#
On the Descriptive Complexity of Color Coding^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- Our Contributions I: A Syntactic Characterization of Color Coding.

- Our Contributions II: New First-Order Quantifier Elimination Rules.

- Our Contributions III: Slicewise Descriptions and Variable Set Sizes.

#### 1.1. Related Work

#### 1.2. Organization of this Paper

## 2. Describing Parameterized Problems

#### 2.1. Logical Terminology

**Lemma**

**1.**

**Proof.**

#### 2.2. Describing Parameterized Problems

#### 2.3. Parameterized Circuits

- $(\mathcal{A},k)\in Q$ if and only if ${C}_{\left|x\right|,k}\left(x\right)=1$ where x is a binary encoding of $\mathcal{A}$ and $\left|x\right|$ is the length of the encoding,
- the size of each ${C}_{n,k}$ is at most $f\left(k\right)\xb7{n}^{c}$ for some function f and some constant c that is independent of both n and k,
- the depth of each ${C}_{n,k}$ is bounded by some constant that is again independent of both n and k, and
- the circuit family satisfies a dlogtime-uniformity condition. (Since the complex questions around circuit uniformity are not of special importance for the present paper, we keep its discussion short and just remark that dlogtime-uniformity roughly means that arithmetic formulas suffice to describe the circuits.)

**Fact**

**1**

**Theorem**

**1.**

**Proof.**

#### 2.4. Bounded Rank Reductions

**Definition**

**1.**

- each ${f}_{k}$ is a first-order query from τ-structures to ${\tau}^{\prime}$-structures and
- each ${\iota}_{k,{k}^{\prime}}$ is a τ-formula

- 1.
- for each $(\mathcal{A},k)\in \mathrm{STRUC}\left[\tau \right]\times \mathbb{N}$ there is exactly one ${k}^{\prime}\in \mathbb{N}$, denoted by ${\iota}_{k}\left(\mathcal{A}\right)$ in the following, such that $\mathcal{A}\vDash {\iota}_{k,{k}^{\prime}}$,
- 2.
- there is a mapping ${\iota}^{*}:\mathbb{N}\to \mathbb{N}$ such that for all $\mathcal{A}\in \mathrm{STRUC}\left[\tau \right]$ we have ${\iota}_{k}\left(\mathcal{A}\right)\le {\iota}^{*}\left(k\right)$,
- 3.
- $(\mathcal{A},k)\in Q$ if and only if $\left(\right)open="("\; close=")">{f}_{k}\left(\mathcal{A}\right),{\iota}_{k}\left(\mathcal{A}\right)$, and
- 4.
- the quantifier rank of all ${\iota}_{k,{k}^{\prime}}$ and of all formulas inside the ${f}_{k}$ and of the widths of each ${f}_{k}$ is bounded by a constant c.

**Lemma**

**2.**

- 1.
- ${max}_{k}qr\left({\varphi}_{k}\right)={max}_{k}qr\left({\varphi}_{k}^{\prime}\right)+O\left(1\right)$ and
- 2.
- ${max}_{k}\left(\right)open="|"\; close="|">bound\left({\varphi}_{k}\right)+O\left(1\right)$.

**Proof.**

## 3. Syntactic Properties Allowing Color Coding

#### 3.1. Formulas with Color Predicates

**Theorem**

**2.**

- 1.
- For all $\mathcal{A}\in \mathrm{STRUC}\left[\tau \right]$ we have $\mathcal{A}\vDash {\varphi}^{\prime}$ if and only if there is a k-coloring $\mathcal{B}$ of $\mathcal{A}$ with $\mathcal{B}\vDash \varphi $.
- 2.
- $qr\left({\varphi}^{\prime}\right)=qr\left(\varphi \right)+O\left(1\right)$.
- 3.
- $\left(\right)open="|"\; close="|">bound\left({\varphi}^{\prime}\right)+O\left(1\right)$.

