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Recent Advances in Positive-Instance Driven Graph Searching^{ †}

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## Abstract

**:**

## 1. Introduction

**Contribution I: A Simple Description of Tamaki’s First Algorithm.**We describe Tamaki’s algorithm as a well-known graph searching game. This provides a link to known theory and allows us to analyze the algorithm in depth.**Contribution II. Extending Tamaki’s Algorithm to Other Parameters**The game theoretic point of view allows us to extend the algorithm naturally to various other parameters—including pathwidth and treedepth.**Contribution III: A Novel Randomized Data Structure.**The bottleneck in positive-instance driven algorithms is the enumeration of already computed solutions. We present a lazily constructed randomized data structure that, in contrast to existing data structures for this task, provides a guarantee that certain useless solutions are not enumerated with high probability.

#### 1.1. Related Work

#### 1.2. Organization of This Paper

#### 1.3. Difference to the Conference Paper

## 2. Preliminaries: Graphs and Their Decompositions

#### Graph Decompositions

## 3. A Gentle Introduction to Positive-Instance Driven Graph Searching

#### 3.1. Graph Searching

- The searchers pick a vertex $v\in C$ on which they want to place the next searcher. We say they clean the vertex v.
- The fugitive responds by picking a component ${C}^{\prime}$ of $G[C\setminus \{v\left\}\right]$. The contaminated area is reduced to ${C}^{\prime}$ and the game proceeds only on this subgraph.

**Fact**

**1**

**([61]).**

#### 3.2. Simple Positive-Instance Driven Graph Searching

#### 3.3. Alternative Characterization

#### 3.4. Execution Modes

## 4. A Unifying Take on Positive-Instance Driven Graph Searching

- The searchers perform one of the following:
- Place a searcher on a contaminated vertex;
- Remove a searcher from a vertex;
- Reveal the current position of the fugitive.

- The fugitive responds as follows:
- If the searchers place or remove a searcher, the fugitive adapts her connected component by adding or removing the vertex, respectively. (This may join multiple components or disconnect the current component, in which case the fugitive selects one of the resulting connected components).
- If the searchers perform a reveal, the fugitive responds by uncovering her current connected component C. The contaminated area is reduced to C.

#### 4.1. Entering the Arena and the Colosseum

#### 4.2. Simplifying the Game

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

#### 4.3. Building the Colosseum

#### 4.4. Fighting in the Pit

**Definition**

**1**(Universal Consistent)

**.**

**Example**

**1.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 4.5. Distance Queries in Edge-Alternating Graphs

**Definition**

**2**(Edge-Alternating Distance)

**.**

**Lemma**

**4.**

**Proof**

**of**

**Lemma**

**4.**

**Theorem**

**2.**

**Proof.**

**treewidth:**To solve treewidth, it is sufficient to find any edge-alternating path from the vertex ${C}_{s}=V(G)$ to a vertex in Q. We can find a path by choosing ${\omega}_{E}$ and ${\omega}_{A}$ as $(x,y)\mapsto 0$, and by setting ${c}_{0}=0$.**pathwidth:**In the pathwidth game, the searchers are not allowed to perform any reveal [76]. Hence, universal edges cannot be used and we set ${\omega}_{A}$ to $(x,y)\mapsto \infty $. By setting ${\omega}_{E}$ to $(x,y)\mapsto 0$ and ${c}_{0}=0$, we again only need to find some path from $V(G)$ to Q with weight less than ∞.**treedepth:**In the game for treedepth, the searchers are not allowed to remove a placed searcher again [61]. Hence, the searchers can only use k existential edges. Choosing ${\omega}_{E}$ as $(x,y)\mapsto 1$, ${\omega}_{A}$ as $(x,y)\mapsto 0$, and ${c}_{0}=1$ is sufficient. We have to search a path of weight at most k.**$\mathit{q}$-branched treewidth:**For q-branched treewidth, we wish to use at most q reveals [62]. By choosing ${\omega}_{E}$ as $(x,y)\mapsto 0$, ${\omega}_{A}$ as $(x,y)\mapsto 1$, and ${c}_{0}=0$, we have to search for a path of weight at most q.**dependency treewidth:**This parameter is, in essence, defined via graph searching game that is equal to the game we study with some fly- and reveal-moves forbidden. Forbidding a move can be achieved by setting the weight of the corresponding edge to ∞ and by searching for an edge-alternating path of weight less than ∞. □

#### 4.6. Extending the Algorithm to Directed Treewidth

**Observation**

**1.**

**Corollary**

**1.**

**Proof.**

## 5. Color Coding Sieves

**Definition**

**3.**

- $C\u22d6{C}^{\prime}$;
- $|{N}_{G}(C\cup {C}^{\prime})|\le k$;
- $C\cap {C}^{\prime}=\varnothing $, ${N}_{G}(C)\cap {C}^{\prime}=\varnothing $, and $C\cap {N}_{G}({C}^{\prime})=\varnothing $.

