Bounded Tree Width Research Articles

Vertex partitioning problems on graphs with bounded tree width

In an undirected graph, a matching cut is a partition of vertices into two sets such that the edges across the sets induce a matching. The Matching Cut problem is the problem of deciding whether a given graph has a matching cut. Let H be a fixed undirected graph. A vertex coloring of an undirected input graph G is said to be an H - Free Coloring if none of the color classes contain H as an induced subgraph. The H - Free Chromatic Number of G is the minimum number of colors required for an H - Free Coloring of G . Both The Matching Cut problem and the H - Free Coloring problem can be expressed using a monadic second-order logic (MSOL) formula and hence is solvable in linear time for graphs with bounded tree-width. However, this approach leads to a running time of f ( | | φ | | , t ) n O ( 1 ) , where | | φ | | is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. The dependency of f ( | | φ | | , t ) on | | φ | | can be as bad as a tower of exponentials. In this paper, we provide explicit combinatorial FPT algorithms for Matching Cut problem and H - Free Coloring problem, parameterized by the tree-width of G . The single exponential FPT algorithm for the Matching Cut problem answers an open question posed by Kratsch and Le (2016). The techniques used in the paper are also used to provide an FPT algorithm for a variant of H -free coloring, where H is forbidden as a subgraph (not necessarily induced) in the color classes of G .

Algorithmic aspects of outer independent Roman domination in graphs

Let [Formula: see text] be a connected graph. A function [Formula: see text] is called a Roman dominating function if every vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. If further the set [Formula: see text] is an independent set, then [Formula: see text] is called an outer independent Roman dominating function (OIRDF). Let [Formula: see text] and [Formula: see text]. Then [Formula: see text] is called the outer independent Roman domination number of [Formula: see text]. In this paper, we prove that the decision problem for [Formula: see text] is NP-complete for chordal graphs. We also show that [Formula: see text] is linear time solvable for threshold graphs and bounded tree width graphs. Moreover, we show that the domination and outer independent Roman domination problems are not equivalent in computational complexity aspects.

Arc-routing for winter road maintenance

The winter road maintenance arc-routing is recognized as a notoriously hard problem not only from the algorithmic point of view. This paper lays down foundations of theoretical understanding of our new winter road maintenance optimization for the Plzeň region of the Czech Republic which has been implemented by the regional authorities since the winter of 2019–20. Our approach is not, contrary to most of existing work, based on the integer and linear programming machinery. We concentrate on studying arc-routing on trees. This is practical since routes of single vehicles can be well represented by trees, and allows algorithms and complementary hardness results. We then extend the approach to the bounded tree width graphs. This leads to considering planar graphs which well abstract the realistic road networks. We formalize important aspects of the winter road maintenance problem which were not formalized before, e.g., public complaints. The number of complaints from public against the winter road maintenance is a quantitative measure of the quality of the service which is focused on, e.g., in media or in election campaigns. A fear of ’complaints’ is a fact every optimizer must deal with. Hence, a formal model of public complaints and its inclusion in the optimization is vital. Our formalization of the winter road maintenance is robust in the sense that it relates to well-known extensively studied concepts of discrete mathematics like graph cutting and splitting of necklaces.

Canonizing Graphs of Bounded Tree Width in Logspace

Graph canonization is the problem of computing a unique representative, a canon, from the isomorphism class of a given graph. This implies that two graphs are isomorphic exactly if their canons are equal. We show that graphs of bounded tree width can be canonized by logarithmic-space (logspace) algorithms. This implies that the isomorphism problem for graphs of bounded tree width can be decided in logspace. In the light of isomorphism for trees being hard for the complexity class logspace, this makes the ubiquitous class of graphs of bounded tree width one of the few classes of graphs for which the complexity of the isomorphism problem has been exactly determined.

A Gentle Introduction to Applications of Algorithmic Metatheorems for Space and Circuit Classes

Algorithmic metatheorems state that if a problem can be described in a certain logic and the inputs are structured in a certain way, then the problem can be solved with a certain amount of resources. As an example, by Courcelle’s Theorem, all monadic second-order (“in a certain logic”) properties of graphs of bounded tree width (“structured in a certain way”) can be solved in linear time (“with a certain amount of resources”). Such theorems have become valuable tools in algorithmics: if a problem happens to have the right structure and can be described in the right logic, they immediately yield a (typically tight) upper bound on the time complexity of the problem. Perhaps even more importantly, several complex algorithms rely on algorithmic metatheorems internally to solve subproblems, which considerably broadens the range of applications of these theorems. This paper is intended as a gentle introduction to the ideas behind algorithmic metatheorems, especially behind some recent results concerning space and circuit classes, and tries to give a flavor of the range of their applications.

