International Symposium on Parameterized and Exact Computation最新文献

From Data Completion to Problems on Hypercubes: A Parameterized Analysis of the Independent Set Problem 从数据完成到超立方体上的问题：独立集问题的参数化分析

International Symposium on Parameterized and Exact Computation Pub Date : 2024-07-15 DOI: 10.4230/LIPIcs.IPEC.2023.16

E. Eiben, R. Ganian, Iyad A. Kanj, S. Ordyniak, Stefan Szeider

{"title":"From Data Completion to Problems on Hypercubes: A Parameterized Analysis of the Independent Set Problem","authors":"E. Eiben, R. Ganian, Iyad A. Kanj, S. Ordyniak, Stefan Szeider","doi":"10.4230/LIPIcs.IPEC.2023.16","DOIUrl":"https://doi.org/10.4230/LIPIcs.IPEC.2023.16","url":null,"abstract":"Several works have recently investigated the parameterized complexity of data completion problems, motivated by their applications in machine learning, and clustering in particular. Interestingly, these problems can be equivalently formulated as classical graph problems on induced subgraphs of powers of partially-defined hypercubes. In this paper, we follow up on this recent direction by investigating the Independent Set problem on this graph class, which has been studied in the data science setting under the name Diversity. We obtain a comprehensive picture of the problem's parameterized complexity and establish its fixed-parameter tractability w.r.t. the solution size plus the power of the hypercube. Given that several such FO-definable problems have been shown to be fixed-parameter tractable on the considered graph class, one may ask whether fixed-parameter tractability could be extended to capture all FO-definable problems. We answer this question in the negative by showing that FO model checking on induced subgraphs of hypercubes is as difficult as FO model checking on general graphs.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"15 5","pages":"16:1-16:14"},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141648148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On the Parameterized Complexity of Multiway Near-Separator 论多路近分隔器的参数化复杂性

International Symposium on Parameterized and Exact Computation Pub Date : 2023-10-06 DOI: 10.48550/arXiv.2310.04332

B. Jansen, S. K. Roy

{"title":"On the Parameterized Complexity of Multiway Near-Separator","authors":"B. Jansen, S. K. Roy","doi":"10.48550/arXiv.2310.04332","DOIUrl":"https://doi.org/10.48550/arXiv.2310.04332","url":null,"abstract":"We study a new graph separation problem called Multiway Near-Separator. Given an undirected graph $G$, integer $k$, and terminal set $T subseteq V(G)$, it asks whether there is a vertex set $S subseteq V(G) setminus T$ of size at most $k$ such that in graph $G-S$, no pair of distinct terminals can be connected by two pairwise internally vertex-disjoint paths. Hence each terminal pair can be separated in $G-S$ by removing at most one vertex. The problem is therefore a generalization of (Node) Multiway Cut, which asks for a vertex set for which each terminal is in a different component of $G-S$. We develop a fixed-parameter tractable algorithm for Multiway Near-Separator running in time $2^{O(k log k)} * n^{O(1)}$. Our algorithm is based on a new pushing lemma for solutions with respect to important separators, along with two problem-specific ingredients. The first is a polynomial-time subroutine to reduce the number of terminals in the instance to a polynomial in the solution size $k$ plus the size of a given suboptimal solution. The second is a polynomial-time algorithm that, given a graph $G$ and terminal set $T subseteq V(G)$ along with a single vertex $x in V(G)$ that forms a multiway near-separator, computes a 14-approximation for the problem of finding a multiway near-separator not containing $x$.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"263 1","pages":"28:1-28:18"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139322121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Kernelization for Counting Problems on Graphs: Preserving the Number of Minimum Solutions 图上计数问题的核化：保留最小解的数量

International Symposium on Parameterized and Exact Computation Pub Date : 2023-10-06 DOI: 10.48550/arXiv.2310.04303