#### 5.1. Insert a Block to the Color Coding Sieve

#### 5.2. Query the Color Coding Sieve

**Lemma**

**5.**

**Proof.**

#### 5.3. Optimizing Level-2 Sieves

**Lemma**

**6.**

**Proof**

**of**

**Lemma**

**6**

**Corollary**

**2.**

**Definition**

**4.**

**Theorem**

**3**

**.**For all natural numbers n and k, an $(n,k,2)$-universal coloring family Λ of size $|\Lambda |\le {2}^{O(k)}\xb7{log}^{2}(n)$ can be found in time ${2}^{O(k)}\xb7n\xb7{log}^{2}(n)$.

**Theorem**

**4.**

#### 5.4. Pruning Queries in Level-3 Sieves

## 6. Experimental Evaluation of Color Coding Sieves

#### 6.1. Color Coding Sieves on Hyperbolic Random Graphs

- Vary the used layers and the number of colorings $\gamma $;
- Run the solver on random graphs rather than predefined benchmark sets.

#### 6.2. Sieve-Quality of the Individual Layers

## 7. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Proof of Theorem 2

**treewidth:**Choose ${\omega}_{E}$ and ${\omega}_{A}$ as $(x,y)\mapsto 0$, and set ${c}_{0}=0$.**pathwidth:**Set ${\omega}_{A}$ to $(x,y)\mapsto \infty $, ${\omega}_{E}$ to $(x,y)\mapsto 0$, and ${c}_{0}=0$.**treedepth:**Choosing ${\omega}_{E}$ as $(x,y)\mapsto 1$, ${\omega}_{A}$ as $(x,y)\mapsto 0$, and ${c}_{0}=1$.**$\mathit{q}$-branched treewidth:**Set ${\omega}_{E}$ to $(x,y)\mapsto 0$, ${\omega}_{A}$ to $(x,y)\mapsto 1$, and ${c}_{0}=0$.**dependency treewidth**As for treewidth, but we have to set the weight of some forbidden edges to infinity.