**Proof.**

**Definition**

**2.**

- 1.
- Monotonicity property: Let ${A}_{1},\dots ,{A}_{k}\subseteq A$ and ${B}_{1},\dots ,{B}_{k}\subseteq A$ be sets with ${A}_{i}\subseteq {B}_{i}$ for all $i\in \{1,\dots ,k\}$. Then $\mathcal{A}\vDash \gamma ({A}_{1},\dots ,{A}_{k},{a}_{1},\dots ,{a}_{m})$ implies $\mathcal{A}\vDash \gamma ({B}_{1},\dots ,{B}_{k},{a}_{1},\dots ,{a}_{m})$.
- 2.
- Small witness property: If there are sets ${B}_{1},\dots ,{B}_{k}\subseteq A$ such that we have $\mathcal{A}\vDash \gamma ({B}_{1},\dots ,{B}_{k},{a}_{1},\dots ,{a}_{m})$, then there are sets ${A}_{i}\subseteq {B}_{i}$ whose sizes $|{A}_{i}|$ depend only on γ for $i\in \{1,\dots ,k\}$, such that $\mathcal{A}\vDash \gamma ({A}_{1},\dots ,{A}_{k},{a}_{1},\dots ,{a}_{m})$.

**Lemma**

**3.**

- There is a k-coloring $\mathcal{B}$ of $\mathcal{A}$ with $\mathcal{B}\vDash \varphi $.
- $\mathcal{A}\vDash {\bigvee}_{g:\{0,\dots ,{k}^{2}-1\}\to \{1,\dots ,k\}}\exists p\exists q\left({\varphi}^{g}(p,q)\right)$.

**Corollary**

**1.**

- 1.
- $\exists ({C}_{1}\dot{\cup}\cdots \dot{\cup}{C}_{k})\left(\varphi \right)\equiv {\varphi}^{\prime}$ (on finite structures).
- 2.
- $qr\left({\varphi}^{\prime}\right)=qr\left(\varphi \right)+O\left(1\right)$.
- 3.
- $\left(\right)open="|"\; close="|">bound\left({\varphi}^{\prime}\right)+O\left(1\right)$.

#### 3.2. Formulas with Weak Quantifiers

**Definition**

**3.**

- 1.
- some universal binding inside ϕ depends on x or
- 2.
- there is a subformula $\alpha \wedge \beta $ of ϕ such that both α and β contain x in literals that are not of the form $x\ne y$ for some variable y.

- 1.
- some existential binding inside ϕ depends on x or
- 2.
- there is a subformula $\alpha \vee \beta $ such that both α and β contain x in literals that are not of the form $x=y$ for some variable y.

**Theorem**

**3.**

- 1.
- ${\varphi}^{\prime}\equiv \varphi $ (on finite structures),
- 2.
- $qr\left({\varphi}^{\prime}\right)=3\xb7strong-qr\left(\varphi \right)+arity\left(\tau \right)+O\left(1\right)$, and
- 3.
- $\left(\right)open="|"\; close="|">bound\left({\varphi}^{\prime}\right)+arity\left(\tau \right)+O\left(1\right)$.

**Proof of Theorem**

**3.**

- Strong literals are literals that do not contain any weak variables.
- Weak equalities are literals of the form $\dot{x}=y$ involving exactly one strong variable that is existentially bound inside the weak variable’s scope:$$\exists \dot{x}(\dots \exists y(\dots \dot{x}=y\dots )\dots ).$$
- Weak inequalities are literals of the form $\dot{x}\ne \dot{y}$ for two weak variables.

**Claim**

**1.**

**Proof.**

**Corollary**

**2.**

- All weak quantifiers come in blocks, and each such block either directly follows a universal quantifier or follows a disjunction after a universal quantifier. In particular, on any root-to-leaf path in the syntax tree of $\varphi $ between any two blocks of weak quantifiers there is at least one universal quantifier.
- All blocks of weak quantifiers have the form$$\begin{array}{c}\hfill \exists {\dot{x}}_{{i}_{1}}\cdots \exists {\dot{x}}_{{i}_{k}}\left(\right)open="("\; close=")">distinct({P}_{1},\dots ,{P}_{l})\wedge \alpha \end{array}$$