On Minimum Bisection and Related Partition Problems in Graphs with Bounded Tree Width

Minimum Bisection denotes the NP-hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the number of edges between these two sets. We consider this problem in bounded degree graphs with a given tree decomposition (T,X) and prove an upper bound for their minimum bisection width in terms of the structure and width of (T,X). When (T,X) is provided as input, a bisection satisfying our bound can be computed in time proportional to the encoding length of (T,X). Furthermore, our result can be generalized to k-section, which is known to be APX-hard even when restricted to trees with bounded degree.

Hitting sets online and unique-max coloring

We consider the problem of hitting sets online. The hypergraph is known in advance, and the hyperedges to be stabbed are input one-by-one in an online fashion. The online algorithm must stab each hyperedge upon arrival. The best known competitive ratio for hitting sets online by Alon et al. (2009) is O(logn⋅logm) for general hypergraphs, where n and m denote the number of vertices and the number of hyperedges, respectively.In this paper we provide the following results:1. We consider hypergraphs in which the union of two intersecting hyperedges is also a hyperedge. Our main result for such hypergraphs is as follows: We consider a recently studied hypergraph coloring notion referred to as “unique-maximum coloring” and show that the competitive ratio of the online hitting set problem is at most the unique-maximum chromatic number and at least this number minus one.2. Given a graph G=(V,E), let H=(V,R) denote the hypergraph whose hyperedges are subsets U⊆V such that the induced subgraph G[U] is connected. We establish a new connection between the best competitive ratio for the online hitting set problem in H and a well studied graph coloring notion referred to as the “vertex ranking number” of G. This connection states that these two parameters are equal. Moreover, this equivalence is constructive. As a corollary, we obtain optimal online hitting set algorithms for many such hypergraphs including those realized by planar graphs, graphs with bounded tree width, trees, etc. This improves the best previously known general bound of Alon et al. (2009).3. We also consider two geometrically defined hypergraphs. The first one is defined by subsets of a given set of n points in the Euclidean plane that are induced by half-planes. The second hypergraph is defined by subsets of a given set of n points in the plane induced by unit discs. For these hypergraphs, the competitive ratio obtained by Alon et al. is O(log2n). For both cases, we introduce an algorithm with O(logn)-competitive ratio. We also show that any online algorithm for both settings has a competitive ratio of Ω(logn), and hence our algorithms are optimal.

Classical spin systems and the quantum stabilizer formalism: General mappings and applications

We present general mappings between classical spin systems and quantum physics. More precisely, we show how to express partition functions and correlation functions of arbitrary classical spin models as inner products between quantum stabilizer states and product states, thereby generalizing mappings for some specific models established in our previous work [M. Van den Nest et al., Phys. Rev. Lett. 98, 117207 (2007)]. For Ising- and Potts-type models with and without external magnetic field, we show how the entanglement features of the corresponding stabilizer states are related to the interaction pattern of the classical model, while the choice of product states encodes the details of the interaction. These mappings establish a link between the fields of classical statistical mechanics and quantum information theory, which we utilize to transfer techniques and methods developed in one field to gain insight into the other. For example, we use quantum information techniques to recover well known duality relations and local symmetries of classical models in a simple way and provide new classical simulation methods to simulate certain types of classical spin models. We show that in this way all inhomogeneous models of q-dimensional spins with pairwise interaction pattern specified by a graph of bounded tree width can be simulated efficiently. Finally, we show relations between classical spin models and measurement-based quantum computation.

Case-factor diagrams for structured probabilistic modeling

We introduce a probabilistic formalism handling both Markov random fields of bounded tree width and probabilistic context-free grammars. Our models are based on case-factor diagrams (CFDs) which are similar to binary decision diagrams (BDDs) but are more concise for circuits of bounded tree width. A probabilistic model consists of a CFD defining a feasible set of Boolean assignments and a weight (or cost) for each individual Boolean variable. We give versions of the inside–outside algorithm and the Viterbi algorithm for these models.

The complexity of homomorphism and constraint satisfaction problems seen from the other side

We give a complexity theoretic classification of homomorphism problems for graphs and, more generally, relational structures obtained by restricting the left hand side structure in a homomorphism. For every class C of structures, let HOM(C,−) be the problem of deciding whether a given structure A ∈C has a homomorphism to a given (arbitrary) structure ß. We prove that, under some complexity theoretic assumption from parameterized complexity theory, HOM(C,−) is in polynomial time if and only if C has bounded tree width modulo homomorphic equivalence. Translated into the language of constraint satisfaction problems, our result yields a characterization of the tractable structural restrictions of constraint satisfaction problems. Translated into the language of database theory, it implies a characterization of the tractable instances of the evaluation problem for conjunctive queries over relational databases.