#### Appendix A.1. Computing Branched Tree Decompositions

**Claim**

**A1.**

**Proof.**

**Fact**

**A1.**

#### Appendix A.1.1. From Tree Decompositions to Edge-Alternating Paths

- If i has exactly one child j, we can find a path ${P}_{1}$ of existential edges leading from C to a configuration ${C}_{1}$ with $N({C}_{1})\subseteq \iota (i)\cap \iota (j)$. Moreover, we can also find a path ${P}_{2}$ of existential edges from ${C}_{1}$ to a configuration ${C}_{2}$ with $N({C}_{2})\subseteq \iota (j)$. The path ${P}_{1}$ will be constructed by iteratively removing all vertices $v\in C$ with $N(v)\cap [\iota (i)\setminus \iota (j)]\ne \varnothing $. For the remaining vertices ${C}_{1}$, we have $N({C}_{1})\subseteq \iota (i)\cap \iota (j)$. If all configurations that we aim to visit on ${P}_{1}$ exist, the corresponding edges also exist by definition. Assume that we are in some configuration ${C}^{\prime}$ with $N({C}^{\prime})\cap [\iota (i)\setminus \iota (j)]\ne \varnothing $ and want to remove a vertex $v\in {C}^{\prime}$ with $N(v)\cap [\iota (i)\setminus \iota (j)]\ne \varnothing $, but ${C}^{\prime}\setminus \left\{v\right\}\notin V(\mathrm{colosseum}(G,k+1))$. By definition of $\mathrm{colosseum}(G,k+1)$, this means that $|N({C}^{\prime}\setminus \left\{v\right\})|\ge k+2$. As we wanted to remove v, we have $N(v)\cap \iota (i)\ne \varnothing $. As $N({C}^{\prime}\setminus \left\{v\right\})\subseteq N({C}^{\prime})\cup \left\{v\right\}$ and $|N({C}^{\prime}\setminus \left\{v\right\})|\ge k+2$, we know that there is some $u\in {C}^{\prime}$ with $v\in N(u)$. Fact Appendix A.1 implies that $v\in \iota (i)\cap \iota (j)$, a contradiction and, hence, all configurations in ${P}_{1}$ exist.Similarly, we construct ${P}_{2}$ by iteratively removing all vertices in $\iota (j)$ from ${C}_{1}$. It is easy to see that the neighborhood of the visited configurations will always be a subset of $\iota (j)$ and, hence, all configurations on this path exist.We have arrived at a configuration ${C}_{2}$ with $N({C}_{2})\subseteq \iota (j)$ and due to Fact A1:$$\begin{array}{c}\hfill {C}_{2}\subseteq \left[\bigcup _{{j}^{\prime}\in \mathrm{desc}(j)}\iota ({j}^{\prime})\right]\setminus \iota (j).\end{array}$$
- If node i has a set of children J with $\left|J\right|\ge 2$, we will use universal edges. Let $\mathcal{C}$ be the connected components of $G[{\bigcup}_{j\in \mathrm{desc}(i)}\iota (j)\setminus \iota (i)]$. We claim that for each component $\Gamma \in \mathcal{C}$ there is a unique index $j(\Gamma )\in J$ such that $\Gamma \cap \iota (j(\Gamma ))\ne \varnothing $. If no such index exists, we have $\iota (j)=\iota (i)$. We can iteratively remove such bags $\iota (j)$ until this cannot happen anymore. If two indices ${j}_{1},{j}_{2}\in J$ exist with $\iota ({j}_{1})\cap \Gamma \ne \varnothing $ and $\iota ({j}_{2})\cap \Gamma \ne \varnothing $, the connectivity property implies that $\iota (i)\cap \Gamma \ne \varnothing $, a contradiction to our assumption. Hence, for each component $\Gamma $, we follow the universal edge to $\Gamma $ and then proceed as above: first, we find a path ${P}_{1}$ of existential edges from $\Gamma $ to a configuration ${\Gamma}_{1}$ with $N({\Gamma}_{1})\subseteq \iota (i)\cap \iota (j(\Gamma ))$ and then a path ${P}_{2}$ of existential edges from ${\Gamma}_{1}$ to a configuration ${\Gamma}_{2}$ with $N({\Gamma}_{2})\subseteq \iota (j(\Gamma ))$. The same arguments as above imply that all configurations on these paths exist and that we arrive at a configuration ${\Gamma}_{2}$ with $N({\Gamma}_{2})\subseteq \iota (j(\Gamma ))$ and:$$\begin{array}{c}\hfill {\Gamma}_{2}\subseteq \left[\bigcup _{{j}^{\prime}\in \mathrm{desc}(j(\Gamma ))}\iota ({j}^{\prime})\right]\setminus \iota (j(\Gamma )).\end{array}$$