**Claim**

**2.**

- 1.
- $\mathcal{A}\vDash \exists {\dot{x}}_{1}\dots \exists {\dot{x}}_{k}\left(\right)open="("\; close=")">distinct({P}_{1},\dots ,{P}_{l})\wedge \alpha $.
- 2.
- There are elements ${a}_{1},\dots ,{a}_{k}\in \left|\mathcal{A}\right|$ with $\mathcal{A}\vDash \alpha ({a}_{1},\dots ,{a}_{k})$ and such that ${a}_{p}\ne {a}_{q}$ whenever ${\dot{x}}_{p}\in {P}_{i}$, ${\dot{x}}_{q}\in {P}_{j}$, and $i\ne j$.
- 3.
- There is an l-coloring $\mathcal{B}$ of $\mathcal{A}$ such that $\mathcal{B}\vDash {\alpha}^{\prime}$.

**Proof.**

- We added new variables and quantifiers to ${\varphi}^{\prime}$ compared to $\varphi $ during the first transformation steps, but the number we added depended only on the signature $\tau $ (it was the maximum arity of relations in $\tau $ plus possibly 2).
- We then removed all weak variables from $\varphi $ in ${\varphi}^{\prime}$.
- We added some variables to ${\varphi}^{\prime}$ each time we applied Theorem 2 to a block $\beta $. The number of variables we added is constant since Theorem 2 adds only a constant number of variables and since we can always reuse the same set of variables each time the theorem is applied.
- We also added some quantifiers to ${\varphi}^{\prime}$ each time we applied Theorem 2, which increases the quantifier rank of ${\varphi}^{\prime}$ compared to $\varphi $ by more than a constant. However, the essential quantifiers we add are $\exists p\exists q$ and these are always added directly after a universal quantifier or directly after a disjunction after a universal quantifier. Since the strong quantifier rank of $\varphi $ is at least the quantifier rank of $\varphi $ where we only consider the universal quantifiers (the “universal quantifier rank”), the two added nested quantifiers per universal quantifiers can add to the quantifier rank of ${\varphi}^{\prime}$ at most twice the universal quantifier rank.

**Theorem**

**4.**

- 1.
- ${\varphi}^{\prime}\equiv \varphi $ (on finite structures),
- 2.
- $qr\left({\varphi}^{\prime}\right)=3\xb7strong-qr\left(\varphi \right)+arity\left(\tau \right)+O\left(1\right)$, and
- 3.
- $\left(\right)open="|"\; close="|">bound\left({\varphi}^{\prime}\right)+arity\left(\tau \right)+O\left(1\right)$.

**Proof.**

**Claim**

**3.**

**Proof.**

## 4. Syntactic Proofs and Natural Problems

#### 4.1. Syntactic Tools: New Operators

**Notation**

**Strong-qr: $1+strong-qr\left(\varphi \right)$**

**Semantics**- There are k distinct ${a}_{1},\dots ,{a}_{k}\in \left|\mathcal{A}\right|$ with $\mathcal{A}\vDash \varphi \left({a}_{i}\right)$ for all i.
**Expansion**- $\exists {\dot{x}}_{1}\cdots \exists {\dot{x}}_{k}{\bigwedge}_{i\ne j}{\dot{x}}_{i}\ne {\dot{x}}_{j}\wedge {\bigwedge}_{i=1}^{k}\exists x(x={\dot{x}}_{i}\wedge \varphi \left(x\right))$

**Notation**

**Strong-qr: $1+strong-qr\left(\varphi \right)$**

**Semantics**- There are at most k distinct ${a}_{1},\dots ,{a}_{k}\in \left|\mathcal{A}\right|$ with $\mathcal{A}\vDash \varphi \left({a}_{i}\right)$ for all i.
**Expansion**- $\forall {\dot{x}}_{1}\cdots \forall {\dot{x}}_{k+1}{\bigvee}_{i\ne j}{\dot{x}}_{i}={\dot{x}}_{j}\vee {\bigvee}_{i=1}^{k+1}\forall x(x\ne {\dot{x}}_{i}\vee \neg \varphi \left(x\right))$ ($\equiv \neg {\exists}^{\ge k+1}x\left(\varphi \left(x\right)\right)$)