B. Jansen, Bart van der Steenhoven

{"title":"Kernelization for Counting Problems on Graphs: Preserving the Number of Minimum Solutions","authors":"B. Jansen, Bart van der Steenhoven","doi":"10.48550/arXiv.2310.04303","DOIUrl":"https://doi.org/10.48550/arXiv.2310.04303","url":null,"abstract":"A kernelization for a parameterized decision problem $mathcal{Q}$ is a polynomial-time preprocessing algorithm that reduces any parameterized instance $(x,k)$ into an instance $(x',k')$ whose size is bounded by a function of $k$ alone and which has the same yes/no answer for $mathcal{Q}$. Such preprocessing algorithms cannot exist in the context of counting problems, when the answer to be preserved is the number of solutions, since this number can be arbitrarily large compared to $k$. However, we show that for counting minimum feedback vertex sets of size at most $k$, and for counting minimum dominating sets of size at most $k$ in a planar graph, there is a polynomial-time algorithm that either outputs the answer or reduces to an instance $(G',k')$ of size polynomial in $k$ with the same number of minimum solutions. This shows that a meaningful theory of kernelization for counting problems is possible and opens the door for future developments. Our algorithms exploit that if the number of solutions exceeds $2^{mathsf{poly}(k)}$, the size of the input is exponential in terms of $k$ so that the running time of a parameterized counting algorithm can be bounded by $mathsf{poly}(n)$. Otherwise, we can use gadgets that slightly increase $k$ to represent choices among $2^{O(k)}$ options by only $mathsf{poly}(k)$ vertices.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"1 1","pages":"27:1-27:15"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139322558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Twin-Width of Graphs with Tree-Structured Decompositions 具有树状结构分解的图的孪生宽度

International Symposium on Parameterized and Exact Computation Pub Date : 2023-08-28 DOI: 10.4230/LIPIcs.IPEC.2023.25

Irene Heinrich, Simon Rassmann

{"title":"Twin-Width of Graphs with Tree-Structured Decompositions","authors":"Irene Heinrich, Simon Rassmann","doi":"10.4230/LIPIcs.IPEC.2023.25","DOIUrl":"https://doi.org/10.4230/LIPIcs.IPEC.2023.25","url":null,"abstract":"The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 (Bonnet et. al. 2020), a mass of new results has appeared relating twin width to group theory, model theory, combinatorial optimization, and structural graph theory. We take a detailed look at the interplay between the twin-width of a graph and the twin-width of its components under tree-structured decompositions: We prove that the twin-width of a graph is at most twice its strong tree-width, contrasting nicely with the result of (Bonnet and D'epr'es 2022), which states that twin-width can be exponential in tree-width. Further, we employ the fundamental concept from structural graph theory of decomposing a graph into highly connected components, in order to obtain an optimal linear bound on the twin-width of a graph given the widths of its biconnected components. For triconnected components we obtain a linear upper bound if we add red edges to the components indicating the splits which led to the components. Extending this approach to quasi-4-connectivity, we obtain a quadratic upper bound. Finally, we investigate how the adhesion of a tree decomposition influences the twin-width of the decomposed graph.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"32 1","pages":"25:1-25:17"},"PeriodicalIF":0.0,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139348729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Obstructions to faster diameter computation: Asteroidal sets 阻碍更快的直径计算:小行星集合

International Symposium on Parameterized and Exact Computation Pub Date : 2022-09-26 DOI: 10.48550/arXiv.2209.12438

G. Ducoffe

{"title":"Obstructions to faster diameter computation: Asteroidal sets","authors":"G. Ducoffe","doi":"10.48550/arXiv.2209.12438","DOIUrl":"https://doi.org/10.48550/arXiv.2209.12438","url":null,"abstract":"An extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let $Ext_{alpha}$ be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than $alpha$ pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every $m$-edge graph in $Ext_{alpha}$ can be computed in deterministic ${cal O}(alpha^3 m^{3/2})$ time. We then improve the runtime to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive $+1$-approximation of all vertex eccentricities in deterministic ${cal O}(alpha^2 m)$ time. This is in sharp contrast with general $m$-edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in ${cal O}(m^{2-epsilon})$ time for any $epsilon>0$. As important special cases of our main result, we derive an ${cal O}(m^{3/2})$-time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an ${cal O}(k^3m^{3/2})$-time algorithm for this problem on graphs of asteroidal number at most $k$. We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121634349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On Sparse Hitting Sets: from Fair Vertex Cover to Highway Dimension 关于稀疏命中集:从公平顶点覆盖到高速公路维度

