Subgraph Problem Research Articles

Shiny GATOM: omics-based identification of regulated metabolic modules in atom transition networks.

Multiple high-throughput omics techniques provide different angles on systematically quantifying and studying metabolic regulation of cellular processes. However, an unbiased analysis of such data and, in particular, integration of multiple types of data remains a challenge. Previously, for this purpose we developed GAM web-service for integrative metabolic network analysis. Here we describe an updated pipeline GATOM and the corresponding web-service Shiny GATOM, which takes as input transcriptional and/or metabolomic data and finds a metabolic subnetwork most regulated between the two conditions of interest. GATOM features a new metabolic network topology based on atom transition, which significantly improves interpretability of the analysis results. To address computational challenges arising with the new network topology, we introduce a new variant of the maximum weight connected subgraph problem and provide a corresponding exact solver. To make the used networks up-to-date we upgraded the KEGG-based network construction pipeline and developed one based on the Rhea database, which allows analysis of lipidomics data. Finally, we simplified local installation, providing R package mwcsr for solving relevant graph optimization problems and R package gatom, which implements the GATOM pipeline. The web-service is available at https://ctlab.itmo.ru/shiny/gatom and https://artyomovlab.wustl.edu/shiny/gatom.

Multicolor Turán numbers

We consider a natural generalization of Turán's forbidden subgraph problem and the Ruzsa-Szemerédi problem by studying the maximum number exF(n,G) of edge-disjoint copies of a fixed graph F can be placed on an n-vertex ground set without forming a subgraph G whose edges are from different F-copies. We determine the pairs {F,G} for which the order of magnitude of exF(n,G) is quadratic and prove several asymptotic results using various tools from the regularity lemma and supersaturation to graph packing results.

HiveRel: hexagons visualization for relationship-based knowledge acquisition

The growing abundance in complex network data models is constantly increasing the challenges for non-expert users who perform an effective exploratory search in large data collections. In such domains, users search for entities related to a topic of interest and acquire knowledge by investigating the relationships between these entities. Designers, in turn, are challenged by the need to provide tools that enable convenient search and exploration to facilitate productive performance on the task. For this purpose, we introduce HiveRel, a search system that presents search results as tiled hexagons on a map-like surface with center-out relevance ordering and allows on-demand display of relationships between search results. HiveRel’s user interface is based on theoretical principles that reflect how users acquire knowledge through relationships. For the search mechanism, we provide a set of information retrieval definitions leading to the formalization of the Maximal n-Bounded Exploration Subgraph problem and present an implementation of a greedy heuristic algorithm that provides non-optimal solutions to this problem. We develop a proof of concept version of HiveRel. We evaluate it in two user studies that compare users’ performance using HiveRel to standard web search over a range of search knowledge acquisition tasks and two different domains. The results indicate that despite the lack of familiarity with the new system, users were generally more accurate and as fast using HiveRel, and provided positive evaluations for the search experience.

Characterization of saturated graphs related to pairs of disjoint matchings

For a finite graph G, we study the maximum 2-edge colorable subgraph problem and a related ratio μ(G) ν(G), where ν(G) is the matching number of G, and μ(G) is the size of the largest matching in any pair (H,H′) of disjoint matchings maximizing |H|+|H′| (equivalently, forming a maximum 2-edge colorable subgraph). Previously, it was shown that 4 5≤μ(G) ν(G)≤1, and the class of graphs achieving 4 5 was completely characterized. In this paper, we first show that graph decompositions into paths and even cycles provide a new way to study these parameters. We then use this technique to characterize the graphs achieving μ(G) ν(G)=1 among all graphs that can be covered by a certain choice of a maximum matching and H, H′ as above.

Exact algorithms for budgeted prize-collecting covering subgraph problems

We introduce a class of budgeted prize-collecting covering subgraph problems. For an input graph with prizes on the vertices and costs on the edges, the aim of these problems is to find a connected subgraph such that the cost of its edges does not exceed a given budget and its collected prize is maximum. A vertex prize is collected when the vertex is visited. Moreover, a prize is also collected if the vertex is covered, where an unvisited vertex is covered by a visited one if the latter belongs to the former’s covering set. A capacity limit is imposed on the number of vertices that can be covered by the same visited vertex. Potential application areas include network design and intermodal transportation. We develop a branch-and-cut framework and a Benders decomposition for the exact solution of the problems in this class. We observe that the former algorithm results in shorter computational times on average, but also that the latter can outperform the former for specific instance settings. Finally, we validate our algorithmic frameworks for the cases where the subgraph is a cycle and a tree, and for these two cases we also identify novel symmetry-breaking inequalities.