**Notation**

**Strong-qr: $1+strong-qr\left(\varphi \right)$**

**Semantics**- There are exactly k distinct ${a}_{1},\dots ,{a}_{k}\in \left|\mathcal{A}\right|$ with $\mathcal{A}\vDash \varphi \left({a}_{i}\right)$ for all i.
**Expansion**- ${\exists}^{\ge k}x\left(\varphi \left(x\right)\right)\wedge {\exists}^{\le k}x\left(\varphi \left(x\right)\right)$

**Notation**

**Strong-qr: $1+strong-qr\left(\varphi \right)$**

**Semantics**- Let ${a}_{1},\dots ,{a}_{k}\in \left|\mathcal{A}\right|$ be the assignments to the ${\stackrel{\u02da}{x}}_{i}$ (note that they need not be distinct). Then $\{{a}_{1},\dots ,{a}_{k}\}=\left(\right)open="\{"\; close="\}">a\in \left|\mathcal{A}\right||\mathcal{A}\vDash \varphi \left(a\right)$ must hold.
**Expansion**- ${\bigwedge}_{i=1}^{k}\exists x\left(\right)open="("\; close=")">x={\stackrel{\u02da}{x}}_{i}\wedge \varphi \left(x\right)$ // ensure $\{{\stackrel{\u02da}{x}}_{1},\dots ,{\stackrel{\u02da}{x}}_{k}\}\subseteq \{x\mid \varphi \left(x\right)\}$${\bigvee}_{s=1}^{k}({\exists}^{=s}x\left(\varphi \left(x\right)\right)\wedge $ // bind s to $\left|\right\{x\mid \varphi \left(x\right)\left\}\right|$${\bigvee}_{I\subseteq \{1,...,k\},\left|I\right|=s}{\bigwedge}_{i,j\in I,i\ne j}{\stackrel{\u02da}{x}}_{i}\ne {\stackrel{\u02da}{x}}_{j})$. // ensure $\left|\right\{{\stackrel{\u02da}{x}}_{1},\dots ,{\stackrel{\u02da}{x}}_{k}\left\}\right|\ge s$

**Notation**

**Strong-qr: $1+strong-qr\left(\varphi \right)+arity\left(\tau \right)$**

**Semantics**- The set $I=\{a\in |\mathcal{A}|\mid \mathcal{A}\vDash \varphi (a\left)\right\}$ has size exactly k and $\mathcal{A}\left[I\right]\in Q$.
**Expansion**- Assuming for simplicity that $\tau $ contains only ${E}^{2}$ as non-arithmetic predicate:

**Notation**

**Strong-qr: $1+strong-qr\left(\varphi \right)+arity\left(\tau \right)$**

**Semantics**- The set $I=\{a\in |\mathcal{A}|\mid \mathcal{A}\vDash \varphi (a\left)\right\}$ has size at most k and $\mathcal{A}\left[I\right]\in Q$.
**Expansion**- ${\bigvee}_{s=0}^{k}{\mathrm{INDUCED}}^{\mathrm{size}=s}\{x\mid \varphi \left(x\right)\}\in Q$

#### 4.2. Bounded Strong-Rank Description of Vertex Cover

**Proof.**

#### 4.3. Bounded Strong-Rank Description of Hitting Set

**Proof**

**of Theorem 6.**

**Claim**

**4.**

**Proof.**

**Definition**

**4.**

- 1.
- $S(l,0)=c$.
- 2.
- $S(l,0)\cup S(l,1)\cup \dots \cup S(l,L)\in E$.
- 3.
- $S(l,i)\cap S(l,j)=\varnothing $ for $0\le i<j\le L$, but $S(l,i)\ne \varnothing $ for $i\in \{1,\dots ,L\}$.
- 4.
- Let $z\in \{1,\dots ,L\}$ be the smallest number such that ${l}^{z}\ne {m}^{z}$, that is, z is the depth where the path from r to l and the path from r to m diverge for the first time. Then $S(l,z)\cap S(m,z)=\varnothing $ must hold.