#### Appendix A.1.2. From Edge-Alternating Paths to Tree Decompositions

- If $q=0$, the path P does not use any universal edges. Let $\pi ={\pi}_{1},\dots ,{\pi}_{s}$ be any classical directed path from the initial configuration V to some winning configuration $\left\{{v}^{*}\right\}$ in $\mathrm{pit}(G,k+1)$ that only uses vertices from P. As the initial configuration is ${\pi}_{1}=V$, the winning configuration is ${\pi}_{s}=\left\{{v}^{*}\right\}$, and there are only existential edges $(C,{C}^{\prime})$ with $|{C}^{\prime}|=|C|-1$ in $\mathrm{pit}(G,k+1)$, we know that $|{\pi}_{i}|=|V|-i+1$, and thus $s=\left|V\right|$. We say that vertex $v\in V$ is removed at time i, if $v\in {\bigcap}_{j=1}^{i}{\pi}_{j}$ and $v\notin {\bigcup}_{j=i+1}^{\left|V\right|}{\pi}_{j}$. We also say that ${v}^{*}$ was removed at time $\left|V\right|$. For $i=1,\dots ,\left|V\right|$, let ${v}_{i}$ be the vertex removed at time i.We will now construct a 0-branched tree decomposition $(T,\iota )$, i. e., a path decomposition. As T is a path, let ${t}_{1},\dots ,{t}_{\left|V\right|}$ be the vertices on the path in their respective ordering with root ${t}_{1}$. We set $\iota ({t}_{i})=N({\pi}_{i})\cup \left\{{v}_{i}\right\}$. For $i=1,\dots ,\left|V\right|-1$, there is an existential edge leading from ${\pi}_{i}$ to ${\pi}_{i+1}$ and thus $|N({\pi}_{i})|\le k$. As ${\pi}_{\left|V\right|}=\left\{{v}_{\left|V\right|}\right\}$ is a winning configuration, we also have $|N({\pi}_{\left|V\right|})|\le k$. Hence, the resulting decomposition T has width at most k. As T is a path, it is also 0-branched.We now need to verify that $(T,\iota )$ is indeed a valid tree decomposition. As every vertex v is removed at some time i, we have $v={v}_{i}$ and thus $v\in \iota ({t}_{i})$. Hence, every vertex is in some bag. Let $\{{v}_{i},{v}_{{i}^{\prime}}\}$ be any edge with $i<{i}^{\prime}$. As ${v}_{{i}^{\prime}}\in {\pi}_{{i}^{\prime}}$ and ${v}_{i}\notin {\pi}_{{i}^{\prime}}$, we have ${v}_{i}\in N({\pi}_{{i}^{\prime}})$ and thus $\{{v}_{i},{v}_{{i}^{\prime}}\}\subseteq N({\pi}_{{i}^{\prime}})\cup \left\{{v}_{{i}^{\prime}}\right\}=\iota ({t}_{{i}^{\prime}})$. Hence, every edge is in some bag. Finally, let ${v}_{i}\in V$. Clearly, as ${v}_{i}\in {\pi}_{1}$, ${v}_{i}\in {\pi}_{2}$,…, ${v}_{i}\in {\pi}_{i-1}$, the first bag where ${v}_{i}$ might appear is $\iota ({t}_{i})$. Let ${v}_{{i}^{\prime}}\in N({v}_{i})$ be the neighbor of ${v}_{i}$ that is removed at the latest time. If ${i}^{\prime}<i$, we have $N({v}_{i})\cap {\bigcup}_{j=i+1}^{\left|V\right|}{\pi}_{j}=\varnothing $ and ${v}_{i}$ thus only appears in $\iota ({t}_{i})$. If $i<{i}^{\prime}$, then ${v}_{i}\in {\bigcap}_{j=i+1}^{{i}^{\prime}}N({\pi}_{j})$ and hence ${v}_{i}\in {\bigcap}_{j=i+1}^{{i}^{\prime}}\iota ({t}_{j})$.
- Now, assume that $q\ge 1$ and that we can construct for every ${q}^{\prime}<q$ a ${q}^{\prime}$-branched tree decomposition of width at most k from any ${q}^{\prime}$-branched edge-alternating path P in $\mathrm{pit}(G,k+1)$. Consider the directed acyclic subgraph H in $\mathrm{pit}(G,k+1)$ induced by P. A configuration $C\in V(H)$ is called a universal configuration, if ${N}_{A}(C)\subseteq V(H)$ and a top-level universal configuration with respect to some directed path $\pi $ if C is the first universal configuration on $\pi $. Note that we can reduce P in such a way that all directed paths $\pi $ from the initial configuration V to some winning configuration $\left\{{v}^{*}\right\}$ in H have the same top-level universal configuration, call it ${C}^{*}$. Let $V={\pi}_{1},\dots ,{\pi}_{i}={C}^{*}$ be the shared existential path from V to ${C}^{*}$ in H and let ${N}_{A}({C}^{*})=\{{C}_{1},\dots ,{C}_{\ell}\}$ be the universal children of ${C}^{*}$. Note that $\{{C}_{1},\dots ,{C}_{\ell}\}\subseteq P$ due to the definition of an edge-alternating path. For each child ${C}_{j}$, the edge-alternating path P contains a directed path ${\pi}^{(j)}$ from ${C}_{j}$ to some final configuration in $\mathrm{pit}(G,k+1)$. Furthermore, each ${\pi}^{(j)}$ contains at most ${q}^{\prime}\le q-1$ universal edges (otherwise, P would not be q-branched). Hence, by induction hypothesis, we can construct a ${q}^{\prime}$-branched tree decomposition $({T}^{(j),{\iota}^{(j)}})$ for the subgraph induced by the vertices contained in the path ${\pi}^{(j)}$ with root ${r}^{(j)}$.Now, we use the same construction as above to construct a path $({T}^{\prime}=({t}_{1}^{\prime},\dots ,{t}_{i}^{\prime}),{\iota}^{\prime})$ from ${\pi}_{1},\dots ,{\pi}_{i}$ and for each path ${\pi}^{(j)}$ we add the root ${r}^{(j)}$ of the ${q}^{\prime}$-branched tree decomposition $({T}^{(j)},\iota (j))$ as a child of bag ${t}_{i}$ to obtain our final tree decomposition $(T,\iota )$. As there is a universal edge from ${C}^{*}$ to ${C}_{j}$, we know that ${C}_{j}$ is a component of ${C}^{*}$. As all $({T}^{(j)},\iota (j))$ are valid $q-1$-branched tree decompositions of width at most k, we can thus conclude that $(T,\iota )$ is a valid q-branched tree decomposition of width k.

