Treewidth computations I. Upper bounds

For more and more applications, it is important to be able to compute the treewidth of a given graph and to find tree decompositions of small width reasonably fast.This paper gives an overview of several upper bound heuristics that have been proposed and tested for the problem of determining the treewidth of a graph and finding tree decompositions. Each of the heuristics produces tree decompositions whose width may be larger than the optimal width. However, experiments show that in many cases, the heuristics give tree decompositions whose width is close to the exact treewidth of the input graphs.     

Treewidth Lower Bounds with Brambles

In this paper we present a new technique for computing lower bounds for graph treewidth. Our technique is based on the fact that the treewidth of a graph G is the maximum order of a bramble of G minus one. We give two algorithms: one for general graphs, and one for planar graphs. The algorithm for planar graphs is shown to give a lower bound for both the treewidth and branchwidth that is at most a constant factor away from the optimum. For both algorithms, we report on extensive computational experiments that show that the algorithms often give excellent lower bounds, in particular when applied to (close to) planar graphs. This work was partially supported by the Netherlands Organisation for Scientific Research NWO (project Treewidth and Combinatorial Optimisation) and partially by the DFG research group “Algorithms, Structure, Randomness” (Grant number GR 883/9-3, GR 883/9-4).     

Algorithmic Meta-theorems for Restrictions of Treewidth

Possibly the most famous algorithmic meta-theorem is Courcelle??s theorem, which states that all MSO-expressible graph properties are decidable in linear time for graphs of bounded treewidth. Unfortunately, the running time??s dependence on the formula describing the problem is in general a tower of exponentials of unbounded height, and there exist lower bounds proving that this cannot be improved even if we restrict ourselves to deciding FO logic on trees. We investigate whether this parameter dependence can be improved by focusing on two proper subclasses of the class of bounded treewidth graphs: graphs of bounded vertex cover and graphs of bounded max-leaf number. We prove stronger algorithmic meta-theorems for these more restricted classes of graphs. More specifically, we show it is possible to decide any FO property in both of these classes with a singly exponential parameter dependence and that it is possible to decide MSO logic on graphs of bounded vertex cover with a doubly exponential parameter dependence. We also prove lower bound results which show that our upper bounds cannot be improved significantly, under widely believed complexity assumptions. Our work addresses an open problem posed by Michael Fellows.     

New closed-form bounds on the partition function

Estimating the partition function is a key but difficult computation in graphical models. One approach is to estimate tractable upper and lower bounds. The piecewise upper bound of Sutton et al. is computed by breaking the graphical model into pieces and approximating the partition function as a product of local normalizing factors for these pieces. The tree reweighted belief propagation algorithm (TRW-BP) by Wainwright et al. gives tighter upper bounds. It optimizes an upper bound expressed in terms of convex combinations of spanning trees of the graph. Recently, Globerson et al. gave a different, convergent iterative dual optimization algorithm TRW-GP for the TRW objective. However, in many practical applications, particularly those that train CRFs with many nodes, TRW-BP and TRW-GP are too slow to be practical. Without changing the algorithm, we prove that TRW-BP converges in a single iteration for associative potentials, and give a closed form for the solution it finds. The closed-form solution obviates the need for complex optimization. We use this result to develop new closed-form upper bounds for MRFs with arbitrary pairwise potentials. Being closed-form, they are much faster to compute than TRW-based bounds. We also prove similar convergence results for loopy belief propagation (LBP) and use it to obtain closed-form solutions to the LBP pseudomarginals and approximation to the partition function for associative potentials. We then use recent results proved by Wainwright et al for binary MRFs to obtain closed-form lower bounds on the partition function. We then develop novel lower bounds for arbitrary associative networks. We report on experiments with synthetic and real-world graphs. Our new upper bounds are considerably tighter than the piecewise bounds in practice. Moreover, we can compute our bounds on several graphs where TRW-BP does not converge. Our novel lower bound, in spite of being closed-form and much faster to compute, outperforms more complicated popular algorithms for computing lower bounds like mean-field on densely connected graphs by wide margins although it does worse on sparsely connected graphs like chains.     

Diameter and Treewidth in Minor-Closed Graph Families

It is known that any planar graph with diameter D has treewidth O(D) , and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a minor-closed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to bounded-genus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph families.     

