Graph Separation Research Articles

Improved rate-distance trade-offs for quantum codes with restricted connectivity

Abstract For quantum error-correcting codes to be realizable, it is important that the qubits subject to the code constraints exhibit some form of limited connectivity. The works of Bravyi and Terhal (2009 New J. Phys. 11 043029) (BT) and Bravyi et al (2010 Phys. Rev. Lett. 104 050503) (BPT) established that geometric locality constrains code properties—for instance [ [ n , k , d ] ] quantum codes defined by local checks on the D-dimensional lattice must obey k d 2 / ( D − 1 ) ⩽ O ( n ) . Baspin and Krishna (2022 Quantum 6 711) studied the more general question of how the connectivity graph associated with a quantum code constrains the code parameters. These trade-offs apply to a richer class of codes compared to the BPT and BT bounds, which only capture geometrically-local codes. We extend and improve this work, establishing a tighter dimension-distance trade-off as a function of the size of separators in the connectivity graph. We also obtain a distance bound that covers all stabilizer codes with a particular separation profile, rather than only LDPC codes.

On Weighted Graph Separation Problems and Flow Augmentation

On Weighted Graph Separation Problems and Flow Augmentation

Non-Existence of Annular Separators in Geometric Graphs

Non-Existence of Annular Separators in Geometric Graphs

On finding separators in temporal split and permutation graphs

We study the NP-hard Temporal(s,z)-Separation problem on temporal graphs, which are graphs with fixed vertex sets but edge sets that change over discrete time steps. We tackle Temporal(s,z)-Separation by restricting the layers (i.e., graphs characterized by edges that are present at a certain point in time) to specific graph classes. We restrict the layers of the temporal graphs to be either all split graphs or all permutation graphs (both being perfect graph classes) and provide both intractability and tractability results. In particular, we show that in general Temporal(s,z)-Separation remains NP-hard both on temporal split and temporal permutation graphs, but we also spot promising islands of fixed-parameter tractability particularly based on parameterizations that measure the amount of “change over time”.

Clique-Based Separators for Geometric Intersection Graphs

Let F be a set of n objects in the plane and let mathcal {G}^{times }(F) be its intersection graph. A balanced clique-based separator of mathcal {G}^{times }(F) is a set mathcal {mathcal {S}} consisting of cliques whose removal partitions mathcal {G}^{times }(F) into components of size at most delta n, for some fixed constant delta <1. The weight of a clique-based separator is defined as sum _{Cin mathcal {mathcal {S}}}log (|C|+1). Recently De Berg et al. (SIAM J. Comput. 49: 1291-1331. 2020) proved that if S consists of convex fat objects, then mathcal {G}^{times }(F) admits a balanced clique-based separator of weight O(sqrt{n}). We extend this result in several directions, obtaining the following results. (i) Map graphs admit a balanced clique-based separator of weight O(sqrt{n}), which is tight in the worst case. (ii) Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n^{2/3}log n). If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to O(sqrt{n}log n). (iii) Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^{2/3}log n). (iv) Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight O(sqrt{n}+rlog (n/r)), which is tight in the worst case. These results immediately imply sub-exponential algorithms for Maximum Independent Set (and, hence, Vertex Cover), for Feedback Vertex Set, and for q-Coloring for constant q in these graph classes.

Connectivity constrains quantum codes

Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D-dimensional hyperbolic space.

Cyclability in graph classes

A subset T⊆V(G) of vertices of a graph G is said to be cyclable if G has a cycle C containing every vertex of T, and for a positive integer k, a graph G is k-cyclable if every set of vertices of size at most k is cyclable. The Terminal Cyclability problem asks, given a graph G and a set T of vertices, whether T is cyclable, and the k-Cyclability problem asks, given a graph G and a positive integer k, whether G is k-cyclable. These problems are generalizations of the classical Hamiltonian Cycle problem. We initiate the study of these problems for graph classes that admit polynomial algorithms for Hamiltonian Cycle. We show that Terminal Cyclability can be solved in linear time for interval graphs, bipartite permutation graphs and cographs. Moreover, we construct certifying algorithms that either produce a solution, that is a cycle, or output a graph separator that certifies a no-answer. We use these results to show that k-Cyclability can be solved in polynomial time when restricted to the aforementioned graph classes.