On the colored Tutte polynomial of a graph of bounded treewidth

We observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601–622] for the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak [An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39–54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005) 276–290], using a different algorithm based on logical techniques.

Coloured Tutte polynomials and Kauffman brackets for graphs of bounded tree width

Tutte polynomials are important graph invariants with rich applications in combinatorics, topology, knot theory, coding theory and even physics. The Tutte polynomial T ( G , X , Y ) is a polynomial in Z [ X , Y ] which depends on a graph G. Computing the coefficients of T ( G , X , Y ) , and even evaluating T ( G , X , Y ) at specific points ( x , y ) is ♯ P hard by a result of Jaeger et al. (Math. Proc. Cambridge Philos. Soc. 108 (1989) 35). On the other hand, Andrzejak (Discrete Math. 190 (1998) 39–54) and Noble (Combin. Probab. Comput. 7 (1998) 307–321) have shown independently, that, if G is a graph of bounded tree width, computing T ( G , X , Y ) can be done in polynomial time. We extend this result to the signed Tutte polynomials introduced in 1989 by Kauffman and the colored Tutte polynomials introduced in 1999 by Bollobas and Riordan. This allows us to prove similar results for the Jones polynomials and Kauffman brackets for knots and links which have a signed graph presentation of bounded tree width. Jones polynomials and Kauffman polynomials are the most prominent invariants of knot theory. For alternating links, they are easily computable from the Tutte polynomials of the signed graph representing the link by a result of Thistlethwaite (1988). For general links one has to use the colored Tutte polynomial instead. Knots and links can be presented as labeled planar graphs. The tree width of a link L is defined as the tree width of its graphical presentation D ( L ) as crossing diagrams. We show that for (not necessarily alternating) knots and links of tree width at most k, even the Kauffman square bracket [ L ] introduced by Bollobas and Riordan can be computed in polynomial time. Hence, the classical Kauffman bracket 〈 L 〉 and the Jones polynomial of links of tree width at most k are computable in polynomial time. Our proof is based on, but extends considerably previous work by B. Courcelle, U. Rotics and the author. It also gives a new proof of the result for Tutte polynomials and generalizes to a wide class of polynomials defined as generating functions definable in Monadic Second Order Logic with order, but invariant under it.

Iterative-free program analysis

Program analysis is the heart of modern compilers. Most control flow analyses are reduced to the problem of finding a fixed point in a certain transition system, and such fixed point is commonly computed through an iterative procedure that repeats tracing until convergence.This paper proposes a new method to analyze programs through recursive graph traversals instead of iterative procedures, based on the fact that most programs (without spaghetti GOTO) have well-structured control flow graphs, graphs with bounded tree width . Our main techniques are; an algebraic construction of a control flow graph, called SP Term , which enables control flow analysis to be defined in a natural recursive form, and the Optimization Theorem , which enables us to compute optimal solution by dynamic programming.We illustrate our method with two examples; dead code detection and register allocation. Different from the traditional standard iterative solution, our dead code detection is described as a simple combination of bottom-up and top-down traversals on SP Term. Register allocation is more interesting, as it further requires optimality of the result. We show how the Optimization Theorem on SP Terms works to find an optimal register allocation as a certain dynamic programming.

Farrell polynomials on graphs of bounded tree width

We consider various classes of graph polynomials and study their computational complexity. Our main focus is on Farrell polynomials and generating functions of graph properties. All these polynomials have a wide range of applications in combinatorics, but also in physics, chemistry, and biology. In general, the worst-case complexity of most these polynomials is known to be NP-hard, or even ♯ P -hard. We show that, if these polynomials satisfy a definability condition in the formalisms of monadic second-order logic, then they can be computed in polynomial time if restricted to graphs of tree width at most k. In other words, they are fixed-parameter tractable (FPT) with parameter the tree width of the input graph.

Predecessor and Permutation Existence Problems for Sequential Dynamical Systems.