On the Incompressibility of Monotone DNFs

We prove optimal lower bounds for multilinear circuits and for monotone circuits with bounded depth. These lower bounds state that, in order to compute certain functions, these circuits need exactly as many OR gates as the respective DNFs. The proofs exploit a property of the functions that is based solely on prime implicant structure. Due to this feature, the lower bounds proved also hold for approximations of the considered functions that are similar to slice functions. Known lower bound arguments cannot handle these kinds of approximations. In order to show limitations of our approach, we prove that cliques of size n - 1 can be detected in a graph with n vertices by monotone formulas with O(log n) OR gates. Our lower bound for multilinear circuits improves a lower bound due to Borodin, Razborov and Smolensky for nondeterministic read-once branching programs computing the clique function.     

Infra-chromatic bound for exact maximum clique search

Many efficient exact branch and bound maximum clique solvers use approximate coloring to compute an upper bound on the clique number for every subproblem. This technique reasonably promises tight bounds on average, but never tighter than the chromatic number of the graph.Li and Quan, 2010, AAAI Conference, p. 128–133 describe a way to compute even tighter bounds by reducing each colored subproblem to maximum satisfiability problem (MaxSAT). Moreover they show empirically that the new bounds obtained may be lower than the chromatic number.Based on this idea this paper shows an efficient way to compute related “infra-chromatic” upper bounds without an explicit MaxSAT encoding. The reported results show some of the best times for a stand-alone computer over a number of instances from standard benchmarks.     

A novel discretization scheme for the close enough traveling salesman problem

This paper addresses a variant of the Euclidean traveling salesman problem in which the traveler visits a node if it passes through the neighborhood set of that node. The problem is known as the close-enough traveling salesman problem. We introduce a new effective discretization scheme that allows us to compute both a lower and an upper bound for the optimal solution. Moreover, we apply a graph reduction algorithm that significantly reduces the problem size and speeds up computation of the bounds. We evaluate the effectiveness and the performance of our approach on several benchmark instances. The computational results show that our algorithm is faster than the other algorithms available in the literature and that the bounds it provides are almost always more accurate.     

Lower bounds on precedence-constrained scheduling for parallel processors

We consider two general precedence-constrained scheduling problems that have wide applicability in the areas of parallel processing, high performance compiling, and digital system synthesis. These problems are intractable so it is important to be able to compute tight bounds on their solutions. A tight lower bound on makespan scheduling can be obtained by replacing precedence constraints with release and due dates, giving a problem that can be efficiently solved. We demonstrate that recursively applying this approach yields a bound that is provably tighter than other known bounds, and experimentally shown to achieve the optimal value at least 90.3% of the time over a synthetic benchmark.We compute the best known lower bound on weighted completion time scheduling by applying the recent discovery of a new algorithm for solving a related scheduling problem. Experiments show that this bound significantly outperforms the linear programming-based bound. We have therefore demonstrated that combinatorial algorithms can be a valuable alternative to linear programming for computing tight bounds on large scheduling problems.     

Treewidth,pathwidth and cospan decompositions with applications to graph-accepting tree automata

We will revisit the categorical notion of cospan decompositions of graphs and compare it to the well-known notions of path decomposition and tree decomposition from graph theory. More specifically, we will define several types of cospan decompositions with appropriate width measures and show that these width measures coincide with pathwidth and treewidth. Such graph decompositions of small width are used to efficiently decide graph properties, for instance via graph automata. Hence we will give an application by defining graph-accepting tree automata, thus integrating previous work by Courcelle into the setting of cospan decompositions. Furthermore we will show that regardless of whether we consider path or tree decompositions, we arrive at the same notion of recognizability.     

Safe Reduction Rules for Weighted Treewidth

Several sets of reductions rules are known for preprocessing a graph when computing its treewidth. In this paper we give reduction rules for a weighted variant of treewidth, motivated by the analysis of algorithms for probabilistic networks. We present two general reduction rules that are safe for weighted treewidth. They generalise many of the existing reduction rules for treewidth. Experimental results show that these reduction rules can significantly reduce the problem size for several instances of real-life probabilistic networks.     

