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APPROX

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

We consider probabilistic embedding of metric spaces into ultra-metrics (or equivalently to a constant factor, into hierarchically separated trees) to minimize the expected distortion of any pairwise distance. Such embeddings have been widely used in network design and online algorithms. Our main result is a polynomial time algorithm that approximates the optimal distortion on any instance to within a constant factor. We achieve this via a novel LP formulation that reduces this problem to a probabilistic version of uniform metric labeling.

Kamesh Munagala, Govind S. Sankar, and Erin Taylor. Probabilistic Metric Embedding via Metric Labeling. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 2:1-2:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{munagala_et_al:LIPIcs.APPROX/RANDOM.2023.2, author = {Munagala, Kamesh and Sankar, Govind S. and Taylor, Erin}, title = {{Probabilistic Metric Embedding via Metric Labeling}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {2:1--2:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.2}, URN = {urn:nbn:de:0030-drops-188279}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.2}, annote = {Keywords: Metric Embedding, Approximation Algorithms, Ultrametrics} }

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**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

In the general AntiFactor problem, a graph G and, for every vertex v of G, a set X_v ⊆ ℕ of forbidden degrees is given. The task is to find a set S of edges such that the degree of v in S is not in the set X_v. Standard techniques (dynamic programming plus fast convolution) can be used to show that if M is the largest forbidden degree, then the problem can be solved in time (M+2)^{tw}⋅n^{O(1)} if a tree decomposition of width tw is given. However, significantly faster algorithms are possible if the sets X_v are sparse: our main algorithmic result shows that if every vertex has at most x forbidden degrees (we call this special case AntiFactor_x), then the problem can be solved in time (x+1)^{O(tw)}⋅n^{O(1)}. That is, AntiFactor_x is fixed-parameter tractable parameterized by treewidth tw and the maximum number x of excluded degrees.
Our algorithm uses the technique of representative sets, which can be generalized to the optimization version, but (as expected) not to the counting version of the problem. In fact, we show that #AntiFactor₁ is already #W[1]-hard parameterized by the width of the given decomposition. Moreover, we show that, unlike for the decision version, the standard dynamic programming algorithm is essentially optimal for the counting version. Formally, for a fixed nonempty set X, we denote by X-AntiFactor the special case where every vertex v has the same set X_v = X of forbidden degrees. We show the following lower bound for every fixed set X: if there is an ε > 0 such that #X-AntiFactor can be solved in time (max X+2-ε)^{tw}⋅n^{O(1)} given a tree decomposition of width tw, then the Counting Strong Exponential-Time Hypothesis (#SETH) fails.

Dániel Marx, Govind S. Sankar, and Philipp Schepper. Anti-Factor Is FPT Parameterized by Treewidth and List Size (But Counting Is Hard). In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{marx_et_al:LIPIcs.IPEC.2022.22, author = {Marx, D\'{a}niel and Sankar, Govind S. and Schepper, Philipp}, title = {{Anti-Factor Is FPT Parameterized by Treewidth and List Size (But Counting Is Hard)}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {22:1--22:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.22}, URN = {urn:nbn:de:0030-drops-173780}, doi = {10.4230/LIPIcs.IPEC.2022.22}, annote = {Keywords: Anti-Factor, General Factor, Treewidth, Representative Sets, SETH} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

In the General Factor problem, we are given an undirected graph G and for each vertex v ∈ V(G) a finite set B_v of non-negative integers. The task is to decide if there is a subset S ⊆ E(G) such that deg_S(v) ∈ B_v for all vertices v of G. Define the max-gap of a finite integer set B to be the largest d ≥ 0 such that there is an a ≥ 0 with [a,a+d+1] ∩ B = {a,a+d+1}. Cornuéjols showed in 1988 that if the max-gap of all sets B_v is at most 1, then the decision version of General Factor is polynomial-time solvable. This result was extended 2018 by Dudycz and Paluch for the optimization (i.e. minimization and maximization) versions. We present a general algorithm counting the number of solutions of a certain size in time #2 (M+1)^{tw}^{𝒪(1)}, given a tree decomposition of width tw, where M is the maximum integer over all B_v. By using convolution techniques from van Rooij (2020), we improve upon the previous (M+1)^{3tw}^𝒪(1) time algorithm by Arulselvan et al. from 2018.
We prove that this algorithm is essentially optimal for all cases that are not trivial or polynomial time solvable for the decision, minimization or maximization versions. Our lower bounds show that such an improvement is not even possible for B-Factor, which is General Factor on graphs where all sets B_v agree with the fixed set B. We show that for every fixed B where the problem is NP-hard, our (max B+1)^tw^𝒪(1) algorithm cannot be significantly improved: assuming the Strong Exponential Time Hypothesis (SETH), no algorithm can solve B-Factor in time (max B+1-ε)^tw^𝒪(1) for any ε > 0. We extend this bound to the counting version of B-Factor for arbitrary, non-trivial sets B, assuming #SETH.
We also investigate the parameterization of the problem by cutwidth. Unlike for treewidth, having a larger set B does not appear to make the problem harder: we give a 2^cutw^𝒪(1) algorithm for any B and provide a matching lower bound that this is optimal for the NP-hard cases.

Dániel Marx, Govind S. Sankar, and Philipp Schepper. Degrees and Gaps: Tight Complexity Results of General Factor Problems Parameterized by Treewidth and Cutwidth. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 95:1-95:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{marx_et_al:LIPIcs.ICALP.2021.95, author = {Marx, D\'{a}niel and Sankar, Govind S. and Schepper, Philipp}, title = {{Degrees and Gaps: Tight Complexity Results of General Factor Problems Parameterized by Treewidth and Cutwidth}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {95:1--95:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.95}, URN = {urn:nbn:de:0030-drops-141647}, doi = {10.4230/LIPIcs.ICALP.2021.95}, annote = {Keywords: General Factor, General Matching, Treewidth, Cutwidth} }