Sandwich Problem Research Articles

Partitions and well-coveredness: The graph sandwich problem

A graph G is well-covered if every maximal independent set of G is maximum. A (k,ℓ)-partition of a graph G is a partition of its vertex set into k independent sets and ℓ cliques. A graph is (k,ℓ)-well-covered if it is well-covered and admits a (k,ℓ)-partition. The recognition of (k,ℓ)-well-covered graphs is polynomial-time solvable for the cases (0,1), (0,2), (1,0), (1,1), (1,2), and (2,0), and hard, otherwise. In the graph sandwich problem for a property Π, we are given a pair of graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2, and asked whether there is a graph G=(V,E) with E1⊆E⊆E2, such that G satisfies the property Π. The problem of recognizing whether a graph G satisfies a property Π is equivalent to the particular graph sandwich problem where E1=E2. In this paper, we study the graph sandwich problem for the property of being (k,ℓ)-well-covered. We present some structural characterizations and extending previous studies on the recognition of (k,ℓ)-well-covered graphs, we prove that Graph Sandwich for(k,ℓ)-well-coveredness is polynomial-time solvable when (k,ℓ)∈{(0,1),(1,0),(1,1),(0,2)}. Besides, we show that it is NP-complete for the property of being (1,2)-well-covered.

Bipartite completion of colored graphs avoiding chordless cycles of given lengths

We consider a well-known restriction of the graph sandwich problem: Given a graph G with a proper vertex coloring, determine if there is a completion of G (formed by adding edges to G while maintaining the proper coloring) that has property P . We are interested in completions of G that are bipartite graphs without induced cycles of prescribed lengths. It is known that deciding whether there is a chordal bipartite completion, namely one without induced cycles on six or more vertices, is NP-complete when the input graph is colored with an arbitrary number of colors. We consider the case when the input graph is 3-colored and show that the problem of deciding whether there is a bipartite completion that avoids induced cycles C 6 , C 8 , ⋯ , C 2 p , for fixed p ≥ 3 , is NP-complete. In contrast, we characterize chordal bipartite completions of 3-colored graphs, and based on this, show that deciding whether a 3-colored graph can be completed to be chordal bipartite is solvable in O( m + n α ( n ) ) time. When the input admits a chordal bipartite completion, a size- n representation of a chordal bipartite completion can be constructed within the same time bound. It follows from our results that for every fixed k ≥ 3 , and for every fixed p ≥ 3 , deciding whether a k -colored graph can be completed to be bipartite and ( C 6 , C 8 , … , C 2 p ) -free is NP-complete. Also, the corresponding graph sandwich problems are hard.

Characterizations, probe and sandwich problems on [formula omitted]-cographs

A cograph is a graph without an induced 3-edge path. A graph is (k,ℓ) if its vertex set can be partitioned into at most k independent sets and ℓ cliques. (k,ℓ)-cographs already have a forbidden induced subgraphs characterization, but no structural characterization is known, except for (1,1)-cographs, i.e. threshold graphs. In this paper, we present a structural characterization and a decomposition theorem for (2,1)-cographs and, consequently, for (1,2)-cographs, leading to linear time recognition algorithms for both classes. Since recognizing a graph is a very important tool to solve several other problems, besides providing these characterizations, we applied them while dealing with two closely related problems: graph sandwich problems and probe problems. graph sandwich problems for propertyΠ (Π-sp) were introduced by Golumbic et al. as a natural generalization of recognition problems. Probe problems were defined by Zhang and can be analyzed in two versions, partitioned, which is a particular case of a sandwich problem, and non-partitioned. Our main result on probe problems shows that the probe (2,1)-cograph problem, in both versions, is polynomial-time solvable. Moreover, we also prove that, although the cograph-sp and the threshold-sp are polynomial-time solvable problems, the (2,1)-cograph-sp and the join of two thresholds-sp are NP-complete problems. As a corollary, we have that the (1,2)-cograph-sp is NP-complete as well. Using these results, we can fully classify P vs NP-complete dichotomy for the (k,ℓ)-cograph-sp, for fixed integers k,ℓ.

A General Method for Forbidden Induced Subgraph Sandwich Problem NP-completeness

We consider the sandwich problem, a generalization of the recognition problem introduced by Golumbic and Shamir (1993), with respect to classes of graphs defined by excluding induced subgraphs. The Π graph sandwich problem asks, for a pair of graphs G1 = (V, E1) and G2 = (V, E2) with E1 ⊆ E2, whether there exists a graph G = (V, E) with E1 ⊆ E ⊆ E2 that satisfies property Π. We consider the property of being H-free, where H is a fixed graph. Using a new variant of the SAT problem, we present a general framework to establish the NP-completeness of the sandwich problem for several H-free graph classes which generalizes the previous strategy for the class of Hereditary clique-Helly graphs. We also provide infinite families of 3-connected special forbidden induced subgraphs for which each forbidden induced subgraph sandwich problem is NP-complete.