Finding Optimal Triangulations Parameterized by Edge Clique Cover

We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover (texttt {cc}) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem texttt {cc} is at most the number of taxa, in fractional hypertreewidth texttt {cc} is at most the number of hyperedges, and in treewidth of Bayesian networks texttt {cc} is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most 2^texttt {cc}, the number of potential maximal cliques is at most 3^texttt {cc}, and these objects can be listed in times O^*(2^texttt {cc}) and O^*(3^texttt {cc}), respectively, even when no edge clique cover is given as input; the O^*(cdot ) notation omits factors polynomial in the input size. These enumeration algorithms imply O^*(3^texttt {cc}) time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give O^*(4^m) time and O^*(3^m) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size texttt {cc}' is given as a part of the input we give O^*(2^{texttt {cc}'}) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an O^*(2^n) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities O^*(9^{texttt {cc}'}) and O^*(9^{texttt {cc}+ O(log ^2 texttt {cc})}) for problems in this framework.

MUSE: Multimodal Separators for Efficient Route Planning in Transportation Networks

Many algorithms compute shortest-path queries in mere microseconds on continental-scale networks. Most solutions are, however, tailored to either road or public transit networks in isolation. To fully exploit the transportation infrastructure, multimodal algorithms are sought to compute shortest paths combining various modes of transportation. Nonetheless, current solutions still lack performance to efficiently handle interactive queries under realistic network conditions where traffic jams, public transit cancelations, or delays often occur. We present a multimodal separators–based algorithm (MUSE), a new multimodal algorithm based on graph separators to compute shortest travel time paths. It partitions the network into independent, smaller regions, enabling fast and scalable preprocessing. The partition is common to all modes and independent of traffic conditions so that the preprocessing is only executed once. MUSE relies on a state automaton that describes the sequence of modes to constrain the shortest path during the preprocessing and the online phase. The support of new sequences of mobility modes only requires the preprocessing of the cliques, independently for each partition. We also augment our algorithm with heuristics during the query phase to achieve further speedups with minimal effect on correctness. We provide experimental results on France’s multimodal network containing the pedestrian, road, bicycle, and public transit networks.

On Local Input-Output Stability of Nonlinear Feedback Systems via Local Graph Separation

On Local Input-Output Stability of Nonlinear Feedback Systems via Local Graph Separation

Hamming-shifting graph of genomic short reads: Efficient construction and its application for compression.

Graphs such as de Bruijn graphs and OLC (overlap-layout-consensus) graphs have been widely adopted for the de novo assembly of genomic short reads. This work studies another important problem in the field: how graphs can be used for high-performance compression of the large-scale sequencing data. We present a novel graph definition named Hamming-Shifting graph to address this problem. The definition originates from the technological characteristics of next-generation sequencing machines, aiming to link all pairs of distinct reads that have a small Hamming distance or a small shifting offset or both. We compute multiple lexicographically minimal k-mers to index the reads for an efficient search of the weight-lightest edges, and we prove a very high probability of successfully detecting these edges. The resulted graph creates a full mutual reference of the reads to cascade a code-minimized transfer of every child-read for an optimal compression. We conducted compression experiments on the minimum spanning forest of this extremely sparse graph, and achieved a 10 − 30% more file size reduction compared to the best compression results using existing algorithms. As future work, the separation and connectivity degrees of these giant graphs can be used as economical measurements or protocols for quick quality assessment of wet-lab machines, for sufficiency control of genomic library preparation, and for accurate de novo genome assembly.

Efficient fastest-path computations for road maps

In the age of real-time online traffic information and GPS-enabled devices, fastest-path computations between two points in a road network modeled as a directed graph, where each directed edge is weighted by a “travel time” value, are becoming a standard feature of many navigation-related applications. To support this, very efficient computation of these paths in very large road networks is critical. Fastest paths may be computed as minimal-cost paths in a weighted directed graph, but traditional minimal-cost path algorithms based on variants of the classical Dijkstra algorithm do not scale well, as in the worst case they may traverse the entire graph. A common improvement, which can dramatically reduce the number of graph vertices traversed, is the A* algorithm, which requires a good heuristic lower bound on the minimal cost. We introduce a simple, but very effective, heuristic function based on a small number of values assigned to each graph vertex. The values are based on graph separators and are computed efficiently in a preprocessing stage. We present experimental results demonstrating that our heuristic provides estimates of the minimal cost superior to those of other heuristics. Our experiments show that when used in the A* algorithm, this heuristic can reduce the number of vertices traversed by an order of magnitude compared to other heuristics.

Polynomially Bounding the Number of Minimal Separators in Graphs: Reductions, Sufficient Conditions, and a Dichotomy Theorem

A graph class is said to be tame if graphs in the class have a polynomially bounded number of minimal separators. Tame graph classes have good algorithmic properties, which follow, for example, from an algorithmic metatheorem of Fomin, Todinca, and Villanger from 2015. We show that a hereditary graph class $\mathcal{G}$ is tame if and only if the subclass consisting of graphs in $\mathcal{G}$ without clique cutsets is tame. This result and Ramsey's theorem lead to several types of sufficient conditions for a graph class to be tame. In particular, we show that any hereditary class of graphs of bounded clique cover number that excludes some complete prism is tame, where a complete prism is the Cartesian product of a complete graph with a $K_2$. We apply these results, combined with constructions of graphs with exponentially many minimal separators, to develop a dichotomy theorem separating tame from non-tame graph classes within the family of graph classes defined by sets of forbidden induced subgraphs with at most four vertices.

Choosability with separation of cycles and outerplanar graphs

We consider the following list coloring with separation problem of graphs: Given a graph $G$ and integers $a,b$, find the largest integer $c$ such that for any list assignment $L$ of $G$ with $|L(v)|\le a$ for any vertex $v$ and $|L(u)\cap L(v)|\le c$ for any edge $uv$ of $G$, there exists an assignment $\varphi$ of sets of integers to the vertices of $G$ such that $\varphi(u)\subset L(u)$ and $|\varphi(v)|=b$ for any vertex $v$ and $\varphi(u)\cap \varphi(v)=\emptyset$ for any edge $uv$. Such a value of $c$ is called the separation number of $(G,a,b)$. We also study the variant called the free-separation number which is defined analogously but assuming that one arbitrary vertex is precolored. We determine the separation number and free-separation number of the cycle and derive from them the free-separation number of a cactus. We also present a lower bound for the separation and free-separation numbers of outerplanar graphs of girth $g\ge 5$.

Sublinear Separators in Intersection Graphs of Convex Shapes

We give a natural sufficient condition for an intersection graph of compact convex sets in $\mathbb{R}^d$ to have a balanced separator of sublinear size. This condition generalizes several previous results on sublinear separators in intersection graphs. Furthermore, the argument used to prove the existence of sublinear separators is based on a connection with generalized coloring numbers which has not been previously explored in geometric settings.