International Symposium on Parameterized and Exact Computation Pub Date : 2022-08-30 DOI: 10.48550/arXiv.2208.14132

Johannes Blum, Y. Disser, A. Feldmann, Siddharth Gupta, Anna Zych-Pawlewicz

{"title":"On Sparse Hitting Sets: from Fair Vertex Cover to Highway Dimension","authors":"Johannes Blum, Y. Disser, A. Feldmann, Siddharth Gupta, Anna Zych-Pawlewicz","doi":"10.48550/arXiv.2208.14132","DOIUrl":"https://doi.org/10.48550/arXiv.2208.14132","url":null,"abstract":"We consider the Sparse Hitting Set (Sparse-HS) problem, where we are given a set system $(V,mathcal{F},mathcal{B})$ with two families $mathcal{F},mathcal{B}$ of subsets of $V$. The task is to find a hitting set for $mathcal{F}$ that minimizes the maximum number of elements in any of the sets of $mathcal{B}$. Our focus is on determining the complexity of some special cases of Sparse-HS with respect to the sparseness $k$, which is the optimum number of hitting set elements in any set of $mathcal{B}$. For the Sparse Vertex Cover (Sparse-VC) problem, $V$ is given by the vertex set of a graph, and $mathcal{F}$ is its edge set. We prove NP-hardness for sparseness $kgeq 2$ and polynomial time solvability for $k=1$. We also provide a polynomial-time $2$-approximation for any $k$. A special case of Sparse-VC is Fair Vertex Cover (Fair-VC), where the family $mathcal{B}$ is given by vertex neighbourhoods. For this problem we prove NP-hardness for constant $k$ and provide a polynomial-time $(2-frac{1}{k})$-approximation. This is better than any approximation possible for Sparse-VC or Vertex Cover (under UGC). We then consider two problems derived from Sparse-HS related to the highway dimension, a graph parameter modelling transportation networks. Most algorithms for graphs of low highway dimension compute solutions to the $r$-Shortest Path Cover ($r$-SPC) problem, where $r>0$, $mathcal{F}$ contains all shortest paths of length between $r$ and $2r$, and $mathcal{B}$ contains all balls of radius $2r$. There is an XP algorithm that computes solutions to $r$-SPC of sparseness at most $h$ if the input graph has highway dimension $h$, but the existence if an FPT algorithm was open. We prove that $r$-SPC and also the related $r$-Highway Dimension ($r$-HD) problem are both W[1]-hard. Furthermore, we prove that $r$-SPC admits a polynomial-time $O(log n)$-approximation.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126579455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2

On the parameterized complexity of symmetric directed multicut 对称定向多路切的参数化复杂度

International Symposium on Parameterized and Exact Computation Pub Date : 2022-08-18 DOI: 10.48550/arXiv.2208.09017

E. Eiben, Clément Rambaud, Magnus Wahlstrom

{"title":"On the parameterized complexity of symmetric directed multicut","authors":"E. Eiben, Clément Rambaud, Magnus Wahlstrom","doi":"10.48550/arXiv.2208.09017","DOIUrl":"https://doi.org/10.48550/arXiv.2208.09017","url":null,"abstract":"We study the problem Symmetric Directed Multicut from a parameterized complexity perspective. In this problem, the input is a digraph $D$, a set of cut requests $C={(s_1,t_1),ldots,(s_ell,t_ell)}$ and an integer $k$, and the task is to find a set $X subseteq V(D)$ of size at most $k$ such that for every $1 leq i leq ell$, $X$ intersects either all $(s_i,t_i)$-paths or all $(t_i,s_i)$-paths. Equivalently, every strongly connected component of $D-X$ contains at most one vertex out of $s_i$ and $t_i$ for every $i$. This problem is previously known from research in approximation algorithms, where it is known to have an $O(log k log log k)$-approximation. We note that the problem, parameterized by $k$, directly generalizes multiple interesting FPT problems such as (Undirected) Vertex Multicut and Directed Subset Feedback Vertex Set. We are not able to settle the existence of an FPT algorithm parameterized purely by $k$, but we give three partial results: An FPT algorithm parameterized by $k+ell$; an FPT-time 2-approximation parameterized by $k$; and an FPT algorithm parameterized by $k$ for the special case that the cut requests form a clique, Symmetric Directed Multiway Cut. The existence of an FPT algorithm parameterized purely by $k$ remains an intriguing open possibility.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"47 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126049113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 3

