ [Bourgeois et al. 2009, 2011 & Cygan et al. 2008]. For maximum independent set and chromatic number, these bounds were complemented by $\exp(n^{1-o(1)}/r^{1+o(1)})$ lower bounds (under the Exponential Time Hypothesis (ETH)) [Chalermsook et al., 2013 & Laekhanukit, 2014 (Ph.D. Thesis)]. Our results show that the naturally-looking $O^*(2^{n/r})$ bounds are not tight for all these problems. The key to these algorithmic results is a sparsification procedure, allowing the use of better approximation algorithms for bounded degree graphs. For obtaining the first two results, we introduce a new randomized branching rule. Finally, we show a connection between PCP parameters and exponential-time approximation algorithms. This connection together with our independent set algorithm refute the possibility to overly reduce the size of Chan's PCP [Chan, 2016]. It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture [Dinur 2016 & Manurangsi and Raghavendra, 2016].

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