Layer-wise relevance propagation of InteractionNet explains protein\u2013ligand interactions at the atom level

Development of deep-learning models for intermolecular noncovalent (NC) interactions between proteins and ligands has great potential in the chemical and pharmaceutical tasks, including structure–activity relationship and drug design. It still remains an open question how to convert the three-dimensional, structural information of a protein–ligand complex into a graph representation in the graph neural networks (GNNs). It is also difficult to know whether a trained GNN model learns the NC interactions properly. Herein, we propose a GNN architecture that learns two distinct graphs—one for the intramolecular covalent bonds in a protein and a ligand, and the other for the intermolecular NC interactions between the protein and the ligand—separately by the corresponding covalent and NC convolutional layers. The graph separation has some advantages, such as independent evaluation on the contribution of each convolutional step to the prediction of dissociation constants, and facile analysis of graph-building strategies for the NC interactions. In addition to its prediction performance that is comparable to that of a state-of-the art model, the analysis with an explainability strategy of layer-wise relevance propagation shows that our model successfully predicts the important characteristics of the NC interactions, especially in the aspect of hydrogen bonding, in the chemical interpretation of protein–ligand binding.

Choosability with union separation of triangle-free planar graphs

For a graph G and a positive integer k, a k-list assignment of G is a function L on the vertices of G such that for each vertex v∈V(G), |L(v)|≥k. Let s be a nonnegative integer. Then L is a (k,k+s)-list assignment of G if |L(u)∪L(v)|≥k+s for each edge uv. If for each (k,k+s)-list assignment L of G, G admits a proper coloring φ such that φ(v)∈L(v) for each v∈V(G), then we say G is (k,k+s)-choosable. This refinement of choosability is called choosability with union separation by Kumbhat et al. (2018), who showed that all planar graphs are (3,11)-choosable and (4,9)-choosable. In this paper, we prove that every triangle-free planar graph is (3,7)-choosable. We also prove that every planar graph with girth at least 5 is (2,7)-choosable.

Mutual conditional independence and its applications to model selection in Markov networks

The fundamental concepts underlying Markov networks are the conditional independence and the set of rules called Markov properties that translate conditional independence constraints into graphs. We introduce the concept of mutual conditional independence in an independent set of a Markov network, and we prove its equivalence to the Markov properties under certain regularity conditions. This extends the notion of similarity between separation in graph and conditional independence in probability to similarity between the mutual separation in graph and the mutual conditional independence in probability. Model selection in graphical models remains a challenging task due to the large search space. We show that mutual conditional independence property can be exploited to reduce the search space. We present a new forward model selection algorithm for graphical log-linear models using mutual conditional independence. We illustrate our algorithm with a real data set example. We show that for sparse models the size of the search space can be reduced from mathcal {O} (n^{3}) to mathcal {O}(n^{2}) using our proposed forward selection method rather than the classical forward selection method. We also envision that this property can be leveraged for model selection and inference in different types of graphical models.

Choosability with union separation of planar graphs without cycles of length 4

Abstract For a graph G and a positive integer k, a k-list assignment of G is a function L on the vertices of G such that for each vertex v ∈ V(G), |L(v)| ≥ k. Let s be a nonnegative integer. Then L is a ( k , k + s ) -list assignment of G if | L ( u ) ∪ L ( v ) | ≥ k + s for each edge uv. If for each ( k , k + s ) -list assignment L of G, G admits a proper coloring φ such that φ(v) ∈ L(v) for each v ∈ V(G), then we say G is ( k , k + s ) -choosable. This refinement of choosability is called choosability with union separation by Kumbhat, Moss and Stolee, who showed that all planar graphs are (3, 11)-choosable. In this paper, we prove that every planar graph without cycles of length 4 is (3,6)-choosable.

Submodel Decomposition for Solving Limited Memory Influence Diagrams (Student Abstract)

This paper presents a systematic way of decomposing a limited memory influence diagram (LIMID) to a tree of single-stage decision problems, or submodels and solving it by message passing. The relevance in LIMIDs is formalized by the notion of the partial evaluation of the maximum expected utility, and the graph separation criteria for identifying submodels follow. The submodel decomposition provides a graphical model approach for updating the beliefs and propagating the conditional expected utilities for solving LIMIDs with the worst-case complexity bounded by the maximum treewidth of the individual submodels.