Split It! Unpacking the Equipartitioning Learning Trajectory

Working with a mathematics coach and university researchers in a K-4 lesson study, teachers increase their understanding of student abilities in a fair-share sandwich problem.

Chordal-(2,1) graph sandwich problem with boundary conditions

In this work, we consider the graph sandwich problem for a property Π, a decision problem proposed by Golumbic, Kaplan, and Shamir as follows: given two graphs G1 = (V, E1) and G2 = (V, E2), the question is whether there exists a graph G = (V, E) such that E1 ⊆ E ⊆ E2 and G satisfies Π. For many graph classes, this problem was settled to be NP-complete. For this reason, a different kind of approach has been proposed: instead of focusing only on property Π, we can also require special properties Πi for the input graphs Gi, i = 1, 2, resulting in a problem that generalizes graph sandwich problems, called graph sandwich problems with boundary conditions. This problem is denoted by a triple (Π1, Π, Π2)-sp. We deal with the property Π of being a chordal-(2,1)-graph when it is given that G2 is a graph of a class for which there is a polynomial bound on the number of cliques, say pnc. chordal-(2,1)-sp is known to be NP-complete but, when requiring G2 to be of a class in pnc, we prove that (*, chordal-(2,1), pnc)-sp is polynomially time solvable.

Sandwich and probe problems for excluding paths

Let Pk denote an induced path on k vertices. For k≥5, we show that the Pk-free sandwich problem, partitioned probe problem, and unpartitioned probe problem are NP-complete. For k≤4, it is known that the Pk-free sandwich problem, partitioned probe problem, and unpartitioned probe problem are in P.

The Sandwich Problem for Decompositions and Almost Monotone Properties

We consider the graph sandwich problem and introduce almost monotone properties, for which the sandwich problem can be reduced to the recognition problem. We show that the property of containing a graph in $${\mathcal {C}}$$ as an induced subgraph is almost monotone if $${\mathcal {C}}$$ is the set of thetas, the set of pyramids, or the set of prisms and thetas. We show that the property of containing a hole of length $$\equiv j \mod n$$ is almost monotone if and only if $$j \equiv 2 \mod n$$ or $$n \le 2$$ . Moreover, we show that the imperfect graph sandwich problem, also known as the Berge trigraph recognition problem, can be solved in polynomial time. We also study the complexity of several graph decompositions related to perfect graphs, namely clique cutset, (full) star cutset, homogeneous set, homogeneous pair, and 1-join, with respect to the partitioned and unpartitioned probe problems. We show that the clique cutset and full star cutset unpartitioned probe problems are NP-hard. We show that for these five decompositions, the partitioned probe problem is in P, and the homogeneous set, 1-join, 1-join in the complement, and full star cutset in the complement unpartitioned probe problems can be solved in polynomial time as well.

On the proof of the thin sandwich conjecture in arbitrary dimensions

In this paper we show the validity, under certain geometric conditions, of Wheeler’s thin sandwich conjecture for higher dimensional theories of gravity. We extend the results for the 3-dimensional case in Phys. Rev. D 48, 3596–3599 (1993) in two ways. On the one hand, we show that the results presented in Phys. Rev. D 48, 3596–3599 (1993) are valid in arbitrary dimensions, and on the other hand, we show that the geometric hypotheses needed for the proofs can always be satisfied, which constitutes in itself a new result for the 3-dimensional case. In this way, we show that on any compact n-dimensional manifold, n≥3, there is an open set in the space of all possible initial data where the thin sandwich problem is well-posed.

Completing colored graphs to meet a target property

We consider the problem of deciding whether a k-colored graph can be completed to have a given property. We establish that, when k is not fixed, the completion problems for (Helly) circular-arc graphs, even (Helly) proper or (Helly) unit circular-arc graphs, are NP-complete. When k is fixed, in the case of completion to a (Helly) circular-arc graph, (Helly) proper circular-arc graph, or (Helly) unit circular-arc graph we fully classify the complexities of the problems. We also show that deciding whether a 3-colored graph can be completed to be strongly chordal can be done in O(n2) time. As a corollary of our results, the sandwich problems for Helly circular-arc graphs, Helly proper circular-arc graphs, and Helly unit circular-arc graphs are NP-complete.

