A contraction-recursive algorithm for treewidth

Q: What is the significance of tw(G) = k + 1?

When tw(G) = k + 1, it indicates that there exists a minimal contraction H of G, where tw(H) = k + 1. This contraction certifies the treewidth of G and extends the scope of instances for which the exact treewidth can be computed in practice. The recursive calls on G/e help in determining the treewidth and identifying the minimal contraction H. This approach is crucial in settling the question of whether tw(G) <= k and plays a significant role in exact treewidth computation.

Q: Which sections should be read first?

To quickly grasp the main ideas and contributions of this paper, it is suggested to read Section 3 - Main algorithm, Section 6 - Uncontracting PMCs, Section 7 - Contracting PMCs, and Section 11 - Experiments (about 9 pages in total including the introduction), together with some parts of the preliminaries section as needed. Section 4 describes some details of the local search algorithm we use, namely heuristic PID. Sections 8, 9, and 10 describe additional techniques for speeding up the main algorithm. Section 12 offers some concluding remarks. The source code of the implementation of our algorithm used in the experiments is available at https://github.com/twalgor/RTW.

Q: What is a clique in a graph?

A clique in a graph is a vertex set C V (G) where G[C] forms a complete graph. This means that every vertex in the set C is adjacent to every other vertex in the set. In other words, a clique is a subset of vertices in a graph where every pair of vertices is connected by an edge. Cliques are important in graph theory as they represent fully connected subgraphs and are used in various applications such as social network analysis, bioinformatics, and computer science. Identifying cliques in a graph can help in understanding the structure and properties of the graph, as well as in solving optimization problems and finding patterns within the graph.

Q: What is a minimal a-b separator?

A minimal a-b separator is a separator in a graph G that is an a-b separator and no proper subset of it is an a-b separator. It ensures that there is no path between vertices a and b in the graph G without the separator S. This concept is crucial in graph theory and helps in understanding the connectivity and structure of graphs. Minimal separators play a significant role in various graph algorithms and applications, such as network analysis, clustering, and optimization problems. They are used to identify critical components or edges that, when removed, disconnect the graph into separate components. Minimal separators are also essential in the study of graph minors and graph mincuts, where they help in finding the minimum cuts and separators in a graph. Overall, minimal separators provide valuable insights into the connectivity and structure of graphs, aiding in the analysis and optimization of complex networks.

Q: How does BT dynamic programming compute tw P(G) for an arbitrary subset P of P(G)?

BT dynamic programming computes tw P(G) for an arbitrary subset P of P(G) by formulating recurrences and evaluating them for blocks in increasing order of cardinality. It obtains tw P(G) = tw P(G, V(G)). The algorithm traces back the recurrences to generate a tree-decomposition T T P(G) with width tw P(G). This approach provides an upper bound on tw(G) and helps in determining the treewidth of a graph. Tamaki's PID algorithm further utilizes this recurrence to generate relevant blocks and PMCs, aiding in the decision problem of whether tw(G) <= k for given G and k.

Q: What are contractors in graph theory?

Contractors in graph theory are partitions of a graph's vertex set into connected sets. They are used to extend the notation of a contraction by an edge to a contraction by multiple edges. A contractor g of a graph G is a mapping from V(G) to {1, 2, ..., m}, where m is the index set of the partition g. The vertex set of the contracted graph G/g is {1, 2, ..., m}, and g(v) represents the vertex in G/g to which vertex v is contracted. For each vertex w in G/g, g-1(w) denotes the part of the partition g that contracts to w. The union of a subset U of vertices in G/g is represented as g-1(U). Contractors provide a convenient way to represent and manipulate graphs with multiple edges between vertices.

Q: How does the RTW algorithm improve the upper bound in the treewidth algorithm?

The RTW algorithm improves the upper bound in the treewidth algorithm by repeatedly refining the upper bound through a local search in the space of sets of PMCs. It starts with a greedy upper bound and uses the RTW subalgorithm to iteratively improve the upper bound. The algorithm decides if the treewidth of the graph G is less than or equal to k by calling RTW with the graph G, the current upper bound k, and the set of PMCs P. If the treewidth is less than or equal to k, the algorithm returns YES with a certificate P' such that the treewidth of P' is less than or equal to k. If the treewidth is greater than k, the algorithm continues to explore the space of sets of PMCs by contracting edges and generating partial tree-decompositions. The algorithm uses a heuristic approach to quickly generate a tree-decomposition of width less than or equal to k. The RTW algorithm communicates with external upper bound heuristics through operations like requiring, importing, and contracting. It iterates over edges, contracts them, and updates the set of PMCs. If the algorithm finds a tree-decomposition of width less than or equal to k, it returns YES with the certificate constructed by the HPID instance. Otherwise, it continues until all edges are explored or the treewidth is determined to be less than or equal to k. The RTW algorithm effectively refines the upper bound by exploring the space of sets of PMCs and generating partial tree-decompositions.

Q: How does s. width() function in relation to tw(G) and k?

The s. width() function returns the width of a graph G, denoted as tw(G). It ensures a YES result when tw(G) is less than or equal to k, and a NO result when a minimal contraction H of G is found with tw(H) equal to k+1. The correctness of this algorithm can be proven by induction and relies on the performance of procedures like expand, contractP M Cs, and uncontractP M Cs. The efficiency of the algorithm heavily depends on these procedures, as the for loop may exit after trying a few edges if tw(G) <= k, and s. finish() will be called only if tw(G) = k + 1 and tw(G/e) <= k for every edge e. In the worst-case scenario, the for loop would run to the end, making the recursion meaningless. Efforts are focused on developing effective methods for these procedures to enhance the algorithm's efficiency.

Question

1. What is the impact of treewidth on combinatorial algorithms?

2. What is the approach for improving upper bounds on treewidth?

3. What is the significance of tw(G) = k + 1?

4. Which sections should be read first?

Accepted Answer

The impact of treewidth on combinatorial algorithms is profound. There are a huge number of NP-hard graph problems that are known to be tractable when parameterized by treewidth. These algorithms typically have a running time of f(k)nO(1), where n is the number of vertices, k is the treewidth of the given graph, and f is typically an exponential function. Treewidth computation plays a crucial role in designing efficient algorithms for these problems. Triangulations of graphs also play an important role in treewidth computation, as they provide a way to enumerate all relevant tree-decompositions. Practical algorithms have been developed to efficiently compute treewidth, such as BT dynamic programming and Tamaki's PID variant, which have shown significant improvements in performance compared to previously implemented algorithms.

Accepted Answer

The approach for improving upper bounds on treewidth involves using tree-decompositions and potential maximal cliques (PMCs). A set of PMCs is represented by a set of tree-decompositions, denoted as T P(G). For each PMC P(G), the set T P(G) contains tree-decompositions of width tw(G). The BT dynamic programming algorithm can be used to find a tree-decomposition in T P(G) that minimizes various width measures based on bag weights. By using appropriate bag weights, the algorithm can decide if tw(G) <= k and find a tree-decomposition with width tw(G) <= k. The algorithm certifies the YES answer by P P(G) with tw P(G) <= k. It uses heuristic methods to find such P and recursively solves the question if tw(G/e) <= k for edge e of G. The uncontraction of tree-decompositions in T P(G/e) with width <= k is also useful for the heuristic upper bound method. The algorithm interleaves the execution of a local search algorithm with recursive calls on G/e for edges e of G and injects PMCs obtained from the results of the recursive calls. This process ends in either finding a tree-decomposition with tw(G) <= k, the local search succeeding in finding P with tw(G) <= k, or the process continuing with recursive calls on G/e.

Accepted Answer

When tw(G) = k + 1, it indicates that there exists a minimal contraction H of G, where tw(H) = k + 1. This contraction certifies the treewidth of G and extends the scope of instances for which the exact treewidth can be computed in practice. The recursive calls on G/e help in determining the treewidth and identifying the minimal contraction H. This approach is crucial in settling the question of whether tw(G) <= k and plays a significant role in exact treewidth computation.

Accepted Answer

To quickly grasp the main ideas and contributions of this paper, it is suggested to read Section 3 - Main algorithm, Section 6 - Uncontracting PMCs, Section 7 - Contracting PMCs, and Section 11 - Experiments (about 9 pages in total including the introduction), together with some parts of the preliminaries section as needed. Section 4 describes some details of the local search algorithm we use, namely heuristic PID. Sections 8, 9, and 10 describe additional techniques for speeding up the main algorithm. Section 12 offers some concluding remarks. The source code of the implementation of our algorithm used in the experiments is available at https://github.com/twalgor/RTW.

Accepted Answer

A clique in a graph is a vertex set C V (G) where G[C] forms a complete graph. This means that every vertex in the set C is adjacent to every other vertex in the set. In other words, a clique is a subset of vertices in a graph where every pair of vertices is connected by an edge. Cliques are important in graph theory as they represent fully connected subgraphs and are used in various applications such as social network analysis, bioinformatics, and computer science. Identifying cliques in a graph can help in understanding the structure and properties of the graph, as well as in solving optimization problems and finding patterns within the graph.

Accepted Answer

A minimal a-b separator is a separator in a graph G that is an a-b separator and no proper subset of it is an a-b separator. It ensures that there is no path between vertices a and b in the graph G without the separator S. This concept is crucial in graph theory and helps in understanding the connectivity and structure of graphs. Minimal separators play a significant role in various graph algorithms and applications, such as network analysis, clustering, and optimization problems. They are used to identify critical components or edges that, when removed, disconnect the graph into separate components. Minimal separators are also essential in the study of graph minors and graph mincuts, where they help in finding the minimum cuts and separators in a graph. Overall, minimal separators provide valuable insights into the connectivity and structure of graphs, aiding in the analysis and optimization of complex networks.

Accepted Answer

BT dynamic programming computes tw P(G) for an arbitrary subset P of P(G) by formulating recurrences and evaluating them for blocks in increasing order of cardinality. It obtains tw P(G) = tw P(G, V(G)). The algorithm traces back the recurrences to generate a tree-decomposition T T P(G) with width tw P(G). This approach provides an upper bound on tw(G) and helps in determining the treewidth of a graph. Tamaki's PID algorithm further utilizes this recurrence to generate relevant blocks and PMCs, aiding in the decision problem of whether tw(G) <= k for given G and k.

Accepted Answer

Contractors in graph theory are partitions of a graph's vertex set into connected sets. They are used to extend the notation of a contraction by an edge to a contraction by multiple edges. A contractor g of a graph G is a mapping from V(G) to {1, 2, ..., m}, where m is the index set of the partition g. The vertex set of the contracted graph G/g is {1, 2, ..., m}, and g(v) represents the vertex in G/g to which vertex v is contracted. For each vertex w in G/g, g-1(w) denotes the part of the partition g that contracts to w. The union of a subset U of vertices in G/g is represented as g-1(U). Contractors provide a convenient way to represent and manipulate graphs with multiple edges between vertices.

Accepted Answer

The RTW algorithm improves the upper bound in the treewidth algorithm by repeatedly refining the upper bound through a local search in the space of sets of PMCs. It starts with a greedy upper bound and uses the RTW subalgorithm to iteratively improve the upper bound. The algorithm decides if the treewidth of the graph G is less than or equal to k by calling RTW with the graph G, the current upper bound k, and the set of PMCs P. If the treewidth is less than or equal to k, the algorithm returns YES with a certificate P' such that the treewidth of P' is less than or equal to k. If the treewidth is greater than k, the algorithm continues to explore the space of sets of PMCs by contracting edges and generating partial tree-decompositions. The algorithm uses a heuristic approach to quickly generate a tree-decomposition of width less than or equal to k. The RTW algorithm communicates with external upper bound heuristics through operations like requiring, importing, and contracting. It iterates over edges, contracts them, and updates the set of PMCs. If the algorithm finds a tree-decomposition of width less than or equal to k, it returns YES with the certificate constructed by the HPID instance. Otherwise, it continues until all edges are explored or the treewidth is determined to be less than or equal to k. The RTW algorithm effectively refines the upper bound by exploring the space of sets of PMCs and generating partial tree-decompositions.

Accepted Answer

The s. width() function returns the width of a graph G, denoted as tw(G). It ensures a YES result when tw(G) is less than or equal to k, and a NO result when a minimal contraction H of G is found with tw(H) equal to k+1. The correctness of this algorithm can be proven by induction and relies on the performance of procedures like expand, contractP M Cs, and uncontractP M Cs. The efficiency of the algorithm heavily depends on these procedures, as the for loop may exit after trying a few edges if tw(G) <= k, and s. finish() will be called only if tw(G) = k + 1 and tw(G/e) <= k for every edge e. In the worst-case scenario, the for loop would run to the end, making the recursion meaningless. Efforts are focused on developing effective methods for these procedures to enhance the algorithm's efficiency.

Accepted Answer

The most important difference between the original PID algorithm and the HPID algorithm lies in the methods used to turn tree-decompositions into rooted tree-decompositions. In the original PID algorithm, the choice of roots heavily depends on the total order C V I T 2 0 1 6 assumed on V (G). However, in the HPID algorithm, the choice of roots is made to depend less on the vertex order, making it fairer for vertices. This ensures better interactions of HPID with other upper bound components through PMCs (partial maximally consistent sets). Additionally, the HPID algorithm focuses on generating only small feasible blocks, except for V (G) itself, where a block B is considered small if there exists another block B' with N G (B') = N G (B) and B' > B. This approach simplifies the process of checking if V (G) is feasible, as it is sufficient to generate only small feasible blocks instead of all feasible blocks.

Accepted Answer

To construct D H (B), we start by observing that every B B is small. For each small block B, we construct a directed tree D H (B) on P(H, B) with sink X B. We inductively construct N H (B) X B by considering the set of caps C B, which satisfy N H (B) X and X B N H (B). The subgraph of W H induced by C B has a sink X B since W H is acyclic. For each block B' B(X B , B), we have B' B and C C(N H (B ' ), X B) > B ' for each B ' B(X B , B), indicating B' is small. By the induction hypothesis, we have a directed tree D H (B ' ) on P(H, B ' ) with N H (B ' ) X B ' for each B ' B(X B , B). Combining D H (B ' ), B ' B(X, B), and an edge from each X B ' to X B, we obtain the desired directed tree D H (B).

Accepted Answer

Minimalizing tree-decompositions serves two main purposes. Firstly, it allows for the representation of a set of tree-decompositions using a set of Partially Minimal Cliques (PMCs). When a tree-decomposition is obtained through a method that may produce non-minimal tree-decompositions, minimalization is performed to ensure all bags are PMCs. Secondly, minimalization can reduce the width of the tree-decomposition. There are two procedures for minimalization: minimalize(T), which uses a standard triangulation minimalization algorithm, and minimalize_optimally(T), which finds a minimalization of T with the smallest width. Although minimalization is NP-hard, an efficient algorithm exists that works well in practice. This algorithm involves constructing a set of admissible minimal separators for T, and then applying the SemiPID variant of BT dynamic programming to obtain a tree-decomposition of the smallest width using only admissible minimal separators. The admissibility constraint significantly reduces the number of minimal separators, resulting in faster enumeration and SemiPID parts compared to the general case without such constraints.

Accepted Answer

The purpose of uncontracting PMCs is to develop an algorithm for procedure uncontractP M Cs(G, P, e). This algorithm generalizes the procedure to uncontractP M Cs(G, P, g), where the third argument is a general contractor of G. The algorithm aims to find tree-decompositions T T P that minimize w(g -1 (T )). This is achieved through BT dynamic programming over T P (G/g), using bag weights defined by o(U ) = |g -1 (U )| - 1. The algorithm also considers the possibility of reducing the weight of g -1 (T ) by minimalization. The main steps of the algorithm are described in Algorithm 3.

Accepted Answer

The purpose of Algorithm 3 Procedure uncontractP M Cs(Π, G, γ) is to return P ' P(G) that results from uncontracting P and then minimizing the weight function on V(G/g) defined by equations 2 and 3. The algorithm uses BT dynamic programming to obtain tree-decompositions T i of G/g, where w(T i , o) = tw P (G, o). For each i, the algorithm minimizes_optimally(g -1 (T i )) and returns the set of bags T ' i. This procedure ensures the efficient uncontracting and minimization of the weight function, ultimately resulting in an optimized P(G).

