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.Comment: To appear in the 31st Annual European Symposium on Algorithms (ESA 2023

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⋅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 ⋅ (log log n)²/log³n)-approximation algorithm by Feige (2004), this implies an O(tw⋅(log log tw)³/log³tw)-approximation algorithm

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 an integer r>1, and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of 1. r for maximum independent set in O∗(exp(O~(n/rlog2r+rlog2r))) time, 2. r for chromatic number in O∗(exp(O~(n/rlogr+rlog2r))) time, 3. (2−1/r) for minimum vertex cover in O∗(exp(n/rΩ(r))) time, and 4. (k−1/r) for minimum k-hypergraph vertex cover in O∗(exp(n/(kr)Ω(kr))) time. (Throughout, O~ and O∗ omit polyloglog(r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were O∗(2n/r) (Bourgeois et al. i

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 an integer r>1, and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of 1. r for maximum independent set in O∗(exp(O~(n/rlog2r+rlog2r))) time, 2. r for chromatic number in O∗(exp(O~(n/rlogr+rlog2r))) time, 3. (2−1/r) for minimum vertex cover in O∗(exp(n/rΩ(r))) time, and 4. (k−1/r) for minimum k-hypergraph vertex cover in O∗(exp(n/(kr)Ω(kr))) time. (Throughout, O~ and O∗ omit polyloglog(r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were O∗(2n/r) (Bourgeois et al. in Discret Appl Math 159(17):1954–1970, 2011; Cygan et al. in Exponential-time approximation of hard problems, 2008). For maximum independent set and chromatic number, these bounds were complemented by exp(n1−o(1)/r1+o(1)) lower bounds (under the Exponential Time Hypothesis (ETH)) (Chalermsook et al. in Foundations of computer science, FOCS, pp. 370–379, 2013; Laekhanukit in Inapproximability of combinatorial problems in subexponential-time. Ph.D. thesis, 2014). Our results show that the naturally-looking O∗(2n/r) bounds are not tight for all these problems. The key to these results is a sparsification procedure that reduces a problem to a bounded-degree variant, allowing the use of approximation algorithms for bounded-degree graphs. To obtain 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 in J. ACM 63(3):27:1–27:32, 2016). It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture (Dinur in Electron Colloq Comput Complex (ECCC) 23:128, 2016; Manurangsi and Raghavendra in A birthday repetition theorem and complexity of approximating dense CSPs, 2016)

Streaming algorithms for geometric Steiner forest

We consider an important generalization of the Steiner tree problem, the Steiner forest problem, in the Euclidean plane: the input is a multiset X ⊆ R^2, partitioned into k color classes C1, C2, . . . , Ck ⊆ X. The goal is to find a minimum-cost Euclidean graph G such that every color class Ci is connected in G. We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to X. Each input point x ∈ X arrives with its color color(x) ∈ [k], and as usual for dynamic geometric streams, the input is restricted to the discrete grid {0, . . . , ∆}^2. We design a single-pass streaming algorithm that uses poly(k · log ∆) space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio α2 (currently 1.1547 ≤ α2 ≤ 1.214). This approximation guarantee matches the state of the art bound for streaming Steiner tree, i.e., when k = 1. Our approach relies on a novel combination of streaming techniques, like sampling and linear sketching, with the classical Arora-style dynamic-programming framework for geometric optimization problems, which usually requires large memory and has so far not been applied in the streaming setting. We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite approximation requires Ω(k) bits of space

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum