Sandwich Problem Research Articles

Tunneling cracks in the adhesive layer of an orthotropic sandwich structure

A tunneling crack confined in the orthotropic adhesive layer bonded to orthotropic substrates under steady state conditions was examined. The problem was formulated using the modified Stroh formalism and orthotropy rescaling technique. The energy release rate for the crack in the sandwich structure was obtained from a solution of the transformed sandwich structure composed of an orthotropic adhesive layer and isotropic substrates. The dimensionless energy release rate for the transformed sandwich problem depends only on four material parameters. Finite element analysis was performed to determine the changes in energy release rate on the four material parameters. The effects of the material parameters on the energy release rate are discussed for various combinations. Periodic tunneling cracks were also considered to examine the effect of the crack spacing on the energy release rate.

Problem of time in quantum gravity

AbstractThe Problem of Time occurs because the ‘time’ of GR and of ordinary Quantum Theory are mutually incompatible notions. This is problematic in trying to replace these two branches of physics with a single framework in situations in which the conditions of both apply, e.g. in black holes or in the very early universe. Emphasis in this Review is on the Problem of Time being multi‐faceted and on the nature of each of the eight principal facets. Namely, the Frozen Formalism Problem, Configurational Relationalism Problem (formerly Sandwich Problem), Foliation Dependence Problem, Constraint Closure Problem (formerly Functional Evolution Problem), Multiple Choice Problem, Global Problem of Time, Problem of Beables (alias Problem of Observables) and Spacetime Reconstruction/Replacement Problem. Strategizing in this Review is not just centred about the Frozen Formalism Problem facet, but rather about each of the eight facets. Particular emphasis is placed upon A) relationalism as an underpinning of the facets and as a selector of particular strategies (especially a modification of Barbour relationalism, though also with some consideration of Rovelli relationalism). B) Classifying approaches by the full ordering in which they embrace constrain, quantize, find time/history and find observables, rather than only by partial orderings such as “Dirac‐quantize”. C) Foliation (in)dependence and Spacetime Reconstruction for a wide range of physical theories, strategizing centred about the Problem of Beables, the Patching Approach to the Global Problem of Time, and the role of the question‐types considered in physics. D) The Halliwell‐ and Gambini–Porto–Pullin‐type combined Strategies in the context of semiclassical quantum cosmology.

Sandwich problems on orientations

Abstract The graph sandwich problem for property Π is defined as follows: Given two graphs G 1=(V,E 1) and G 2=(V,E 2) such that E 1⊆E 2, is there a graph G=(V,E) such that E 1⊆E⊆E 2 which satisfies property Π? We propose to study sandwich problems for properties Π concerning orientations, such as Eulerian orientation of a mixed graph and orientation with given in-degrees of a graph. We present a characterization and a polynomial-time algorithm for solving the m-orientation sandwich problem.

(k, ℓ)-Sandwich Problems: why not ask for special kinds of bread?

(k, ℓ)-Sandwich Problems: why not ask for special kinds of bread?

The Simultaneous Representation Problem for Chordal, Comparability and Permutation Graphs

We introduce the simultaneous representation problem , defined for any graph class $\cal C$ characterized in terms of representations, e.g. any class of intersection graphs. Two graphs G 1 and G 2 , sharing some vertices X (and the corresponding induced edges), are said to have a simultaneous representation with respect to a graph class $\cal C$, if there exist representations R 1 and R 2 of G 1 and G 2 that are consistent on X . Equivalently (for the classes $\cal C$ that we consider) there exist edges E *** between G 1 *** X and G 2 *** X such that G 1 *** G 2 *** E *** belongs to class $\cal C$. Simultaneous representation problems are related to graph sandwich problems, probe graph recognition problems and simultaneous planar embeddings and have applications in any situation where it is desirable to consistently represent two related graphs. In this paper we give efficient algorithms for the simultaneous representation problem on chordal, comparability and permutation graphs. These results complement the recent poly-time algorithms for recognizing probe graphs for the above classes and imply that the graph sandwich problem for these classes is solvable for an interesting special case: when the set of optional edges induce a complete bipartite graph. Moreover for comparability and permutation graphs, our results can be extended to solve a generalized version of the simultaneous representation problem when there are k graphs any two of which share a common vertex set X . This generalized version is equivalent to the graph sandwich problem when the set of optional edges induce a k -partite graph.

Mapping ancestral genomes with massive gene loss: a matrix sandwich problem.