Structural Properties and Applications to Sandwich Problem of the two forbidden four-vertex graph classes

Structural Properties and Applications to Sandwich Problem of the two forbidden four-vertex graph classes

On the forbidden induced subgraph probe and sandwich problems

In this work we compare the complexities of graph sandwich problems, partitioned and unpartitioned probe problems. Particularly, we consider the problem of recognizing probe graphs with respect to a class of graphs defined by excluding induced subgraphs. Our main result is: for F-free graphs, where F is a 3-connected, non complete graph, we prove that if F-free graph sandwich problem is NP-complete, then partitioned probeF-free problem and probeF-free problem are NP-complete. We also compare partitioned and unpartitioned versions of probe problems dealing with 2-connected graphs. Besides that, we consider Ck-free and (C4,…,C∣N∣)-free graphs and we prove that both problems in both versions of probe problems are NP-complete. In contrast, we prove that probe (Kr∖e)-free is polynomially solvable, where Kr∖e means a complete graph of size r without one edge. Finally, motivated by probeC4-free, we analyze the complexity of finding a stable set that touches every even cycle of size 2k exactly k times, which we call k-stableC2ktransversal. We prove that finding a k-stable C2k transversal can be done in polynomial time.

Hierarchical models for the static analysis of three-dimensional sandwich beam structures

This paper presents a static analysis of three-dimensional sandwich beam structures. Several higher-order, displacements-based theories as well as classical Euler–Bernoulli’s and Timoshenko’s models are derived by approximating the displacement field over the cross-section in a compact, unified form. The governing differential equations and the boundary conditions are derived in a nuclear form which is representative of a generic term in the displacement field approximation. This approach yields a problem formulation that does not depend upon the kinematic approximation order. The resulting boundary value problem is analytically solved by a Navier-type solution. The analysis is carried out in terms of displacement and stress field due to bending and torsional loads in slender and short sandwich beams. The effect of the skin/core anisotropy ratio is investigated. Results are assessed towards reference three-dimensional finite element solutions showing that the sandwich problems can be accurately yet efficiently solved.

Sandwich problem for [formula omitted]- and [formula omitted]-free multigraphs and its applications to positional games

An n-multigraphG=(V;Ei∣i∈I) is a complete graph G=(V,E) whose edges are covered by n=|I| sets, E=∪i∈IEi, some of which might be empty. If this cover is a partition, then G is called an n-graph. We say that an n-graph G′=(V;Ei′∣i∈I) is an edge subgraph of an n-multigraph G=(V;Ei∣i∈I) if Ei′⊆Ei for all i∈I. We denote by Δ the n-graph on three vertices with three nonempty sets each containing a single edge, and by Π the four-vertex n-graph with two non-empty sets each of which contains the edges of a P4. In this paper, we recognize in polynomial time whether a given n-multigraph G contains a Π- and Δ-free n-subgraph, or not, and if yes, provide a polynomial delay algorithm generating all such subgraphs. The above decision problem can be viewed as a generalization of the sandwich problem for P4-free graphs solved by Golumbic et al. (1995).As a motivation and application, we consider the n-person positional game forms, which are known to be in a one-to-one correspondence with the Π- and Δ-free n-graphs. Given a game form g, making use of the above result, we recognize in polynomial time whether g is a subform of a positional (that is, tight and rectangular) game form and, if yes, we generate with polynomial delay all such positional extensions of g.

Orthology and paralogy constraints: satisfiability and consistency.

BackgroundA variety of methods based on sequence similarity, reconciliation, synteny or functional characteristics, can be used to infer orthology and paralogy relations between genes of a given gene family . But is a given set of orthology/paralogy constraints possible, i.e., can they simultaneously co-exist in an evolutionary history for ? While previous studies have focused on full sets of constraints, here we consider the general case where does not necessarily involve a constraint for each pair of genes. The problem is subdivided in two parts: (1) Is satisfiable, i.e. can we find an event-labeled gene tree G inducing ? (2) Is there such a G which is consistent, i.e., such that all displayed triplet phylogenies are included in a species tree?ResultsPrevious results on the Graph sandwich problem can be used to answer to (1), and we provide polynomial-time algorithms for satisfiability and consistency with a given species tree. We also describe a new polynomial-time algorithm for the case of consistency with an unknown species tree and full knowledge of pairwise orthology/paralogy relationships, as well as a branch-and-bound algorithm in the case when unknown relations are present. We show that our algorithms can be used in combination with ProteinOrtho, a sequence similarity-based orthology detection tool, to extract a set of robust orthology/paralogy relationships.

Problem of time: facets and Machian strategy

The problem of time is that "time" in each of ordinary quantum theory and general relativity are mutually incompatible notions. This causes difficulties in trying to put these two theories together to form a theory of quantum gravity. The problem of time has eight facets in canonical approaches. I clarify that all but one of these facets already occur at the classical level, and reconceptualize and re-name some of these facets as follows. The frozen formalism problem becomes temporal relationalism, the thin sandwich problem becomes configurational relationalism, via the notion of best matching. The problem of observables becomes the problem of beables, and the functional evolution problem becomes the constraint closure problem. I also outline how each of the global and multiple-choice problems of time have their own plurality of facets. This article additionally contains a local resolution to the problem of time at the conceptual level and which is actually realizable for the relational triangle and minisuperspace models. This resolution is, moreover, Machian, and has three levels: classical, semiclassical, and a combined semiclassical-histories-timeless records scheme. I end by delineating the current frontiers of this program toward resolution of the problem of time in the cases of full general relativity and of slightly inhomogeneous cosmology.

