arXiv subjects

Tuukka Korhonen

Publications and source records attributed to Tuukka Korhonen.

At least 37 records · Page 2Linked to original sources

arXiv2023

Fully dynamic approximation schemes on planar and apex-minor-free graphs

The classic technique of Baker [J. ACM '94] is the most fundamental approach for designing approximation schemes on planar, or more generally topologically-constrained graphs, and it has been applied in a myriad of different variants and settings throughout the last 30 years. In this work we propose a dynamic variant of Baker's technique, where instead of finding an approximate solution in a given static graph, the task is to design a data structure for maintaining an approximate solution in a fully dynamic graph, that is, a graph that is changing over time by edge deletions and edge insertions. Specifically, we address the two most basic problems -- Maximum Weight Independent Set and Minimum Weight Dominating Set -- and we prove the following: for a fully dynamic $n$-vertex planar graph $G$, one can: * maintain a $(1-\varepsilon)$-approximation of the maximum weight of an independent set in $G$ with amortized update time $f(\varepsilon)\cdot n^{o(1)}$; and, * under the additional assumption that the maximum degree of the graph is bounded at all times by a constant, also maintain a $(1+\varepsilon)$-approximation of the minimum weight of a dominating set in $G$ with amortized update time $f(\varepsilon)\cdot n^{o(1)}$. In both cases, $f(\varepsilon)$ is doubly-exponential in $\mathrm{poly}(1/\varepsilon)$ and the data structure can be initialized in time $f(\varepsilon)\cdot n^{1+o(1)}$. All our results in fact hold in the larger generality of any graph class that excludes a fixed apex-graph as a minor.

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arXiv2023

On Induced Versions of Menger's Theorem on Sparse Graphs

Let $A$ and $B$ be sets of vertices in a graph $G$. Menger's theorem states that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of at most $k-1$ vertices. Let $Δ$ be the maximum degree of $G$. We show that there exists a function $f(Δ) = (Δ+1)^{Δ^2+1}$, so that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint and pairwise anticomplete paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of at most $k \cdot f(Δ)$ vertices. We also show that the result can be generalized from bounded-degree graphs to graphs excluding a topological minor. On the negative side, we show that no such relation holds on graphs that have degeneracy 2 and arbitrarily large girth, even when $k = 2$. Similar results were obtained independently and concurrently by Hendrey, Norin, Steiner, and Turcotte [arXiv:2309.07905].

math.CO↗

arXiv2023

SharpSAT-TD in Model Counting Competitions 2021-2023

We describe SharpSAT-TD, our submission to the unweighted and weighted tracks of the Model Counting Competition in 2021-2023, which has won in total $6$ first places in different tracks of the competition. SharpSAT-TD is based on SharpSAT [Thurley, SAT 2006], with the primary novel modification being the use of tree decompositions in the variable selection heuristic as introduced by the authors in [CP 2021]. Unlike the version of SharpSAT-TD evaluated in [CP 2021], the current version that is available in https://github.com/Laakeri/sharpsat-td features also other significant modifications compared to the original SharpSAT, for example, a new preprocessor.

cs.AI↗

arXiv2023

A Single-Exponential Time 2-Approximation Algorithm for Treewidth

We give an algorithm that, given an $n$-vertex graph $G$ and an integer $k$, in time $2^{O(k)} n$ either outputs a tree decomposition of $G$ of width at most $2k + 1$ or determines that the treewidth of $G$ is larger than $k$. This is the first 2-approximation algorithm for treewidth that is faster than the known exact algorithms, and in particular improves upon the previous best approximation ratio of 5 in time $2^{O(k)} n$ given by Bodlaender et al. [SIAM J. Comput., 45 (2016)]. Our algorithm works by applying incremental improvement operations to a tree decomposition, using an approach inspired by a proof of Bellenbaum and Diestel [Comb. Probab. Comput., 11 (2002)].