Motivation: Ancestral genomes provide a better way to understand the structural evolution of genomes than the simple comparison of extant genomes. Most ancestral genome reconstruction methods rely on universal markers, that is, homologous families of DNA segments present in exactly one exemplar in every considered species. Complex histories of genes or other markers, undergoing duplications and losses, are rarely taken into account. It follows that some ancestors are inaccessible by these methods, such as the proto–monocotyledon whose evolution involved massive gene loss following a whole genome duplication.Results: We propose a mapping approach based on the combinatorial notion of ‘sandwich consecutive ones matrix’, which explicitly takes gene losses into account. We introduce combinatorial optimization problems related to this concept, and propose a heuristic solver and a lower bound on the optimal solution. We use these results to propose a configuration for the proto-chromosomes of the monocot ancestor, and study the accuracy of this configuration. We also use our method to reconstruct the ancestral boreoeutherian genomes, which illustrates that the framework we propose is not specific to plant paleogenomics but is adapted to reconstruct any ancestral genome from extant genomes with heterogeneous marker content.Availability: Upon request to the authors.Contact: haris.gavranovic@gmail.com; eric.tannier@inria.fr

The P versus NP–complete dichotomy of some challenging problems in graph theory

The Clay Mathematics Institute has selected seven Millennium Problems to motivate research on important classic questions that have resisted solution over the years. Among them is the central problem in theoretical computer science: the P versus NP problem, which aims to classify the possible existence of efficient solutions to combinatorial and optimization problems. The main goal is to determine whether there are questions whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure. In this context, it is important to determine precisely what facet of a problem makes it NP–complete. We shall discuss classes of problems for which dichotomy results do exist: every problem in the class is classified into polynomial or NP–complete. We shall discuss our contribution through the classification of some long-standing problems in important areas of graph theory: perfect graphs, intersection graphs, and structural characterization of graph classes. More precisely, we have shown that Chvátal’s skew partition is polynomial and that Roberts–Spencer’s clique graph is NP–complete. We have also solved the dichotomy for Golumbic–Kaplan–Shamir’s sandwich problem. We shall describe two examples where we can determine the full dichotomy: the edge-colouring problem for graphs with no cycle with a unique chord and the three nonempty part sandwich problem. Some open problems are discussed: the stubborn problem for list partition, the chromatic index of chordal graphs, and the recognition of split clique graphs.

On the forbidden induced subgraph sandwich problem

We consider the sandwich problem, a generalization of the recognition problem introduced by Golumbic et al. (1995) [15], with respect to classes of graphs defined by excluding induced subgraphs. We prove that the sandwich problem corresponding to excluding a chordless cycle of fixed length k is NP-complete. We prove that the sandwich problem corresponding to excluding K r ∖ e for fixed r is polynomial. We prove that the sandwich problem corresponding to 3 P C ( ⋅ , ⋅ ) -free graphs is NP-complete. These complexity results are related to the classification of a long-standing open problem: the sandwich problem corresponding to perfect graphs.

The chain graph sandwich problem

The chain graph sandwich problem asks: Given a vertex set V, a mandatory edge set E 1, and a larger edge set E 2, is there a graph G=(V,E) such that E 1⊆E⊆E 2 with G being a chain graph (i.e., a 2K 2-free bipartite graph)? We prove that the chain graph sandwich problem is NP-complete. This result stands in contrast to (1) the case where E 1 is a connected graph, which has a linear-time solution, (2) the threshold graph sandwich problem, which has a linear-time solution, and (3) the chain probe graph problem, which has a polynomial-time solution.

Complexity issues for the sandwich homogeneous set problem

Graph sandwich problems were introduced by Golumbic et al. (1994) in [12] for DNA physical mapping problems and can be described as follows. Given a property Π of graphs and two disjoint sets of edges E 1 , E 2 with E 1 ⊆ E 2 on a vertex set V , the problem is to find a graph G on V with edge set E s having property Π and such that E 1 ⊆ E s ⊆ E 2 . In this paper, we exhibit a quasi-linear reduction between the problem of finding an independent set of size k ≥ 2 in a graph and the problem of finding a sandwich homogeneous set of the same size k . Using this reduction, we prove that a number of natural (decision and counting) problems related to sandwich homogeneous sets are hard in general. We then exploit a little further the reduction and show that finding efficient algorithms to compute small sandwich homogeneous sets would imply substantial improvement for computing triangles in graphs.

