Degrees and Gaps: Tight Complexity Results of General Factor Problems Parameterized by Treewidth and Cutwidth

Marx, Dániel; Sankar, Govind S.; Schepper, Philipp

doi:10.4230/lipics.icalp.2021.95

Cited by 5 publications

(36 citation statements)

References 32 publications

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“…The proof of Theorem 1.3 follows mostly the ideas of the algorithm for GenFac from Theorem 1.3 in [32]. The main difference is that X is finite and thus X is cofinite.…”

Section: Parameterizing By the Maximum Excluded Degreementioning

confidence: 99%

“…The problems above can be unified under the General Factor (GenFac) problem [12,30,32], where one is given a graph G and an associated list of integers B v for every vertex v of G. The objective is to find a subgraph such that every vertex v has its degree in B v . Cornuéjols [12] showed that GenFac is polynomial-time solvable if every set B v has maximum gap at most 1.…”

Section: Introductionmentioning

confidence: 99%

“…Given the apparent hardness of B-Factor problems in general, Marx et al [32] studied the complexity of the problem on bounded treewidth graphs. Treewidth is a parameter of a graph that indicates how "tree-like" it is.…”

Section: Introductionmentioning

confidence: 99%

“…For a wide range of hard problems, algorithms with running time of the form f (k) • n O (1) exist if the input graph comes with a tree decomposition of width k. Moreover, in many cases, we have good understanding of the best possible form of f (k) in the running time (under suitable complexity assumptions, such as the Exponential Time Hypothesis (ETH) or the Strong Exponential Time Hypothesis (SETH) put forward by Impagliazzo and Paturi [25]). Marx et al [32] studied the B-Factor problem for this viewpoint, parameterized by treewidth for finite, fixed lists B. They showed that a combination of standard dynamic programming techniques with fast subset convolution (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, this form of running time is essentially optimal conditioned on the SETH. Theorem 1.1 (Theorems 1.3-1.6 in [32]). Fix a finite, non-empty B ⊆ N.…”

Section: Introductionmentioning

confidence: 99%

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Anti-Factor is FPT Parameterized by Treewidth and List Size (but Counting is Hard)

Marx¹,

Sankar²,

Schepper³

2021

Preprint

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In the general AntiFactor problem, a graph G is given with a set Xv ⊆ N of forbidden degrees for every vertex v and the task is to find a set S of edges such that the degree of v in S is not in the set Xv. Standard techniques (dynamic programming + 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 Xv are sparse: our main algorithmic result shows that if every vertex has at most x forbidden degrees (we call this special case AntiFactorx), then the problem can be solved in time (x + 1) O(tw) • n O(1) . That is, the AntiFactorx 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 Xv = X of forbidden degrees. We show the following lower bound for every fixed set X: if there is an > 0 such that #X-AntiFactor can be solved in time (max X + 2 − ) tw • n O(1) on a tree decomposition of width tw, then the Counting Strong Exponential-Time Hypothesis (#SETH) fails.

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“…The proof of Theorem 1.3 follows mostly the ideas of the algorithm for GenFac from Theorem 1.3 in [32]. The main difference is that X is finite and thus X is cofinite.…”

Section: Parameterizing By the Maximum Excluded Degreementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Furthermore, this form of running time is essentially optimal conditioned on the SETH. Theorem 1.1 (Theorems 1.3-1.6 in [32]). Fix a finite, non-empty B ⊆ N.…”

Section: Introductionmentioning

confidence: 99%

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Anti-Factor is FPT Parameterized by Treewidth and List Size (but Counting is Hard)

Marx¹,

Sankar²,

Schepper³

2021

Preprint

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Counting list homomorphisms from graphs of bounded treewidth: tight complexity bounds