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arXiv2023

An Improved Parameterized Algorithm for Treewidth

We give an algorithm that takes as input an $n$-vertex graph $G$ and an integer $k$, runs in time $2^{O(k^2)} n^{O(1)}$, and outputs a tree decomposition of $G$ of width at most $k$, if such a decomposition exists. This resolves the long-standing open problem of whether there is a $2^{o(k^3)} n^{O(1)}$ time algorithm for treewidth. In particular, our algorithm is the first improvement on the dependency on $k$ in algorithms for treewidth since the $2^{O(k^3)} n^{O(1)}$ time algorithm given by Bodlaender and Kloks [ICALP 1991] and Lagergren and Arnborg [ICALP 1991]. We also give an algorithm that given an $n$-vertex graph $G$, an integer $k$, and a rational $\varepsilon \in (0,1)$, in time $k^{O(k/\varepsilon)} n^{O(1)}$ either outputs a tree decomposition of $G$ of width at most $(1+\varepsilon)k$ or determines that the treewidth of $G$ is larger than $k$. Prior to our work, no approximation algorithms for treewidth with approximation ratio less than $2$, other than the exact algorithms, were known. Both of our algorithms work in polynomial space.

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arXiv2023

Induced-Minor-Free Graphs: Separator Theorem, Subexponential Algorithms, and Improved Hardness of Recognition

A graph $G$ contains a graph $H$ as an induced minor if $H$ can be obtained from $G$ by vertex deletions and edge contractions. The class of $H$-induced-minor-free graphs generalizes the class of $H$-minor-free graphs, but unlike $H$-minor-free graphs, it can contain dense graphs. We show that if an $n$-vertex $m$-edge graph $G$ does not contain a graph $H$ as an induced minor, then it has a balanced vertex separator of size $O_{H}(\sqrt{m})$, where the $O_{H}(\cdot)$-notation hides factors depending on $H$. More precisely, our upper bound for the size of the balanced separator is $O(\min(|V(H)|^2, \log n) \cdot \sqrt{|V(H)|+|E(H)|} \cdot \sqrt{m})$. We give an algorithm for finding either an induced minor model of $H$ in $G$ or such a separator in randomized polynomial-time. We apply this to obtain subexponential $2^{O_{H}(n^{2/3} \log n)}$ time algorithms on $H$-induced-minor-free graphs for a large class of problems including maximum independent set, minimum feedback vertex set, 3-coloring, and planarization. For graphs $H$ where every edge is incident to a vertex of degree at most 2, our results imply a $2^{O_{H}(n^{2/3} \log n)}$ time algorithm for testing if $G$ contains $H$ as an induced minor. Our second main result is that there exists a fixed tree $T$, so that there is no $2^{o(n/\log^3 n)}$ time algorithm for testing if a given $n$-vertex graph contains $T$ as an induced minor unless the Exponential Time Hypothesis (ETH) fails. Our reduction also gives NP-hardness, which solves an open problem asked by Fellows, Kratochvíl, Middendorf, and Pfeiffer [Algorithmica, 1995], who asked if there exists a fixed planar graph $H$ so that testing for $H$ as an induced minor is NP-hard.

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arXiv2023

New Width Parameters for Independent Set: One-sided-mim-width and Neighbor-depth

We study the tractability of the maximum independent set problem from the viewpoint of graph width parameters, with the goal of defining a width parameter that is as general as possible and allows to solve independent set in polynomial-time on graphs where the parameter is bounded. We introduce two new graph width parameters: one-sided maximum induced matching-width (o-mim-width) and neighbor-depth. O-mim-width is a graph parameter that is more general than the known parameters mim-width and tree-independence number, and we show that independent set and feedback vertex set can be solved in polynomial-time given a decomposition with bounded o-mim-width. O-mim-width is the first width parameter that gives a common generalization of chordal graphs and graphs of bounded clique-width in terms of tractability of these problems. The parameter o-mim-width, as well as the related parameters mim-width and sim-width, have the limitation that no algorithms are known to compute bounded-width decompositions in polynomial-time. To partially resolve this limitation, we introduce the parameter neighbor-depth. We show that given a graph of neighbor-depth $k$, independent set can be solved in time $n^{O(k)}$ even without knowing a corresponding decomposition. We also show that neighbor-depth is bounded by a polylogarithmic function on the number of vertices on large classes of graphs, including graphs of bounded o-mim-width, and more generally graphs of bounded sim-width, giving a quasipolynomial-time algorithm for independent set on these graph classes. This resolves an open problem asked by Kang, Kwon, Strømme, and Telle [TCS 2017].

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arXiv2023

Polynomial-time Approximation of Independent Set Parameterized by Treewidth

We prove the following result about approximating the maximum independent set in a graph. Informally, we show that any approximation algorithm with a ``non-trivial'' approximation ratio (as a function of the number of vertices of the input graph $G$) can be turned into an approximation algorithm achieving almost the same ratio, albeit as a function of the treewidth of $G$. More formally, we prove that for any function $f$, the existence of a polynomial time $(n/f(n))$-approximation algorithm yields the existence of a polynomial time $O(tw \cdot\log{f(tw)}/f(tw))$-approximation algorithm, where $n$ and $tw$ denote the number of vertices and the width of a given tree decomposition of the input graph. By pipelining our result with the state-of-the-art $O(n \cdot (\log \log n)^2/\log^3 n)$-approximation algorithm by Feige (2004), this implies an $O(tw \cdot (\log \log tw)^3/\log^3 tw)$-approximation algorithm.

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arXiv2023

Computing paths of large rank in planar frameworks deterministically

A framework consists of an undirected graph $G$ and a matroid $M$ whose elements correspond to the vertices of $G$. Recently, Fomin et al. [SODA 2023] and Eiben et al. [ArXiV 2023] developed parameterized algorithms for computing paths of rank $k$ in frameworks. More precisely, for vertices $s$ and $t$ of $G$, and an integer $k$, they gave FPT algorithms parameterized by $k$ deciding whether there is an $(s,t)$-path in $G$ whose vertex set contains a subset of elements of $M$ of rank $k$. These algorithms are based on Schwartz-Zippel lemma for polynomial identity testing and thus are randomized, and therefore the existence of a deterministic FPT algorithm for this problem remains open. We present the first deterministic FPT algorithm that solves the problem in frameworks whose underlying graph $G$ is planar. While the running time of our algorithm is worse than the running times of the recent randomized algorithms, our algorithm works on more general classes of matroids. In particular, this is the first FPT algorithm for the case when matroid $M$ is represented over rationals. Our main technical contribution is the nontrivial adaptation of the classic irrelevant vertex technique to frameworks to reduce the given instance to one of bounded treewidth. This allows us to employ the toolbox of representative sets to design a dynamic programming procedure solving the problem efficiently on instances of bounded treewidth.

cs.DS↗

arXiv2023

Two-sets cut-uncut on planar graphs

We study the following Two-Sets Cut-Uncut problem on planar graphs. Therein, one is given an undirected planar graph $G$ and two sets of vertices $S$ and $T$. The question is, what is the minimum number of edges to remove from $G$, such that we separate all of $S$ from all of $T$, while maintaining that every vertex in $S$, and respectively in $T$, stays in the same connected component. We show that this problem can be solved in time $2^{|S|+|T|} n^{O(1)}$ with a one-sided error randomized algorithm. Our algorithm implies a polynomial-time algorithm for the network diversion problem on planar graphs, which resolves an open question from the literature. More generally, we show that Two-Sets Cut-Uncut remains fixed-parameter tractable even when parameterized by the number $r$ of faces in the plane graph covering the terminals $S \cup T$, by providing an algorithm of running time $4^{r + O(\sqrt r)} n^{O(1)}$.

cs.DS↗

arXiv2023

Dynamic treewidth

We present a data structure that for a dynamic graph $G$ that is updated by edge insertions and deletions, maintains a tree decomposition of $G$ of width at most $6k+5$ under the promise that the treewidth of $G$ never grows above $k$. The amortized update time is ${\cal O}_k(2^{\sqrt{\log n}\log\log n})$, where $n$ is the vertex count of $G$ and the ${\cal O}_k(\cdot)$ notation hides factors depending on $k$. In addition, we also obtain the dynamic variant of Courcelle's Theorem: for any fixed property $φ$ expressible in the $\mathsf{CMSO}_2$ logic, the data structure can maintain whether $G$ satisfies $φ$ within the same time complexity bounds. To a large extent, this answers a question posed by Bodlaender [WG 1993].