Accepted Answer

To minimize w(g(T)) in contracting PMCs, an algorithm is used that finds tree-decompositions T T P (G/g) that minimize w(g(T)). This is achieved through BT dynamic programming with a weight function o(U) = 2|g(U)| if g(U) is a PMC of G/g, and o(U) = 2|g(U)| - 1 otherwise. The algorithm then minimizes these tree-decompositions and collects the bags of the minimalized tree-decompositions. Safe separators are used both for preprocessing and during recursion. For preprocessing, minimal separators of a minimal triangulation H of G are tested for sufficient conditions in a theorem. A safe-separator decomposition is obtained by replacing each bag X of a tree-decomposition A with a tree-decomposition of G[X] CC(G\X) K(N G (C)). During recursion, the second sufficient condition in Theorem 2 is useful to find a contractor g such that tw(G/g) = tw(G), allowing safe recursion on G/g. The algorithm constructs a contractor g' and extends the tree-decomposition of tw(G/g') to a tree-decomposition of G, ensuring tw(G/g') <= k. This process results in a contractor P P(G) such that tw P (G) <= k, effectively minimizing w(g(T)) in contracting PMCs.

Accepted Answer

Edge ordering in graph theory is crucial for optimizing algorithms and understanding graph properties. It helps in identifying edges that maintain the same tree width (tw) after graph contraction, as mentioned in the provided section. By ordering edges based on their closeness to an ideal situation, researchers can prioritize edges that contribute to efficient graph traversal and analysis. This ordering technique, derived from safe separators and deficiency calculations, allows for better understanding of graph structures and facilitates the development of advanced graph algorithms. Overall, edge ordering plays a vital role in graph theory research and applications.

Accepted Answer

Suppressed edges play a crucial role in recursive calls by optimizing the process. When an ancestor call on G ' exists, and edge e ' of G ' maps to edge e, if the recursive call on G ' /e ' has been made and tw(G ' /e ' ) <= k, then tw(G/e) <= k. This indicates that G/e is a contraction of G ' /e ', allowing the omission of the recursive call on G/e without compromising correctness. However, for efficiency, it is preferable to obtain the certificate P P(G/e) for tw(G/e) <= k and feed the uncontraction of P to the HPID instance on G. If edge e is suppressed by (G ' , e '), and P ' P(G ' /e ' ) satisfies tw P ' (G ' /e ' ) <= k, then g ' can be obtained as the contractor of G ' /e ' such that G ' /e ' /g ' = G/g/e. By contracting P with g ', we obtain P P(G/e) with tw P (G/e) <= k, optimizing the recursive call process.

Accepted Answer

The primary benchmark for evaluating RTW implementation is the bonus instance set of the exact treewidth track of PACE 2017 algorithm implementation challenge. This set consists of 100 instances, designed to be challenging for future implementations. The instances are hard for the winning solvers of the competition, with about half taking more than one hour to solve and 15 instances taking more than a day or being unsolvable. RTW was tested on these instances with a 10000-second timeout, solving 98 instances, 72 in 100 seconds, and 92 in 1000 seconds. This demonstrates RTW's ability to solve practically solvable instances, extending the scope compared to PID solver, which only solves 68 instances within the timeout. Tamaki's new solver also performs well, solving many instances faster than RTW but with fewer solvable instances in practical time. Further analysis of instances easy for PID but hard for RTW could lead to a unified algorithm.

A contraction-recursive algorithm for treewidth

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Most frequently asked questions

1. What is the impact of treewidth on combinatorial algorithms?

2. What is the approach for improving upper bounds on treewidth?

3. What is the significance of tw(G) = k + 1?

4. Which sections should be read first?

5. What is a clique in a graph?

6. What is a minimal a-b separator?

7. How does BT dynamic programming compute tw P(G) for an arbitrary subset P of P(G)?

8. What are contractors in graph theory?

9. How does the RTW algorithm improve the upper bound in the treewidth algorithm?

10. How does s. width() function in relation to tw(G) and k?

11. What is the main difference between the original PID algorithm and the HPID algorithm in terms of generating partial tree-decompositions?

12. How do we construct D H (B)?

13. Why minimalize tree-decompositions?

14. What is the purpose of uncontracting PMCs?

15. What is the purpose of Algorithm 3 Procedure uncontractP M Cs(Π, G, γ)?

16. How can tree-decompositions minimize w(g(T)) in contracting PMCs?

17. What is the significance of edge ordering in graph theory?

18. What is the significance of suppressed edges in recursive calls?

19. What is the primary benchmark for evaluating RTW implementation?

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