A class of finite discrete dynamical systems, called Sequential Dynamical Systems (SDSs), was introduced in [BR99] as a formal model for analyzing simulation systems. Here, we address the complexity of two basic problems and their generalizations for SDSs.Given an SDS $\mathcal{S}$ and a configuration $\mathcal{C}$, the PREDECESSOR EXISTENCE (or PRE) problem is to determine whether there is a configuration $\mathcal{C}'$ such that $\mathcal{S}$ has a transition from $\mathcal{C}'$ to $\mathcal{C}$. Our results provide separations between efficiently solvable and computationally intractable instances of the PRE problem. For example, we show that the PRE problem can be solved efficiently for SDSs with Boolean state values when the node functions are symmetric and the underlying graph is of bounded tree width. In contrast, we show that allowing just one non-symmetric node function renders the problem $\mathbf{NP}$-complete even when the underlying graph is a star (which has a tree width of 1). Our results extend some of the earlier results by Sutner [Su95] and Green [Gr87] on the complexity of the PREDECESSOR EXISTENCE problem for 1-dimensional cellular automata.Given two configurations $\mathcal{C}$ and $\mathcal{C}'$ of a partial SDS $\mathcal{S}$, the PERMUTATION EXISTENCE (or PME) problem is to determine whether there is a permutation of nodes such that $\mathcal{S}$ has a transition from $\mathcal{C}'$ to $\mathcal{C}$ in one step. We show that the PME problem is $\mathbf{NP}$-complete even when the function associated with each node is a simple-threshold function. We also show that the problem can be solved efficiently for SDSs whose underlying graphs are of bounded degree and bounded tree width. We consider a generalized version (GEN-PME) of the PME problem and show that the problem is $\mathbf{NP}$-complete for SDSs where each node function is NOR and the underlying graph has a maximum node degree of 3. When each node computes the OR function or when each node computes the AND function, we show that the GEN-PME problem is solvable in polynomial time.

Guarded fixed point logics and the monadic theory of countable trees

Different variants of guarded logics (a powerful generalization of modal logics) are surveyed and an elementary proof for the decidability of guarded fixed point logics is presented. In a joint paper with Igor Walukiewicz, we proved that the satisfiability problems for guarded fixed point logics are decidable and complete for deterministic double exponential time (E. Grädel and I. Walukiewicz, Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, pp. 45–54). That proof relies on alternating automata on trees and on a forgetful determinacy theorem for games on graphs with unbounded branching. The exposition given here emphasizes the tree model property of guarded logics: every satisfiable sentence has a model of bounded tree width. Based on the tree model property, we show that the satisfiability problem for guarded fixed point formulae can be reduced to the monadic theory of countable trees (S ωS), or to the μ-calculus with backwards modalities.

Channel Assignment on Nearly Bipartite and Bounded Treewidth Graphs

Channel Assignment on Nearly Bipartite and Bounded Treewidth Graphs

High-Girth Graphs Avoiding a Minor are Nearly Bipartite

Let H be a fixed graph. We show that any H-minor free graph G of high enough girth has circular chromatic number arbitrarily close to two. Equivalently, each such graph G admits a homomorphism to a large odd circuit. In particular, graphs of high girth and of bounded genus, or of bounded tree width, are “nearly bipartite” in this sense. For example, any planar graph of girth at least 16 admits a homomorphism to a pentagon. We also obtain tight bounds on the girth of G in a few specific cases of small forbidden minors H.

The complexity of acyclic conjunctive queries

This paper deals with the evaluation of acyclic Boolean conjunctive queries in relational databases. By well-known results of Yannakakis[1981], this problem is solvable in polynomial time; its precise complexity, however, has not been pinpointed so far. We show that the problem of evaluating acyclic Boolean conjunctive queries is complete for LOGCFL, the class of decision problems that are logspace-reducible to a context-free language. Since LOGCFL is contained in AC1 and NC2, the evaluation problem of acyclic Boolean conjunctive queries is highly parallelizable. We present a parallel database algorithm solving this problem with alogarithmic number of parallel join operations. The algorithm is generalized to computing the output of relevant classes of non-Boolean queries. We also show that the acyclic versions of the following well-known database and AI problems are all LOGCFL-complete: The Query Output Tuple problem for conjunctive queries, Conjunctive Query Containment, Clause Subsumption, and Constraint Satisfaction. The LOGCFL-completeness result is extended to the class of queries of bounded tree width and to other relevant query classes which are more general than the acyclic queries.

Graph Decomposition of Slim Graphs

A Graph G=(V,E) is called k-slim if for every subgraph S=(VS,ES) of G with s=|VS|≥k there exists K⊂VS, |K|=k, such that the vertices of VS\K can be partitioned into two subsets, A and B, such that |A|≤⅔s and |B|≤⅔s and no edge of ES connects a vertex from A and a vertex from B. k-slim graphs contain, in particular, the graphs with tree-width k. In this paper we give an algorithm solving the H-decomposition problem for a large family of graphs H which contains, among others, the stars, the complete graphs, and the complete r-partite graphs where r≥3. The algorithm runs in polynomial time in case the input graph is k-slim, where k is fixed. In particular, our algorithm runs in polynomial time when the input graph has bounded tree width k. Our results supply the first polynomial time algorithm for H-decomposition of connected graphs H having at least 3 edges, in graphs with bounded tree-width.