Chordal- (k,ℓ)and strongly chordal- (k,ℓ)graph sandwich problems

In this work, we consider the graph sandwich decision problem for property Π, introduced by Golumbic, Kaplan and Shamir: given two graphs G1=(V,E1) and G2=(V,E2), the question is to know whether there exists a graph G=(V,E) such that E1⊆E⊆E2 and G satisfies property Π. Particurlarly, we are interested in fully classifying the complexity of this problem when we look to the following properties Π: `G is a chordal- (k,l)-graph' and `G is a strongly chordal- (k,l)-graph', for all k,l. In order to do that, we consider each pair of positive values of k and l, exhibiting correspondent polynomial algorithms, or NP-complete reductions. We prove that the strongly chordal- ( k,l ) graph sandwich problem is NP-complete, for k≥1 and l≥1, and that the chordal- ( k,l ) graph sandwich problem is NP-complete, for positive integers k and l such that k+l ≥ 3. Moreover, we prove that both problems are in P when k or l is zero and k+l ≤ 2. To complete the complexity dichotomy concerning these problems for all nonnegative values of k and l, there still remains the open question of settling the complexity for the case k+l ≥ 3 and one of them is equal to zero.

Characterizing compatibility and agreement of unrooted trees via cuts in graphs.

BackgroundDeciding whether there is a single tree —a supertree— that summarizes the evolutionary information in a collection of unrooted trees is a fundamental problem in phylogenetics. We consider two versions of this question: agreement and compatibility. In the first, the supertree is required to reflect precisely the relationships among the species exhibited by the input trees. In the second, the supertree can be more refined than the input trees.Testing for compatibility is an NP-complete problem; however, the problem is solvable in polynomial time when the number of input trees is fixed. Testing for agreement is also NP-complete, but it is not known whether it is fixed-parameter tractable. Compatibility can be characterized in terms of the existence of a specific kind of triangulation in a structure known as the display graph. Alternatively, it can be characterized as a chordal graph sandwich problem in a structure known as the edge label intersection graph. No characterization of agreement was known.ResultsWe present a simple and natural characterization of compatibility in terms of minimal cuts in the display graph, which is closely related to compatibility of splits. We then derive a characterization for agreement.ConclusionsExplicit characterizations of tree compatibility and agreement are essential to finding practical algorithms for these problems. The simplicity of the characterizations presented here could help to achieve this goal.

The complexity of forbidden subgraph sandwich problems and the skew partition sandwich problem

The Π graph sandwich problem asks, for a pair of graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2, whether there exists a graph G=(V,E) that satisfies property Π and E1⊆E⊆E2. We consider the property of being F-free, where F is a fixed graph. We show that the claw-free graph sandwich and the bull-free graph sandwich problems are both NP-complete, but the paw-free graph sandwich problem is polynomial. This completes the study of all cases where F has at most four vertices. A skew partition of a graph G is a partition of its vertex set into four nonempty parts A,B,C,D such that each vertex of A is adjacent to each vertex of B, and each vertex of C is nonadjacent to each vertex of D. We prove that the skew partition sandwich problem is NP-complete, establishing a computational complexity non-monotonicity.

Unique perfect phylogeny is intractable

A phylogeny is a tree capturing evolution and ancestral relationships of a set of taxa (e.g., species). Reconstructing phylogenies from molecular data plays an important role in many areas of contemporary biological research. A phylogeny is perfect if (in rough terms) it correctly captures all input data. Determining if a perfect phylogeny exists was shown to be intractable in 1992 by Mike Steel (Steel, 1992) [32] and independently by Bodlaender et al. (Bodlaender et al., 1992) [4]. In light of this, a related problem was proposed by Steel (1992) [32]: given a perfect phylogeny, determine if it is the unique perfect phylogeny for the given dataset, where the dataset is provided as a set of quartet (4-leaf) trees. It was suggested that this problem may be more tractable (Steel, 1992) [32], and determining its complexity became known as the Quartet Challenge (Steel, 2012) [33].In this paper, we resolve this question by showing that the problem is CoNP-complete. We prove this by relating perfect phylogenies to satisfying assignments of Boolean formulae. To this end, we cast the question as a chordal sandwich problem. As a particular consequence of our method, we show that the unique minimal chordal sandwich problem is CoNP-complete, and counting minimal chordal sandwiches is #P-complete.