Formal Language Techniques for Space Lower Bounds Philipp Kuinke - - PowerPoint PPT Presentation

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Formal Language Techniques for Space Lower Bounds Philipp Kuinke - - PowerPoint PPT Presentation

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Formal Language Techniques for Space Lower Bounds Philipp Kuinke February 23, 2018 Contained in S anchez Villaamils Phd Thesis 2017 Treewidth

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Formal Language Techniques for Space Lower Bounds

Philipp Kuinke February 23, 2018

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Contained in S´ anchez Villaamil’s Phd Thesis 2017

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Treewidth

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Dynamic Programming

Use treewidth structure to traverse the graph

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Dynamic Programming

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Dynamic Programming

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Dynamic Programming

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Dynamic Programming

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Dynamic Programming

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Dynamic Programming

The runtime of dynamic programming algorithms depends on the table sizes!

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Dynamic Programming

Common properties of DP-algorithms we formalize

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Dynamic Programming

Common properties of DP-algorithms we formalize

  • 1. They do a single pass over the decomposition;
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Dynamic Programming

Common properties of DP-algorithms we formalize

  • 1. They do a single pass over the decomposition;
  • 2. they use O(f (w) logO(1) n) space; and
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Dynamic Programming

Common properties of DP-algorithms we formalize

  • 1. They do a single pass over the decomposition;
  • 2. they use O(f (w) logO(1) n) space; and
  • 3. they do not modify or rearrange the decomposition.
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Dynamic Programming

Definition (DPTM)

A Dynamic Programming Turing Machine (DPTM) is a Turing Machine with an input read-only tape, whose head moves only in one direction and a separate working tape. It only accepts well-formed instances as inputs.

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Boundaried Graphs

Definition

An s-boundaried graph G is a graph with s distinguished vertices, called the boundary.

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Boundaried Graphs

Definition

G1 ⊕ G2 is the disjoint union of two s-boundaried graphs merged at the boundary.

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Boundaried Graphs

Definition

Gs is the set of all s-boundaried graphs.

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Formal Languages

Interpret Problem as a language Π, i.e. G ∈ Π if and only if G is a yes-instance.

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Myhill-Nerode Families

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Myhill-Nerode Families

Definition (Myhill-Nerode family)

A set H ⊆ Gs is an s-Myhill-Nerode family for a DP language Π if

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Myhill-Nerode Families

Definition (Myhill-Nerode family)

A set H ⊆ Gs is an s-Myhill-Nerode family for a DP language Π if

  • 1. For every subset I ⊆ H there exists an s-boundaried

graph GI with bounded size, such that for every H ∈ H it holds that GI ⊕ H ∈ Π ⇔ H ∈ I

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Myhill-Nerode Families

Definition (Myhill-Nerode family)

A set H ⊆ Gs is an s-Myhill-Nerode family for a DP language Π if

  • 1. For every subset I ⊆ H there exists an s-boundaried

graph GI with bounded size, such that for every H ∈ H it holds that GI ⊕ H ∈ Π ⇔ H ∈ I

  • 2. For every H ∈ H it holds that H has bounded size.
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Myhill-Nerode Families

Definition (Myhill-Nerode family)

A set H ⊆ Gs is an s-Myhill-Nerode family for a DP language Π if

  • 1. For every subset I ⊆ H there exists an s-boundaried

graph GI with |GI| = |H| logO(1) H, such that for every H ∈ H it holds that GI ⊕ H ∈ Π ⇔ H ∈ I

  • 2. For every H ∈ H it holds that |H| = |H| logO(1) H.
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Myhill-Nerode Families

GI ⊕ H1 ∈ Π GI ⊕ H2 ∈ Π

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DPTM bounds

Lemma ([S´ anchez Villaamil ’17])

Let ǫ > 0 and Π be a DP decision problem such that for every s there exists an s-Myhill-Nerode family H for Π of size cs and width tw(H) = s. Then no DPTM can decide Π using space O((c − ǫ)k log n), where n is the size of the input and k the treewidth of the input.

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DPTM bounds

Lemma ([S´ anchez Villaamil ’17])

Let ǫ > 0 and Π be a DP decision problem such that for every s there exists an s-Myhill-Nerode family H for Π of size cs/f (s), where f (s) = sO(1) ∩ Θ(1) and width tw(H) = s + o(s). Then no DPTM can decide Π using space O((c − ǫ)k logO(1) n), where n is the size of the input and k the treewidth of the input.

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3-Coloring

◮ Input: A Graph G ◮ k: The treewidth of G ◮ Question: Can G be colored with 3 colors?

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Coloring Gadget

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Coloring Gadget

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Coloring Gadget

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Coloring Gadget

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Coloring Gadget

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Coloring Gadget

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The Graph ΓX

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Enforcing Colorings with HX

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No-Instances

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No-Instances

This is not 3-colorable.

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Yes-Instances

This is 3-colorable.

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Yes-Instances

This is 3-colorable.

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Myhill-Nerode Families

◮ ΓX ⊕ HX ∈ Π

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Myhill-Nerode Families

◮ ΓX ⊕ HX ∈ Π ◮ ΓX ⊕ HX ′ ∈ Π, for (X = X ′)

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Myhill-Nerode Families

◮ ΓX ⊕ HX ∈ Π ◮ ΓX ⊕ HX ′ ∈ Π, for (X = X ′) ◮ GI = ⊕HX ∈IΓX

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Myhill-Nerode Families

◮ ΓX ⊕ HX ∈ Π ◮ ΓX ⊕ HX ′ ∈ Π, for (X = X ′) ◮ GI = ⊕HX ∈IΓX ◮ GI ⊕ HX ∈ Π ⇔ HX ∈ Π

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Myhill-Nerode Families

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Myhill-Nerode Families

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Myhill-Nerode Families

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Myhill-Nerode Families

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Myhill-Nerode Families

  • GI ⊕ HX ∈ Π ⇔ HX ∈ Π
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Myhill-Nerode Families

  • We can generate 3w/6 such graphs.
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Myhill-Nerode Families

  • We can generate a Myhill-Nerode family of index 3w/6.
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Myhill-Nerode Families

  • We cannot use O((3 − ǫ)w · log n) space for a dynamic

programming algorithm.

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Obtained result

Theorem ([S´ anchez Villaamil ’17])

No DPTM solves 3-Coloring on a treewidth-decomposition

  • f width w with space bounded by O((3 − ǫ)w · logO(1) n).
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Further results

Theorem ([S´ anchez Villaamil ’17])

No DPTM solves Vertex Cover on a treewidth-decomposition of width w with space bounded by O((2 − ǫ)w · logO(1) n).

Theorem ([S´ anchez Villaamil ’17])

No DPTM solves Dominating Set on a treewidth-decomposition of width w with space bounded by O((3 − ǫ)w · logO(1) n).

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Not Captured

◮ Compression. ◮ Algebraic techniques. ◮ Preprocessing to compute optimal traversal. ◮ Branching instead of DP

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The end