#### Appendix A.2. Computing Treedepth Decompositions

**Claim**

**A2.**

**Proof**

**.**

#### Appendix A.3. Computing Dependency Treewidth

**Claim**

**A3.**

**Proof**

**.**

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**Figure 1.**Various tree decompositions of an undirected graph $G=(V,E)$ shown at (

**a**). The decompositions justify (

**b**) $\mathrm{tw}(G)\le 1$, (

**c**) $\mathrm{pw}(G)\le 2$, and (

**d**) $\mathrm{td}(G)\le 3$. With respect to q-branched treewidth, the decompositions also justify (

**b**) ${\mathrm{tw}}_{2}(G)\le 1$ and (

**c**) ${\mathrm{tw}}_{0}(G)\le 2$.

**Figure 2.**An illustration of the graph searching game for treedepth on the 8 vertex graph shown at the very top. Vertices that contain a green dot are currently contaminated. The searchers will place a searcher on the vertex with a red circle in the next round. The cleaned vertices (on which a searcher stands) are filled with blue. The arrows indicate the various choices of the fugitive. The diagram proves that 4 searchers have a winning strategy on this specific graph.

**Figure 3.**A directed spider with ${2}^{n}$ legs each of length n that could be the auxiliary graph of a graph searching game—i. e., the vertex set is $\mathcal{B}(G,k)$. Assume s is the start configuration and, for the sake of argument, that t is the sole winning configuration. Then $\mathcal{R}(G,k)$ contains only n elements, while the whole game has ${2}^{n}\xb7n$ configurations.

**Figure 4.**The glue operation: We have already guessed the vertex v and currently consider the set $X=\{\phantom{\rule{0.166667em}{0ex}}({C}_{1},{\rho}_{1}),({C}_{2},{\rho}_{2}),({C}_{3},{\rho}_{3})\phantom{\rule{0.166667em}{0ex}}\}$. Note that all blocks are adjacent to v and are pairwise non-intersecting. The combined neighborhood is highlighted. This area is not allowed to be larger than $k-\mathrm{max}\{{\rho}_{1},{\rho}_{2},{\rho}_{3}\}$, as the searchers must clean it before they can proceed the search on one of the blocks. From this situation, we generate the block $({C}_{1}\cup {C}_{2}\cup {C}_{3}\cup \left\{v\right\},1+\mathrm{max}\{{\rho}_{1},{\rho}_{2},{\rho}_{3}\})$.

**Figure 5.**The discover algorithm computes, given a graph $G=(V,E)$ and an integer $k\in \mathbb{N}$, the auxiliary graph $\mathrm{pit}(G,k)$. Using the positive-instance driven paradigm, only the elements of the pit are explored during this process. The executed subprocedures can be found in Figure 6.

**Figure 6.**Subprocedures used by the discover algorithm. The offer procedure adds a block to the queue if t is not too large. The intersect procedure simply checks if two blocks are compatible (i. e., that they can be glued together), and the discoverEdges procedure identifies the edges of the pit.