Exact Exponential Algorithms for Clustering Problems 聚类问题的精确指数算法

International Symposium on Parameterized and Exact Computation Pub Date : 2022-08-14 DOI: 10.48550/arXiv.2208.06847

F. Fomin, P. Golovach, Tanmay Inamdar, Nidhi Purohit, Saket Saurabh

{"title":"Exact Exponential Algorithms for Clustering Problems","authors":"F. Fomin, P. Golovach, Tanmay Inamdar, Nidhi Purohit, Saket Saurabh","doi":"10.48550/arXiv.2208.06847","DOIUrl":"https://doi.org/10.48550/arXiv.2208.06847","url":null,"abstract":"In this paper we initiate a systematic study of exact algorithms for well-known clustering problems, namely $k$-Median and $k$-Means. In $k$-Median, the input consists of a set $X$ of $n$ points belonging to a metric space, and the task is to select a subset $C subseteq X$ of $k$ points as centers, such that the sum of the distances of every point to its nearest center is minimized. In $k$-Means, the objective is to minimize the sum of squares of the distances instead. It is easy to design an algorithm running in time $max_{kleq n} {n choose k} n^{O(1)} = O^*(2^n)$ ($O^*(cdot)$ notation hides polynomial factors in $n$). We design first non-trivial exact algorithms for these problems. In particular, we obtain an $O^*((1.89)^n)$ time exact algorithm for $k$-Median that works for any value of $k$. Our algorithm is quite general in that it does not use any properties of the underlying (metric) space -- it does not even require the distances to satisfy the triangle inequality. In particular, the same algorithm also works for $k$-Means. We complement this result by showing that the running time of our algorithm is asymptotically optimal, up to the base of the exponent. That is, unless ETH fails, there is no algorithm for these problems running in time $2^{o(n)} cdot n^{O(1)}$. Finally, we consider the\"supplier\"versions of these clustering problems, where, in addition to the set $X$ we are additionally given a set of $m$ candidate centers $F$, and objective is to find a subset of $k$ centers from $F$. The goal is still to minimize the $k$-Median/$k$-Means/$k$-Center objective. For these versions we give a $O(2^n (mn)^{O(1)})$ time algorithms using subset convolution. We complement this result by showing that, under the Set Cover Conjecture, the supplier versions of these problems do not admit an exact algorithm running in time $2^{(1-epsilon) n} (mn)^{O(1)}$.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131639558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Domination and Cut Problems on Chordal Graphs with Bounded Leafage 有界叶弦图的支配与割问题

International Symposium on Parameterized and Exact Computation Pub Date : 2022-08-04 DOI: 10.48550/arXiv.2208.02850

Esther Galby, D. Marx, Philipp Schepper, Roohani Sharma, P. Tale

引用次数: 0

Parameterized Complexity of Streaming Diameter and Connectivity Problems 流直径的参数化复杂度与连通性问题

International Symposium on Parameterized and Exact Computation Pub Date : 2022-07-11 DOI: 10.48550/arXiv.2207.04872

Jelle J. Oostveen, E. J. V. Leeuwen

{"title":"Parameterized Complexity of Streaming Diameter and Connectivity Problems","authors":"Jelle J. Oostveen, E. J. V. Leeuwen","doi":"10.48550/arXiv.2207.04872","DOIUrl":"https://doi.org/10.48550/arXiv.2207.04872","url":null,"abstract":"We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size $k$ allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is $O(log n)$ for any fixed $k$. Underlying these algorithms is a method to execute a breadth-first search in $O(k)$ passes and $O(k log n)$ bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where $Omega(n/p)$ bits of memory is needed for any $p$-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph $H$, for most $H$. For some cases, we can also show one-pass, $Omega(n log n)$ bits of memory lower bounds. We also prove a much stronger $Omega(n^2/p)$ lower bound for Diameter on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size $k$. This yields a kernel of $2k$ vertices (with $O(k^2)$ edges) produced as a stream in $text{poly}(k)$ passes and only $O(k log n)$ bits of memory.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130471712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0