#### 4.4. Bounded Strong-Rank Description of Model Checking for First-Order Logic

**Theorem**

**7**

**Proof.**

#### 4.5. Bounded Strong-Rank Description of Embedding Graphs of Constant Tree Width or Constant Tree Depth

**Theorem**

**8**

**Proof.**

**Claim**

**5.**

- 1.
- $\mathcal{G}\vDash {\varphi}_{H,T,B}$ if and only if H embeds into $\mathcal{G}$ (more precisely, into $\left(\right|\mathcal{G}|,{E}^{\mathcal{G}})$).
- 2.
- $strong-qr\left({\varphi}_{H,T,B}\right)=depth\left(T\right)$.
- 3.
- $\left(\right)open="|"\; close="|">strong-bound\left({\varphi}_{H,T,B}\right)$.

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**An example graph H together with a tree decomposition for it, consisting of the tree T and the bag function B indicated using the small gray mapping arrows. A consistent numbering p is indicated in red.

**Table 1.**Membership of the problems discussed in this paper in the two classes $\text{para-}{\mathrm{AC}}^{0}$ and $\text{para-}{\mathrm{AC}}^{0\uparrow}$ and the formulas whose syntactic structures prove this. Except for clique problem, formulas exist where at least the number of strong variables is independent of the parameter—but it is not always the most “natural” formulas that have this desirable property, see the vertex cover problem.

${\mathrm{p}}_{k}\text{-MATCHING}\in \text{para-}{\mathbf{AC}}^{0}$ |

Problem: Does a graph have a size-k matching? |

Formulas: $\exists {\dot{x}}_{1}\cdots \exists {\dot{x}}_{2k}\left(\right)open="("\; close=")">{\bigwedge}_{i\ne j}{\dot{x}}_{i}\ne {\dot{x}}_{j}\wedge {\bigwedge}_{i=1}^{k}E{\dot{x}}_{2i-1}{\dot{x}}_{2i}$, see Equation (1) |

Descriptive complexity: $strong-qr=0$, $strong-vars=0$ |

${\mathrm{p}}_{k}\text{-TRIANGLE-PACKING}\in \text{para-}{\mathrm{AC}}^{0}$ |

Problem: Does a graph contain k vertex-disjoint triangles? |

Formulas: $\exists {\dot{x}}_{1}\cdots \exists {\dot{x}}_{3k}({\bigwedge}_{i\ne j}{\dot{x}}_{i}\ne {\dot{x}}_{j}\wedge {\bigwedge}_{j=1}^{k}\exists x\exists y\exists z$ ( |

${\dot{x}}_{3j-2}=x\wedge {\dot{x}}_{3j-1}=y\wedge {\dot{x}}_{3j}=z\wedge Exy\wedge Exz\wedge Eyz\left)\right)$, see Equation (2) |

Descriptive complexity: $strong-qr=3$, $strong-vars=3$ |

${\mathrm{p}}_{k}\text{-}H\text{-PACKING}\in \text{para-}{\mathrm{AC}}^{0}$ for a fixed graph $H=\left(V\right(H),E(H\left)\right)$ (instead of triangles) |

Problem: Does a graph contain k vertex-disjoint induced copies of H? |

Formulas: $\exists {\dot{x}}_{1}\cdots \exists {\dot{x}}_{k\left|V\right(H\left)\right|}({\bigwedge}_{i\ne j}{\dot{x}}_{i}\ne {\dot{x}}_{j}\wedge {\bigwedge}_{j=1}^{k}\exists {y}_{1}\dots \exists {y}_{\left|V\right(H\left)\right|}$ ( |

${\bigwedge}_{i=1}^{\left|V\right(H\left)\right|}{y}_{i}={\dot{x}}_{(j-1)k+i}\wedge {\bigwedge}_{(p,q)\in E\left(H\right)}E{y}_{p}{y}_{q}\wedge {\bigwedge}_{(p,q)\notin E\left(H\right)}\neg E{y}_{p}{y}_{q}\left)\right)$ |

Descriptive complexity: $strong-qr=\left|V\right(H\left)\right|$, $strong-vars=\left|V\right(H\left)\right|$ |