**Figure 7.**A color coding sieve for the nine vertex graph shown on the bottom left with $k=4$ and $\gamma =1$. The three levels are illustrated as gray boxes and show the contained sieves of the next level—green pointers indicate which level-i sieve is shown, i. e., the level-2 sieve is ${S}_{1}$ and the level-3 sieve is ${S}_{1}\left[2\right]$ (all the other sieves are not shown). In the center, an insert of $\{2,8\}$ is illustrated: the first sieve inserts it to ${S}_{1}$, the level-2 sieve ${S}_{1}$ inserts it to the level-3 sieve ${S}_{1}\left[2\right]$ (other branches are not shown). The level-2 sieve uses the random partition shown on the graph, that is, $\{2,8\}$ has two orange neighbors. The level-3 sieve ${S}_{1}\left[2\right]$ is an ordinary list before the insert and becomes a trie after the insert, as a random split at vertex 8 occurs. On the right side of the figure, we see all sieves that we have to search through in order to answer the query $(\{3,4,5\})$.

**Figure 8.**Scatter plots that show the performance of PID${}^{\star}$ without color coding sieves compared to the performance of the solver with them enabled ($\gamma =1$ and a randomly generated coloring is used). The left plot contains the DIMACS graph coloring instances, the right plot the PACE 2020 test set. Each point corresponds to an instance; the x-axis is the time needed by the solver using color coding sieves and the y-axis without using this feature. The color of the dot indicates which version was better; it is gray if they are equal. If a solver needed more than 10 min for an instance, the coordinate is set to the red dotted line.

**Figure 9.**A cumulative distribution function plot that shows the performance of PID${}^{\star}$ without color coding sieves (-wo), with just the lazily built set trie (-trie), with the full color coding sieve and $\gamma =1$ (-color-1), and with the full color coding sieve and $\gamma =2$ (-color-2). The experiments are performed on the set of 210 hyperbolic random graphs described in Table 1.

**Figure 10.**Performance $\alpha $ of a level-2 sieve with $\gamma $ colorings on instance $\mathrm{exact}\_036$ from the PACE 2020 benchmark set [45].

**Table 1.**Parameters of the used hyperbolic graph generator [87]. We set the number of vertices to a range in which the instances are tractable but challenging for the solver PID${}^{\star}$. The expected average degree is ${k}_{1}$ for $n\in \{100,110,120\}$, ${k}_{2}$ for $n\in \{130,140\}$, ${k}_{3}$ for $n\in \{150,160\}$, and ${k}_{4}$ for $n\in \{170,180,190,200\}$. For each combination of n and the corresponding ${k}_{i}$, we generated six instances with random seed $s\in \{1,\cdots ,6\}$ yielding 210 instances. The remaining parameters of the generator are set to their default value.

number of nodes | $n\in \{100,110,120,\cdots ,200\}$ |

expected average degree | ${k}_{1}\in \{10,20,30\}$, ${k}_{2}\in {k}_{1}\cup \left\{40\right\}$, ${k}_{3}={k}_{2}\cup \left\{50\right\}$, ${k}_{4}\in \{3,5\}$ |

expected power-law exponent | 2 (default) |

square root of curvature | 1 (default) |

temperature | 0 (default) |

seed | $i\in \{1,2,\cdots ,6\}$ |

Configuration | Meaning |
---|---|

-wo | Color coding sieves are not used. |

-trie | Only the lazy set tries are used. |

-color-1 | All three sieves are used and $\gamma =1$. |

-color-2 | All three sieves are used and $\gamma =2$. |

**Table 3.**Variables used to describe the performance of a sieve. The values t, c and ℓ are considered over a complete run of the solver for a given instance, e. g., summed over various values for the target width k.

Variable | Meaning | Comment |
---|---|---|

t | Total number of blocks. | |

c | Number of compatible blocks. | $c\le t$ |

ℓ | Number of blocks loaded by the sieve. | $c\le \ell \le t$ |

$\alpha $ | Performance as $\alpha =1-\frac{\ell -c}{t-c}$ | $\alpha =0$ for a trivial sieve $\alpha =1$ for a perfect sieve |

**Table 4.**Overview of the performance of PID${}^{\star}$ if a single layer of the color coding sieve is used. The columns n, m, and td describe the number of vertices and edges, as well as the treedepth of the corresponding graph. The columns labeled “Level-i” indicate that a level-i sieve is used (and none of the other layers). In this case, the level-2 sieve is used with $\gamma =1$ and the level-3 sieve is used in its plain version (without any improvements). Columns labeled with $\gamma =c$ indicate that a level-2 sieve with c colorings was used (the “random” colorings were generated with a seeded pseudo-random generator, that is, the first coloring of the $\gamma =3$ sieve is the same as the coloring of the $\gamma =1$ sieve and the first three colorings of the $\gamma =6$ sieve are the ones of the $\gamma =3$ sieve), and the column “imp. Level-3” corresponds to a level-3 sieve with the improvements discussed after Theorem 3. In each row, the sieve with the best performance on this instance is highlighted.