The external constraint 4 nonempty part sandwich problem

List partitions generalize list colourings. Sandwich problems generalize recognition problems. The polynomial dichotomy (NP-complete versus polynomial) of list partition problems is solved for 4-dimensional partitions with the exception of one problem (the list stubborn problem) for which the complexity is known to be quasipolynomial. Every partition problem for 4 nonempty parts and only external constraints is known to be polynomial with the exception of one problem (the 2 K 2 - partition problem) for which the complexity of the corresponding list problem is known to be NP-complete. The present paper considers external constraint 4 nonempty part sandwich problems. We extend the tools developed for polynomial solutions of recognition problems obtaining polynomial solutions for most corresponding sandwich versions. We extend the tools developed for NP-complete reductions of sandwich partition problems obtaining the classification into NP-complete for some external constraint 4 nonempty part sandwich problems. On the other hand and additionally, we propose a general strategy for defining polynomial reductions from the 2 K 2 - partition problem to several external constraint 4 nonempty part sandwich problems, defining a class of 2 K 2 -hard problems. Finally, we discuss the complexity of the Skew Partition Sandwich Problem.

On the complexity of computing treelength

We resolve the computational complexity of determining the treelength of a graph, thereby solving an open problem of Dourisboure and Gavoille, who introduced this parameter, and asked to determine the complexity of recognizing graphs of a bounded treelength Dourisboure and Gavoille (2007) [6]. While recognizing graphs with treelength 1 is easily seen as equivalent to recognizing chordal graphs, which can be done in linear time, the computational complexity of recognizing graphs with treelength 2 was unknown until this result. We show that the problem of determining whether a given graph has a treelength at most k is NP-complete for every fixed k ≥ 2 , and use this result to show that treelength in weighted graphs is hard to approximate within a factor smaller than 3 2 . Additionally, we show that treelength can be computed in time O ∗ ( 1.754 9 n ) by giving an exact exponential time algorithm for the Chordal Sandwich problem and showing how this algorithm can be used to compute the treelength of a graph.

The polynomial dichotomy for three nonempty part sandwich problems

We classify into polynomial time or NP -complete all three nonempty part sandwich problems. This solves the polynomial dichotomy into polynomial time and NP -complete for this class of graph partition problems.

Skew partition sandwich problem is NP-complete

Abstract Sandwich problems generalize graph recognition problems with respect to a property Π. A recognition problem has a graph as input, whereas a sandwich problem has two graphs as input. In a sandwich problem, we look for a third graph, required to satisfy a property Π, whose edge set lies between the edge sets of two given graphs. A skew partition of a graph G = ( V , E ) is a partition of its vertex set V into four nonempty parts A, B, C, D such that each vertex of part A is adjacent to each vertex of part B, and each vertex of part C is nonadjacent to each vertex of part D. Skew cutset generalizes star cutset which in turn generalizes both homogeneous set and clique cutset. Homogeneous set, clique cutset, star cutset, and skew cutset are decompositions arising in perfect graph theory and the recognition of each decomposition is known to be polynomial. Regarding sandwich problems, it is known that homogeneous set sandwich problem is polynomial, clique cutset sandwich problem is NP-complete, and star cutset sandwich problem is polynomial. We prove that skew partition sandwich problem is NP-complete, establishing an interesting computational complexity non-monotonicity.

The Graph Sandwich Problem for [formula omitted]-sparse graphs

The P 4 -sparse Graph Sandwich Problem asks, given two graphs G 1 = ( V , E 1 ) and G 2 = ( V , E 2 ) , whether there exists a graph G = ( V , E ) such that E 1 ⊆ E ⊆ E 2 and G is P 4 -sparse. In this paper we present a polynomial-time algorithm for solving the Graph Sandwich Problem for P 4 -sparse graphs.

The polynomial dichotomy for three nonempty part sandwich problems

Abstract We classify into polynomial time or NP -complete all three nonempty part sandwich problems. This solves the polynomial dichotomy for this class of problems.

Helly property, clique raphs, complementary graph classes, and sandwich problems

Abstract A sandwich problem for property π asks whether there exists a sandwich graph of a given pair of graphs which has the desired property π Graph sandwich problems were first defined in the context of Computational Biology as natural generalizations of recognition problems. We contribute to the study of the complexity of graph sandwich problems by considering the Helly property and complementary graph classes. We obtain a graph class defined by a finite family of minimal forbidden subgraphs for which the sandwich problem is NP-complete. A graph is clique-Helly when its family of cliques satisfies the Helly property. A graph is hereditary clique-Helly when all of its induced subgraphs are clique-Helly. The clique graph of a graph is the intersection graph of the family of its cliques. The recognition problem for the class of clique graphs was a long-standing open problem that was recently solved. We show that the sandwich problems for the graph classes: clique, clique-Helly, hereditary clique-Helly, and clique-Helly nonhereditary are all NP-complete. We propose the study of the complexity of sandwich problems for complementary graph classes as a mean to further understand the sandwich problem as a generalization of the recognition problem.