${\mathrm{p}}_{k}\text{-LONG-PATH}\in \text{para-}{\mathrm{AC}}^{0\uparrow}$ |

Problem: Does a graph contain a path of length k? |

Formulas: $\exists s\exists t\exists x(Esx\wedge \exists {\dot{x}}_{1}({\dot{x}}_{1}=x\wedge \exists y(Exy\wedge \dots ({\dot{x}}_{k}=x\wedge {\bigwedge}_{i\ne j}{\dot{x}}_{i}\ne {\dot{x}}_{j})\dots )))$ |

Descriptive complexity: $strong-qr=k+2$, $strong-vars=4$, see Equation (5) |

${\mathrm{p}}_{k}\text{-VERTEX-COVER}\in \text{para-}{\mathrm{AC}}^{0}$ |

Problem: Does a graph have a size-k vertex cover? |

Formulas 1: $\exists {x}_{1}\cdots \exists {x}_{k}\forall u\forall v\left(\right)open="("\; close=")">Euv\to {\bigvee}_{i}({x}_{i}=u\vee {x}_{i}=v)$ |

Descriptive complexity: $strong-qr=k+2$, $strong-vars=k+2$ |

Formulas 2: Describe the Buss kernelization, see Theorem 5 |

Descriptive complexity: $strong-qr=O\left(1\right)$, $strong-vars=O\left(1\right)$ |

${\mathrm{p}}_{k,d}\text{-HITTING-SET}\in \text{para-}{\mathrm{AC}}^{0}$ |

Problem: Does a hypergraph with maximum edge size d a size-k hitting set? |

Formulas: Describe a kernel based on pseudo-sunflowers, see Theorem 6 |

Descriptive complexity: $strong-qr=O\left(1\right)$, $strong-vars=O\left(1\right)$ |

${\mathrm{p}}_{\psi ,\delta}\text{-MC}\left(\mathrm{FO}\right)\in \text{para-}{\mathrm{AC}}^{0\uparrow}$ |

Problem: Is $\psi $ a model of a graph with maximum degree $\delta $? |

Formulas: Describe the disjointness of balls in the formula resulting from Gaifman’s Theorem using weak variables, see Theorem 7 |

Descriptive complexity: $strong-qr=O(locality-rank(\psi \left)\right)$, $strong-vars=O\left(1\right)$ |

${\mathrm{p}}_{H}{\text{-EMB}}_{\mathrm{td}\left(H\right)\le c}\in \text{para-}{\mathrm{AC}}^{0}$ and ${\mathrm{p}}_{H}{\text{-EMB}}_{\mathrm{tw}\left(H\right)\le c}\in \text{para-}{\mathrm{AC}}^{0\uparrow}$ |

Problem: Can H, of tree depth or width c, be embedded into a graph G? |

Formulas: Bind the vertices of H’s image to weak variables and use a formula whose syntactic structure exactly mimics an optimal tree decomposition of H, see Theorem 8 |

Descriptive complexity: $strong-qr=O(td(H\left)\right)$, $strong-vars=O(tw(H\left)\right)$ |

${\mathrm{p}}_{k}\text{-CLIQUE}\in \mathrm{W}\left[1\right]$ |

Problem: Does a graph contain a k-clique? |

Formulas 1: $\exists {x}_{1}\cdots \exists {x}_{k}\left(\right)open="("\; close=")">{\bigwedge}_{i\ne j}{x}_{i}\ne {x}_{j}\wedge {\bigwedge}_{i\ne j}E{x}_{i}{x}_{j}$, see Equation (3) |

Formulas 2: $\exists {x}_{1}\cdots \exists {x}_{k}\forall u\forall v\left(\right)open="("\; close=")">{\bigvee}_{i\ne j}({x}_{i}=u\wedge {x}_{j}=v)\to Euv$, see Equation (19) |

Descriptive complexity: $strong-qr=k$, $strong-vars=k$ in both cases |

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Bannach, M.; Tantau, T.
On the Descriptive Complexity of Color Coding. *Algorithms* **2021**, *14*, 96.
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Bannach M, Tantau T.
On the Descriptive Complexity of Color Coding. *Algorithms*. 2021; 14(3):96.
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Bannach, Max, and Till Tantau.
2021. "On the Descriptive Complexity of Color Coding" *Algorithms* 14, no. 3: 96.
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