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arXiv2023

Grid Induced Minor Theorem for Graphs of Small Degree

A graph $H$ is an induced minor of a graph $G$ if $H$ can be obtained from $G$ by vertex deletions and edge contractions. We show that there is a function $f(k, d) = O(k^{10} + 2^{d^5})$ so that if a graph has treewidth at least $f(k, d)$ and maximum degree at most $d$, then it contains a $k \times k$-grid as an induced minor. This proves the conjecture of Aboulker, Adler, Kim, Sintiari, and Trotignon [Eur. J. Comb., 98, 2021] that any graph with large treewidth and bounded maximum degree contains a large wall or the line graph of a large wall as an induced subgraph. It also implies that for any fixed planar graph $H$, there is a subexponential time algorithm for maximum weight independent set on $H$-induced-minor-free graphs.

math.CO↗

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.

cs.DS↗

arXiv2022

Shortest Cycles With Monotone Submodular Costs

We introduce the following submodular generalization of the Shortest Cycle problem. For a nonnegative monotone submodular cost function $f$ defined on the edges (or the vertices) of an undirected graph $G$, we seek for a cycle $C$ in $G$ of minimum cost $\textsf{OPT}=f(C)$. We give an algorithm that given an $n$-vertex graph $G$, parameter $\varepsilon > 0$, and the function $f$ represented by an oracle, in time $n^{\mathcal{O}(\log 1/\varepsilon)}$ finds a cycle $C$ in $G$ with $f(C)\leq (1+\varepsilon)\cdot \textsf{OPT}$. This is in sharp contrast with the non-approximability of the closely related Monotone Submodular Shortest $(s,t)$-Path problem, which requires exponentially many queries to the oracle for finding an $n^{2/3-\varepsilon}$-approximation [Goel et al., FOCS 2009]. We complement our algorithm with a matching lower bound. We show that for every $\varepsilon > 0$, obtaining a $(1+\varepsilon)$-approximation requires at least $n^{Ω(\log 1/ \varepsilon)}$ queries to the oracle. When the function $f$ is integer-valued, our algorithm yields that a cycle of cost $\textsf{OPT}$ can be found in time $n^{\mathcal{O}(\log \textsf{OPT})}$. In particular, for $\textsf{OPT}=n^{\mathcal{O}(1)}$ this gives a quasipolynomial-time algorithm computing a cycle of minimum submodular cost. Interestingly, while a quasipolynomial-time algorithm often serves as a good indication that a polynomial time complexity could be achieved, we show a lower bound that $n^{\mathcal{O}(\log n)}$ queries are required even when $\textsf{OPT} = \mathcal{O}(n)$.

cs.DS↗

arXiv2022

Fixed-Parameter Tractability of Maximum Colored Path and Beyond

We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is as follows. We give a randomized algorithm, that given a colored $n$-vertex undirected graph, vertices $s$ and $t$, and an integer $k$, finds an $(s,t)$-path containing at least $k$ different colors in time $2^k n^{O(1)}$. This is the first FPT algorithm for this problem, and it generalizes the algorithm of Björklund, Husfeldt, and Taslaman [SODA 2012] on finding a path through $k$ specified vertices. It also implies the first $2^k n^{O(1)}$ time algorithm for finding an $(s,t)$-path of length at least $k$. Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an $n$-vertex undirected graph $G$, a matroid $M$ whose elements correspond to the vertices of $G$ and which is represented over a finite field of order $q$, a positive integer weight function on the vertices of $G$, two sets of vertices $S,T \subseteq V(G)$, and integers $p,k,w$, and the task is to find $p$ vertex-disjoint paths from $S$ to $T$ so that the union of the vertices of these paths contains an independent set of $M$ of cardinality $k$ and weight $w$, while minimizing the sum of the lengths of the paths. We give a $2^{p+O(k^2 \log (q+k))} n^{O(1)} w$ time randomized algorithm for this problem.