$\mathbf{\alpha}$ of ⋯ | |||||||||
---|---|---|---|---|---|---|---|---|---|

Graph | n | m | td | Level-1 | Level-2 | $\mathbf{\gamma}=3$ | $\mathbf{\gamma}=6$ | Level-3 | imp. Level-3 |

$\mathrm{grid}\_4\times 4$ | 16 | 24 | 7 | $0.00$ | $0.22$ | $0.41$ | 0.63 | $0.25$ | $0.34$ |

$\mathrm{grid}\_5\times 5$ | 25 | 40 | 9 | $0.00$ | $0.20$ | $0.49$ | 0.66 | $0.46$ | $0.65$ |

$\mathrm{grid}\_6\times 6$ | 36 | 60 | 11 | $0.02$ | $0.26$ | $0.52$ | $0.70$ | $0.62$ | 0.81 |

$\mathrm{grid}\_7\times 7$ | 49 | 84 | 13 | $0.07$ | $0.29$ | $0.59$ | $0.78$ | $0.69$ | 0.92 |

$\mathrm{grid}\_8\times 8$ | 649 | 112 | 15 | $0.12$ | $0.31$ | $0.62$ | $0.81$ | $0.72$ | 0.96 |

**Table 5.**Same as Table 4, but on spider-graphs rather than grids.

$\mathbf{\alpha}$ of ⋯ | |||||||||
---|---|---|---|---|---|---|---|---|---|

Graph | n | m | td | Level-1 | Level-2 | $\mathbf{\gamma}=3$ | $\mathbf{\gamma}=6$ | Level-3 | imp. Level-3 |

$\mathrm{spider}\_(10,5,5)$ | 255 | 1760 | 20 | $0.00$ | $0.23$ | $0.38$ | 0.51 | $0.04$ | $0.13$ |

$\mathrm{spider}\_(10,10,5)$ | 505 | 3510 | 25 | $0.03$ | $0.18$ | $0.35$ | 0.49 | $0.05$ | $0.22$ |

$\mathrm{spider}\_(10,10,6)$ | 606 | 5115 | 30 | $0.03$ | $0.16$ | $0.34$ | 0.44 | $0.05$ | $0.19$ |

$\mathrm{spider}\_(20,10,3)$ | 606 | 2403 | 15 | $0.11$ | $0.16$ | $0.38$ | 0.57 | $0.03$ | $0.33$ |

$\mathrm{spider}\_(30,10,5)$ | 1505 | 10510 | 25 | $0.15$ | $0.14$ | $0.31$ | 0.42 | $0.02$ | $0.34$ |

$\mathbf{\alpha}$ of ⋯ | |||||||||
---|---|---|---|---|---|---|---|---|---|

Graph | n | m | td | Level-1 | Level-2 | $\mathbf{\gamma}=3$ | $\mathbf{\gamma}=6$ | Level-3 | imp. Level-3 |

$\mathrm{exact}\_057$ | 50 | 75 | 13 | $0.19$ | $0.31$ | $0.65$ | $0.87$ | $0.77$ | 0.97 |

$\mathrm{exact}\_077$ | 66 | 120 | 12 | $0.10$ | $0.27$ | $0.53$ | $0.75$ | $0.51$ | 0.83 |

$\mathrm{exact}\_078$ | 67 | 152 | 12 | $0.04$ | $0.12$ | $0.26$ | 0.33 | $0.13$ | $0.20$ |

$\mathrm{exact}\_079$ | 68 | 83 | 9 | $0.06$ | $0.27$ | $0.59$ | 0.71 | $0.36$ | $0.69$ |

$\mathrm{exact}\_083$ | 70 | 274 | 22 | $0.16$ | $0.33$ | $0.63$ | $0.79$ | $0.67$ | 0.92 |

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Bannach, M.; Berndt, S.
Recent Advances in Positive-Instance Driven Graph Searching. *Algorithms* **2022**, *15*, 42.
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Bannach M, Berndt S.
Recent Advances in Positive-Instance Driven Graph Searching. *Algorithms*. 2022; 15(2):42.
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2022. "Recent Advances in Positive-Instance Driven Graph Searching" *Algorithms* 15, no. 2: 42.
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