Fat 객체 교차 그래프에서 강인한 서브지수 알고리즘 설계

📝 Abstract

We study the design of robust subexponential algorithms for classical connectivity problems on intersection graphs of similarly sized fat objects in R d . In this setting, each vertex corresponds to a geometric object, and two vertices are adjacent if and only if their objects intersect. We introduce a new tool for designing such algorithms, which we call a λ-linked partition. This is a partition of the vertex set into groups of highly connected vertices. Crucially, such a partition can be computed in polynomial time and does not require access to the geometric representation of the graph. We apply this framework to problems related to paths and cycles in graphs. First, we obtain the first robust ETH-tight algorithms for Hamiltonian Path and Hamiltonian Cycle, running in time 2 O(n 1-1/d ) on intersection graphs of similarly sized fat objects in R d . This resolves an open problem of de Berg et al. [STOC 2018] and completes the study of these problems on geometric intersection graphs from the viewpoint of ETH-tight exact algorithms. We further extend our approach to the parameterized setting and design the first robust subexponential parameterized algorithm for Long Path in any fixed dimension d. More precisely, we obtain a randomized robust algorithm running in time 2 O(k 1-1/d log 2 k) n O(1) on intersection graphs of similarly sized fat objects in R d , where k is the natural parameter. Besides λ-linked partitions, our algorithm also relies on a low-treewidth pattern covering theorem that we establish for geometric intersection graphs, which may be viewed as a refinement of a result of Marx-Pilipczuk [ESA 2017].

💡 Analysis

We study the design of robust subexponential algorithms for classical connectivity problems on intersection graphs of similarly sized fat objects in R d . In this setting, each vertex corresponds to a geometric object, and two vertices are adjacent if and only if their objects intersect. We introduce a new tool for designing such algorithms, which we call a λ-linked partition. This is a partition of the vertex set into groups of highly connected vertices. Crucially, such a partition can be computed in polynomial time and does not require access to the geometric representation of the graph. We apply this framework to problems related to paths and cycles in graphs. First, we obtain the first robust ETH-tight algorithms for Hamiltonian Path and Hamiltonian Cycle, running in time 2 O(n 1-1/d ) on intersection graphs of similarly sized fat objects in R d . This resolves an open problem of de Berg et al. [STOC 2018] and completes the study of these problems on geometric intersection graphs from the viewpoint of ETH-tight exact algorithms. We further extend our approach to the parameterized setting and design the first robust subexponential parameterized algorithm for Long Path in any fixed dimension d. More precisely, we obtain a randomized robust algorithm running in time 2 O(k 1-1/d log 2 k) n O(1) on intersection graphs of similarly sized fat objects in R d , where k is the natural parameter. Besides λ-linked partitions, our algorithm also relies on a low-treewidth pattern covering theorem that we establish for geometric intersection graphs, which may be viewed as a refinement of a result of Marx-Pilipczuk [ESA 2017].

📄 Content

Given a set F of objects in R d , the intersection graph of F is defined as the graph having one vertex for each object in F, and an edge between two vertices whenever the corresponding objects intersect. One of the most extensively studied classes of intersection graphs are the unit disk graphs, obtained when the objects are disks in R 2 of identical radius. In this work, we consider families F of similarly sized β-fat objects in R d (for some constants d, β ≥ 1) which means that for every object O ∈ F, there are two balls B in and B out of R d such that B in ⊆ O ⊆ B out , where B in has radius 1 and B out has radius β.

In their seminal work, de Berg et al.

[4] introduced a general framework for deriving algorithms with running times that are tight under the Exponential Time Hypothesis (ETH) for a broad range of classical problems on intersection graphs of similarly sized fat objects. Their framework encompasses problems such as Maximum Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle. The key idea of their approach is the construction of a partition P = (V 1 , . . . , V t ) of the vertex set of the input graph G satisfying the following properties:

(i) each induced subgraph G[V i ] can be further partitioned into at most κ cliques, and (ii) the quotient graph of the parts, denoted G P , has bounded degree and treewidth O(n 1-1/d ).

Notice that P can be computed without access to the geometric representation of F , and is enough to solve all the aforementioned problems except for Hamiltonian Cycle. Algorithms of this type, which operate solely on the intersection graph and do not require a geometric representation, are referred to as robust algorithms. We stress here that robustness is a substantial advantage since for many classes of intersection graphs such as unit disk graphs, computing a representation is ∃R-complete [11].

However, the algorithm of [4] solving Hamiltonian Cycle requires one additional step: computing a partition of each V i into a bounded number of cliques. For this step, de Berg et al. used the geometric representation. They explicitly left open the question of whether a robust ETH-tight algorithm for Hamiltonian Cycle could be obtained. One possible direction toward such a result would be to design a robust algorithm that computes a partition into a constant number of cliques whenever the input graph is known to contain one. Unfortunately, recent advances on clique partitions of geometric graphs suggest that this task is likely to be difficult [12]. 1 It is worth noting that, in the case of unit disk graphs, such a clique partition can be obtained using the approach of [14].

We circumvent this problem by using a new type of partition into highly connected subgraphs called λ-linked partitions. We show the two following crucial properties : these partitions can be used to solve the considered problems (instead of partitions into cliques) and they can be computed in polynomial time (and, importantly, without relying on the geometric representation) in graphs known to admit a partition into a constant number of cliques.

Combining these findings with the framework of de Berg et al., we obtain the following.

▶ Theorem 1. For every constants d ⩾ 1 and β ⩾ 1 there is a robust algorithm solving Hamiltonian Path (resp. Hamiltonian Cycle) in time 2 O(n 1-1/d ) on intersection graphs of similarly sized β-fat objects in R d .

This result matches the ETH-based lower bound of [4] and resolves an open problem from the same paper.

To further demonstrate the applicability of our techniques, we show in a second part of the paper how they can be used to deal with parameterized version of Hamiltonian Path, that is, Long Path parameterized by the length of the path. For this problem and for Long Cycle, Fomin et al. proved in [9] that there are algorithms running in time 2 (1) on intersection graphs of similarly sized fat objects in R2 . Their approach relies on computing a clique partition of the vertex set and analyzing the resulting clique-grid graph. The idea of using clique partitions in the context of subexponential parameterized algorithms was introduced in [8] and has since become a central tool in the area [16]. However, the clique-grid graph is typically defined using a geometric representation of the input graph. Designing a robust subexponential parameterized algorithm for Long Path in dimensions d ⩾ 3 has remained an open problem. Indeed, while the bidimensionality-based approach of [9] does not generalize to higher dimensions, the high-dimensional approach of [8], based on Baker’s technique [5], again requires access to the geometric representation.

We show that this dependency can be removed by combining two key techniques. The first consists, as in the case of the Hamiltonian Path problem, in replacing the clique partition with a partition into highly connected parts. The second is an adaptation of the low-treewidth pattern covering technique, introduced by Fo

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