Focke¹,

Marx²,

Rzążewski

2022

Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

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The goal of this work is to give precise bounds on the counting complexity of a family of generalized coloring problems (list homomorphisms) on bounded-treewidth graphs. Given graphs 𝐺, 𝐻 , and lists 𝐿(𝑣) ⊆ 𝑉 (𝐻 ) for every 𝑣 ∈ 𝑉 (𝐺), a list homomorphism is a function 𝑓 : 𝑉 (𝐺) → 𝑉 (𝐻 ) that preserves the edges (i.e., 𝑢𝑣 ∈ 𝐸 (𝐺) implies 𝑓 (𝑢) 𝑓 (𝑣) ∈ 𝐸 (𝐻 )) and respects the lists (i.e., 𝑓 (𝑣) ∈ 𝐿(𝑣)). Standard techniques show that if 𝐺 is given with a tree decomposition of width 𝑡, then the number of list homomorphisms can be counted in time |𝑉 (𝐻 )| 𝑡 • 𝑛 O (1) . Our main result is determining, for every fixed graph 𝐻 , how much the base |𝑉 (𝐻 )| in the running time can be improved. For a connected graph 𝐻 we define irr(𝐻 ) in the following way: if 𝐻 has a loop or is nonbipartite, then irr(𝐻 ) is the maximum size of a set 𝑆 ⊆ 𝑉 (𝐻 ) where any two vertices have different neighborhoods; if 𝐻 is bipartite, then irr(𝐻 ) is the maximum size of such a set that is fully in one of the bipartition classes. For disconnected 𝐻 , we define irr(𝐻 ) as the maximum of irr(𝐶) over every connected component 𝐶 of 𝐻 . It follows from earlier results that if irr(𝐻 ) = 1, then the problem of counting list homomorphisms to 𝐻 is polynomial-time solvable, and otherwise it is #P-hard. We show that, for every fixed graph 𝐻 , the number of list homomorphisms from (𝐺, 𝐿) to 𝐻• can be counted in time irr(𝐻 ) 𝑡 • 𝑛 O (1) if a tree decomposition of 𝐺 having width at most 𝑡 is given in the input, and • given that irr(𝐻 ) ≥ 2, cannot be counted in time (irr(𝐻 ) − 𝜀) 𝑡 • 𝑛 O (1) for any 𝜀 > 0, even if a tree decomposition of 𝐺 having width at most 𝑡 is given in the input, unless the Counting Strong Exponential-Time Hypothesis (#SETH) fails.Thereby we give a precise and complete complexity classification featuring matching upper and lower bounds for all target graphs with or without loops.CCS Concepts: • Theory of computation → Parameterized complexity and exact algorithms; • Mathematics of computing → Combinatorics.

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Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs

Focke¹,

Marx²,

Inerney³

et al. 2023

Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

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For a well-studied family of domination-type problems, in bounded-treewidth graphs, we investigate whether it is possible to find faster algorithms. For sets σ, ρ of non-negative integers, a (σ, ρ)-set of a graph G is a set S of vertices such that |N (u) ∩ S| ∈ σ for every u ∈ S, and |N (v) ∩ S| ∈ ρ for every v ̸ ∈ S. The problem of finding a (σ, ρ)-set (of a certain size) unifies common problems like Independent Set, Dominating Set, Independent Dominating Set, and many others.In an accompanying paper, it is proven that, for all pairs of finite or cofinite sets (σ, ρ), there is an algorithm that counts (σ, ρ)-sets in time (c σ,ρ ) tw • n O(1) (if a tree decomposition of width tw is given in the input). Here, c σ,ρ is a constant with an intricate dependency on σ and ρ. Despite this intricacy, we show that the algorithms in the accompanying paper are most likely optimal, i.e., for any pair (σ, ρ) of finite or cofinite sets where the problem is non-trivial, and any ε > 0, a (c σ,ρ − ε) tw • n O(1) -algorithm counting the number of (σ, ρ)-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets σ and ρ, our lower bounds also extend to the decision version, showing that those algorithms are optimal in this setting as well.

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Degrees and Gaps: Tight Complexity Results of General Factor Problems Parameterized by Treewidth and Cutwidth

Cited by 5 publications

References 32 publications

Anti-Factor is FPT Parameterized by Treewidth and List Size (but Counting is Hard)

Anti-Factor is FPT Parameterized by Treewidth and List Size (but Counting is Hard)

Counting list homomorphisms from graphs of bounded treewidth: tight complexity bounds

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs

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