A non‐smooth lower bound on ν

A non‐smooth optimization technique to directly compute a lower bound on the skew structured singular value ν is developed. As corroborated by several real‐world challenging applications, the proposed technique can provide tighter lower bounds when compared with currently available techniques. Moreover, in many cases, the determined lower bound equals the true value of ν. Thanks to the efficiency of the non‐smooth technique, the algorithm can be applied to problems involving even a significant number of uncertain parameters. Another appealing feature of the proposed non‐smooth approach is that the dimension of repeated scalar uncertainties in the overall structured uncertainty matrix has little impact on the computational time. The technique can be used to compute a lower bound on the structured singular value μ as well. Copyright © 2012 John Wiley & Sons, Ltd.     

Lower and upper bounds on time for multiprocessor optimal schedules

The lower and upper bounds on the minimum time needed to process a given directed acyclic task graph for a given number of processors are derived. It is proved that the proposed lower bound on time is not only sharper than the previously known values but also easier to calculate. The upper bound on time, which is useful in determining the worst case behavior of a given task graph, is presented. The lower and upper bounds on the minimum number of processors required to process a given task graph in the minimum possible time are also derived. It is seen with a number of randomly generated dense task graphs that the lower and upper bounds we derive are equal, thus giving the optimal time for scheduling directed acyclic task graphs on a given set of processors     

Lower and Upper Bounds for Linkage Discovery

For a real-valued function f defined on {0,1}n , the linkage graph of f is a hypergraph that represents the interactions among the input variables with respect to f . In this paper, lower and upper bounds for the number of function evaluations required to discover the linkage graph are rigorously analyzed in the black box scenario. First, a lower bound for discovering linkage graph is presented. To the best of our knowledge, this is the first result on the lower bound for linkage discovery. The investigation on the lower bound is based on Yao's minimax principle. For the upper bounds, a simple randomized algorithm for linkage discovery is analyzed. Based on the Kruskal-Katona theorem, we present an upper bound for discovering the linkage graph. As a corollary, we rigorously prove that O(n ²logn) function evaluations are enough for bounded functions when the number of hyperedges is O(n), which was suggested but not proven in previous works. To see the typical behavior of the algorithm for linkage discovery, three random models of fitness functions are considered. Using probabilistic methods, we prove that the number of function evaluations on the random models is generally smaller than the bound for the arbitrary case. Finally, from the relation between the linkage graph and the Walsh coefficients, it is shown that, for bounded functions, the proposed bounds are eventually the bounds for finding the Walsh coefficients.     

Graph embeddings and simplicial maps

An undirected graph is viewed as a simplicial complex. The notion of a graph embedding of a guest graph in a host graph is generalized to the realm of simplicial maps. Dilation is redefined in this more general setting. Lower bounds on dilation for various guest and host graphs are considered. Of particular interest are graphs that have been proposed as communication networks for parallel architectures. Bhattet al. provide a lower bound on dilation for embedding a planar guest graph in a butterfly host graph. Here, this lower bound is extended in two directions. First, a lower bound that applies to arbitrary guest graphs is derived, using tools from algebraic topology. Second, this lower bound is shown to apply to arbitrary host graphs through a new graph-theoretic measure, called bidecomposability. Bounds on the bidecomposability of the butterfly graph and of thek-dimensional torus are determined. As corollaries to the main lower-bound theorem, lower bounds are derived for embedding arbitrary planar graphs, genusg graphs, andk-dimensional meshes in a butterfly host graph.     

Parallel Algorithms for Series Parallel Graphs and Graphs with Treewidth Two 1

In this paper a parallel algorithm is given that, given a graph G=(V,E) , decides whether G is a series parallel graph, and, if so, builds a decomposition tree for G of series and parallel composition rules. The algorithm uses O(log \kern -1pt |E|log ^\ast \kern -1pt |E|) time and O(|E|) operations on an EREW PRAM, and O(log \kern -1pt |E|) time and O(|E|) operations on a CRCW PRAM. The results hold for undirected as well as for directed graphs. Algorithms with the same resource bounds are described for the recognition of graphs of treewidth two, and for constructing tree decompositions of treewidth two. Hence efficient parallel algorithms can be found for a large number of graph problems on series parallel graphs and graphs with treewidth two. These include many well-known problems like all problems that can be stated in monadic second-order logic. Received July 15, 1997; revised January 29, 1999, and June 23, 1999.     