cs.DS↗

arXiv2022

Finding Optimal Triangulations Parameterized by Edge Clique Cover

We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover ($cc$) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem $cc$ is at most the number of taxa, in fractional hypertreewidth $cc$ is at most the number of hyperedges, and in treewidth of Bayesian networks $cc$ is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most $2^{cc}$, the number of potential maximal cliques is at most $3^{cc}$, and these objects can be listed in times $O^*(2^{cc})$ and $O^*(3^{cc})$, respectively, even when no edge clique cover is given as input; the $O^*(\cdot)$ notation omits factors polynomial in the input size. These enumeration algorithms imply $O^*(3^{cc})$ time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give $O^*(4^m)$ time and $O^*(3^m)$ time algorithms, respectively, where $m$ is the number of hyperedges. When an edge clique cover of size $cc'$ is given as a part of the input we give $O^*(2^{cc'})$ time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an $O^*(2^n)$ time algorithm for perfect phylogeny, where $n$ is the number of taxa. We also give polynomial space algorithms with time complexities $O^*(9^{cc'})$ and $O^*(9^{cc + O(\log^2 cc)})$ for problems in this framework.

cs.DS↗

arXiv2021

Fast FPT-Approximation of Branchwidth

Branchwidth determines how graphs, and more generally, arbitrary connectivity (basically symmetric and submodular) functions could be decomposed into a tree-like structure by specific cuts. We develop a general framework for designing fixed-parameter tractable (FPT) 2-approximation algorithms for branchwidth of connectivity functions. The first ingredient of our framework is combinatorial. We prove a structural theorem establishing that either a sequence of particular refinement operations could decrease the width of a branch decomposition or that the width of the decomposition is already within a factor of 2 from the optimum. The second ingredient is an efficient implementation of the refinement operations for branch decompositions that support efficient dynamic programming. We present two concrete applications of our general framework. $\bullet$ An algorithm that for a given $n$-vertex graph $G$ and integer $k$ in time $2^{2^{O(k)}} n^2$ either constructs a rank decomposition of $G$ of width at most $2k$ or concludes that the rankwidth of $G$ is more than $k$. It also yields a $(2^{2k+1}-1)$-approximation algorithm for cliquewidth within the same time complexity, which in turn, improves to $f(k)n^2$ the running times of various algorithms on graphs of cliquewidth $k$. Breaking the "cubic barrier" for rankwidth and cliquewidth was an open problem in the area. $\bullet$ An algorithm that for a given $n$-vertex graph $G$ and integer $k$ in time $2^{O(k)} n$ either constructs a branch decomposition of $G$ of width at most $2k$ or concludes that the branchwidth of $G$ is more than $k$. This improves over the 3-approximation that follows from the recent treewidth 2-approximation of Korhonen [FOCS 2021].

cs.DS↗

arXiv2021

Lower Bounds on Dynamic Programming for Maximum Weight Independent Set

We prove lower bounds on pure dynamic programming algorithms for maximum weight independent set (MWIS). We model such algorithms as tropical circuits, i.e., circuits that compute with $\max$ and $+$ operations. For a graph $G$, an MWIS-circuit of $G$ is a tropical circuit whose inputs correspond to vertices of $G$ and which computes the weight of a maximum weight independent set of $G$ for any assignment of weights to the inputs. We show that if $G$ has treewidth $w$ and maximum degree $d$, then any MWIS-circuit of $G$ has $2^{Ω(w/d)}$ gates and that if $G$ is planar, or more generally $H$-minor-free for any fixed graph $H$, then any MWIS-circuit of $G$ has $2^{Ω(w)}$ gates. An MWIS-formula is an MWIS-circuit where each gate has fan-out at most one. We show that if $G$ has treedepth $t$ and maximum degree $d$, then any MWIS-formula of $G$ has $2^{Ω(t/d)}$ gates. It follows that treewidth characterizes optimal MWIS-circuits up to polynomials for all bounded degree graphs and $H$-minor-free graphs, and treedepth characterizes optimal MWIS-formulas up to polynomials for all bounded degree graphs.

cs.CC↗