Helly property, clique graphs, complementary graph classes, and sandwich problems

A sandwich problem for property Π asks whether there exists a sandwich graph of a given pair of graphs which has the desired property Π. Graph sandwich problems were first defined in the context of Computational Biology as natural generalizations of recognition problems. We contribute to the study of the complexity of graph sandwich problems by considering the Helly property and complementary graph classes. We obtain a graph class defined by a finite family of minimal forbidden subgraphs for which the sandwich problem is NP-complete. A graph is clique-Helly when its family of cliques satisfies the Helly property. A graph is hereditary clique-Helly when all of its induced subgraphs are clique-Helly. The clique graph of a graph is the intersection graph of the family of its cliques. The recognition problem for the class of clique graphs was a long-standing open problem that was recently solved. We show that the sandwich problems for the graph classes: clique, clique-Helly, hereditary clique-Helly, and clique-Helly nonhereditary are all NP-complete. We propose the study of the complexity of sandwich problems for complementary graph classes as a mean to further understand the sandwich problem as a generalization of the recognition problem.

Tree-Related Widths of Graphs and Hypergraphs

A hypergraph pair is a pair $(G,H)$ where G and H are hypergraphs on the same set of vertices. We extend the definitions of hypertree-width [G. Gottlob, N. Leone, and F. Scarcello, J. Comput. System Sci., 64 (2002), pp. 579–627] and generalized hypertree-width [G. Gottlob, N. Leone, and F. Scarcello, J. Comput. System Sci., 66 (2003), pp. 775–808] from hypergraphs to hypergraph pairs. We show that for constant k the problem of deciding whether a hypergraph pair has generalized hypertree-width $\leq k$, is equivalent to the hypergraph sandwich problem (HSP) [A. Lustig and O. Shmueli, J. Algorithms, 30 (1999), pp. 400–422]. It was recently proved in [G. Gottlob, Z. Miklós, and Th. Schwentick, Proceedings of the Symposium on Principles of Database Systems $(PODS\/07)$] that the HSP is NP-complete. For constant k there is a polynomial time algorithm that decides whether a given hypergraph pair has hypertree-width $\leq k$. (For hypertree-width of hypergraphs, this was shown in [G. Gottlob, N. Leone, and F. Scarcello, J. Comput. System Sci., 64 (2002), pp. 579–627].) It follows that the HSP is solvable in polynomial time for inputs $(G,H)$ satisfying: $\operatorname{ghw}(G,H)\leq 1$ if and only if $\operatorname{hw}(G,H)\leq 1$. Besides this practical interest, hypergraph pairs serve as a tool for giving a common proof for the game theoretic characterizations of tree-width [P. D. Seymour and R. Thomas, J. Combin. Theory Ser. B, 58 (1993), pp. 22–33] and hypertree-width [G. Gottlob, N. Leone, and F. Scarcello, J. Comput. System Sci., 66 (2003), pp. 775–808]. Furthermore, they enable us to show a compactness property of generalized hypertree-width for a large class of hypergraphs, the hypergraphs with finite character. Finally, we present two examples showing that neither hypertree-width of hypergraph pairs nor hypertree-width of hypergraphs has the compactness property.

Competitive graph searches

We exemplify an optimization criterion for divide-and-conquer algorithms with a technique called generic competitive graph search. The technique is then applied to solve two problems arising from biocomputing, so-called Common Connected Components and Cograph Sandwich. The first problem can be defined as follows: given two graphs on the same set of n vertices, find the coarsest partition of the vertex set into subsets which induce connected subgraphs in both input graphs. The second problem is an instance of sandwich problems: given a partial subgraph G 1 of G 2 , find a partial subgraph G of G 2 that is partial supergraph of G 1 (sandwich), and that is a cograph. For the former problem our generic algorithm not only achieves the current best known performance on arbitrary graphs and forests, but also improves by a log n factor when the input is made of planar graphs. However, our complexity for intervals graphs is slightly lower than a recent result. For the latter problem, we first study the relationship between the common connected components problem and the cograph sandwich problem, then, using our competitive graph search paradigm, we improve the computation of cograph sandwiches from O ( n ( n + m ) ) down to O ( n + m log 2 n ) , where n is the number of vertices and m of total edges.