Constant-Degree Graph Expansions that Preserve Treewidth

Many hard algorithmic problems dealing with graphs, circuits, formulas and constraints admit polynomial-time upper bounds if the underlying graph has small treewidth. The same problems often encourage reducing the maximal degree of vertices to simplify theoretical arguments or address practical concerns. Such degree reduction can be performed through a sequence of splittings of vertices, resulting in an expansion of the original graph. We observe that the treewidth of a graph may increase dramatically if the splittings are not performed carefully. In this context we address the following natural question: is it possible to reduce the maximum degree to a constant without substantially increasing the treewidth?We answer the above question affirmatively. We prove that any simple undirected graph G=(V,E) admits an expansion G′=(V′,E′) with the maximum degree ≤3 and tw(G′)≤tw(G)+1, where tw(?) is the treewidth of a graph. Furthermore, such an expansion will have no more than 2|E|+|V| vertices and 3|E| edges; it can be computed efficiently from a tree-decomposition of G. We also construct a family of examples for which the increase by 1 in treewidth cannot be avoided.     

Lower bounds for the Quadratic Minimum Spanning Tree Problem based on reduced cost computation

The Minimum Spanning Tree Problem (MSTP) is one of the most known combinatorial optimization problems. It concerns the determination of a minimum edge-cost subgraph spanning all the vertices of a given connected graph. The Quadratic Minimum Spanning Tree Problem (QMSTP) is a variant of the MSTP whose cost considers also the interaction between every pair of edges of the tree. In this paper we review different strategies found in the literature to compute a lower bound for the QMSTP and develop new bounds based on a reformulation scheme and some new mixed 0–1 linear formulations that result from a reformulation–linearization technique (RLT). The new bounds take advantage of an efficient way to retrieve dual information from the MSTP reduced cost computation. We compare the new bounds with the other bounding procedures in terms of both overall strength and computational effort. Computational experiments indicate that the dual-ascent procedure applied to the new RLT formulation provides the best bounds at the price of increased computational effort, while the bound obtained using the reformulation scheme seems to reasonably tradeoff between the bound tightness and computational effort.     

Resource Bounds and Subproblem Independence

There are several reasons why some NP-hard problems can be solved deterministically in 2^{O(n^a)} time for some a, 0 < a < 1. One reason is that the problem can be solved with a nondeterministic procedure which uses only O(n^a) space. A second reason is that each instance of the problem exhibits a high degree of "subproblem independence" (specifically O(n^a) line channelwidth or treewidth). A third is that each instance of the problem can be solved using nondeterministic straight-line programs where the number of variables is only O(n^a). In this paper we show that, after imposing q-linear bounds on the computation time and obliviousness constraints on the nondeterminism, these three reasons are essentially equivalent. (q-Linear functions are similar to functions of the form n (log n)^k.) Specifically, the problem classes associated with these reasons are interchangeable using reductions which are simultaneously polynomial time and q-linear size-bounded. We call these PQ-reductions. The classes of problems solvable by these methods we call NQn^a. Because these classes are defined in a very general way, they include a rich variety of problems. We take this as evidence that these classes have "power index" a (i.e., that 2^{O(n^a)} is also a lower time bound). On the other hand, the easiness of all these problems can be derived from "subproblem independence" considerations, and thus the techniques associated with subproblem independence are seen to be more widely applicable than one might expect.     

New spectral lower bounds on the bisection width of graphs

The communication overhead is a major bottleneck for the execution of a process graph on a parallel computer system. In the case of two processors, the minimization of the communication can be modeled using the graph bisection problem. The spectral lower bound of λ₂|V|/4 for the bisection width of a graph is widely known. The bisection width is equal to λ₂|V|/4 iff all vertices are incident to λ₂/2 cut edges in every optimal bisection.

We present a new method of obtaining tighter lower bounds on the bisection width. This method makes use of the level structure defined by the bisection. We define some global expansion properties and we show that the spectral lower bound increases with this global expansion. Under certain conditions we obtain a lower bound depending on λ₂^β|V| with . We also present examples of graphs for which our new bounds are tight up to a constant factor. As a by-product, we derive new lower bounds for the bisection widths of 3- and 4-regular Ramanujan graphs.