Grundy Distinguishes Treewidth from Pathwidth

Belmonte, Rémy; Kim, Eun Jung; Lampis, Michael; Mitsou, Valia; Otachi, Yota

doi:10.1137/20m1385779

Cited by 3 publications

(1 citation statement)

References 64 publications

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“…As mentioned, Max Min FVS is an example of a wider class of MaxMin problems which have recently attracted much attention in the literature, among the most well-studied of which are Maximum Minimal Vertex Cover [10,11,33] and Upper Dominating Set (which is the standard name for Maximum Minimal Dominating Set) [1,4,20]. Additionally, MaxMin or MinMax versions of cut and separations problems [18,25,29], knapsack problems [21,23], matching problems [13], and coloring problems [5] have also been studied.…”

Section: Introductionmentioning

confidence: 99%

Parameterized Max Min Feedback Vertex Set

Lampis¹,

Melissinos²,

Vasilakis³

2023

Preprint

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Given a graph G and an integer k, Max Min FVS asks whether there exists a minimal set of vertices of size at least k whose deletion destroys all cycles. We present several results that improve upon the state of the art of the parameterized complexity of this problem with respect to both structural and natural parameters. Using standard DP techniques, we first present an algorithm of time tw O(tw) n O(1) , significantly generalizing a recent algorithm of Gaikwad et al. of time vc O(vc) n O(1) , where tw, vc denote the input graph's treewidth and vertex cover respectively. Subsequently, we show that both of these algorithms are essentially optimal, since a vc o(vc) n O(1) algorithm would refute the ETH. With respect to the natural parameter k, the aforementioned recent work by Gaikwad et al. claimed an FPT branching algorithm with complexity 10 k n O(1) . We point out that this algorithm is incorrect and present a branching algorithm of complexity 9.34 k n O(1) .

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Section: Introductionmentioning

confidence: 99%

Parameterized Max Min Feedback Vertex Set

Lampis¹,

Melissinos²,

Vasilakis³

2023

Preprint

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On Approximating Cutwidth and Pathwidth

Bansal,

Katzelnick,

Schwartz

2024

2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)

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The Bounded Pathwidth of Control-Flow Graphs

Conrado,

Goharshady,

Lam

2023

Proc. ACM Program. Lang.

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Pathwidth and treewidth are standard and well-studied graph sparsity parameters which intuitively model the degree to which a given graph resembles a path or a tree, respectively. It is well-known that the control-flow graphs of structured goto-free programs have a tree-like shape and bounded treewidth. This fact has been exploited to design considerably more efficient algorithms for a wide variety of static analysis and compiler optimization problems, such as register allocation, µ-calculus model-checking and parity games, data-flow analysis, cache management, and liftetime-optimal redundancy elimination. However, there is no bound in the literature for the pathwidth of programs, except the general inequality that the pathwidth of a graph is at most O (lg n ) times its treewidth, where n is the number of vertices of the graph. In this work, we prove that control-flow graphs of structured programs have bounded pathwidth and provide a linear-time algorithm to obtain a path decomposition of small width. Specifically, we establish a bound of 2 · d on the pathwidth of programs with nesting depth d . Since real-world programs have small nesting depth, they also have bounded pathwidth. This is significant for a number of reasons: (i) ‍pathwidth is a strictly stronger parameter than treewidth, i.e. ‍any graph family with bounded pathwidth has bounded treewidth, but the converse does not hold; (ii) ‍any algorithm that is designed with treewidth in mind can be applied to bounded-pathwidth graphs with no change; (iii) ‍there are problems that are fixed-parameter tractable with respect to pathwidth but not treewidth; (iv) ‍verification algorithms that are designed based on treewidth would become significantly faster when using pathwidth as the parameter; and (v) ‍it is easier to design algorithms based on bounded pathwidth since one does not have to consider the often-challenging case of merge nodes in treewidth-based dynamic programming. Thus, we invite the static analysis and compiler optimization communities to adopt pathwidth as their parameter of choice instead of, or in addition to, treewidth. Intuitively, control-flow graphs are not only tree-like, but also path-like and one can obtain simpler and more scalable algorithms by relying on path-likeness instead of tree-likeness. As a motivating example, we provide a simpler and more efficient algorithm for spill-free register allocation using bounded pathwidth instead of treewidth. Our algorithm reduces the runtime from O ( n · r 2 · tw · r + 2 · r ) to O ( n · pw · r pw · r + r + 1 ), where n is the number of lines of code, r is the number of registers, pw is the pathwidth of the control-flow graph and tw is its treewidth. We provide extensive experimental results showing that our approach is applicable to a wide variety of real-world embedded benchmarks from SDCC and obtains runtime improvements of 2-3 orders of magnitude. This is because the pathwidth is equal to the treewidth, or one more, in the overwhelming majority of real-world CFGs and thus our algorithm provides an exponential runtime improvement. As such, the benefits of using pathwidth are not limited to the theoretical side and simplicity in algorithm design, but are also apparent in practice.

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Grundy Distinguishes Treewidth from Pathwidth

Cited by 3 publications

References 64 publications

Parameterized Max Min Feedback Vertex Set

Parameterized Max Min Feedback Vertex Set

On Approximating Cutwidth and Pathwidth

The Bounded Pathwidth of Control-Flow Graphs

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