Towards improving Christofides algorithm on fundamental classes by gluing convex combinations of tours

We present a new approach for gluing tours over certain tight, 3-edge cuts. Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles in special graph classes and in proving bounds for 2-edge-connected subgraph problem, but not much was known in this direction for gluing connected multigraphs. We apply this approach to the traveling salesman problem (TSP) in the case when the objective function of the subtour elimination relaxation is minimized by a \(\theta \)-cyclic point: \(x_e \in \{0,\theta , 1-\theta , 1\}\), where the support graph is subcubic and each vertex is incident to at least one edge with x-value 1. Such points are sufficient to resolve TSP in general. For these points, we construct a convex combination of tours in which we can reduce the usage of edges with x-value 1 from the \(\frac{3}{2}\) of Christofides algorithm to \(\frac{3}{2}-\frac{\theta }{10}\) while keeping the usage of edges with fractional x-value the same as Christofides algorithm. A direct consequence of this result is for the Uniform Cover Problem for TSP: In the case when the objective function of the subtour elimination relaxation is minimized by a \(\frac{2}{3}\)-uniform point: \(x_e \in \{0, \frac{2}{3}\}\), we give a \(\frac{17}{12}\)-approximation algorithm for TSP. For such points, this lands us halfway between the approximation ratios \(\frac{3}{2}\) of Christofides algorithm and \(\frac{4}{3}\) implied by the famous “four-thirds conjecture”.

Characterization of QUBO reformulations for the maximum k-colorable subgraph problem

Quantum devices can be used to solve constrained combinatorial optimization (COPT) problems thanks to the use of penalization methods to embed the COPT problem’s constraints in its objective to obtain a quadratic unconstrained binary optimization (QUBO) reformulation of the COPT. However, the particular way in which this penalization is carried out affects the value of the penalty parameters, as well as the number of additional binary variables that are needed to obtain the desired QUBO reformulation. In turn, these factors substantially affect the ability of quantum computers to efficiently solve these constrained COPT problems. This efficiency is the key toward the goal of using quantum computers to solve constrained COPT problems more efficiently than with classical computers. Along these lines, we consider an important constrained COPT problem, namely the maximum k-colorable subgraph (MkCS) problem, in which the aim is to find an induced k-colorable subgraph with maximum cardinality in a given graph. This problem arises in channel assignment in spectrum sharing networks, VLSI design, human genetic research, and cybersecurity. We derive two QUBO reformulations for the MkCS problem and fully characterize the range of the penalty parameters that can be used in the QUBO reformulations. Further, one of the QUBO reformulations of the MkCS problem is obtained without the need to introduce additional binary variables. To illustrate the benefits of obtaining and characterizing these QUBO reformulations, we benchmark different QUBO reformulations of the MkCS problem by performing numerical tests on D-Wave’s quantum annealing devices. These tests also illustrate the numerical power gained by using the latest D-Wave’s quantum annealing device.

On the Hardness and Approximation of the Densest $K$-Subgraph Problem in Parameterized Metric Graphs

On the Hardness and Approximation of the Densest $K$-Subgraph Problem in Parameterized Metric Graphs

On Computing the Hamiltonian Index of Graphs ⋆

The $r$-th iterated line graph $L^{r}(G)$ of a graph $G$ is defined by: (i) $L^{0}(G) = G$ and (ii) $L^{r}(G) = L(L^{(r- 1)}(G))$ for $r > 0$, where $L(G)$ denotes the line graph of $G$. The Hamiltonian Index $h(G)$ of $G$ is the smallest $r$ such that $L^{r}(G)$ has a Hamiltonian cycle. Checking if $h(G) = k$ is NP-hard for any fixed integer $k \geq 0$ even for subcubic graphs $G$. We study the parameterized complexity of this problem with the parameter treewidth, $tw(G)$, and show that we can find $h(G)$ in time $O*((1 + 2^{(\omega + 3)})^{tw(G)})$ where $\omega$ is the matrix multiplication exponent and the $O*$ notation hides polynomial factors in input size. The NP-hard Eulerian Steiner Subgraph problem takes as input a graph $G$ and a specified subset $K$ of terminal vertices of $G$ and asks if $G$ has an Eulerian (that is: connected, and with all vertices of even degree.) subgraph $H$ containing all the terminals. A second result (and a key ingredient of our algorithm for finding $h(G)$) in this work is an algorithm which solves Eulerian Steiner Subgraph in $O*((1 + 2^{(\omega + 3)})^{tw(G)})$ time.

The Balanced Connected Subgraph Problem for Geometric Intersection Graphs

The Balanced Connected Subgraph Problem for Geometric Intersection Graphs

On Directed Densest Subgraph Discovery

Given a directed graph G , the directed densest subgraph (DDS) problem refers to the finding of a subgraph from G , whose density is the highest among all the subgraphs of G . The DDS problem is fundamental to a wide range of applications, such as fraud detection, community mining, and graph compression. However, existing DDS solutions suffer from efficiency and scalability problems: on a 3,000-edge graph, it takes three days for one of the best exact algorithms to complete. In this article, we develop an efficient and scalable DDS solution. We introduce the notion of [ x , y ]-core, which is a dense subgraph for G , and show that the densest subgraph can be accurately located through the [ x , y ]-core with theoretical guarantees. Based on the [ x , y ]-core, we develop exact and approximation algorithms. We further study the problems of maintaining the DDS over dynamic directed graphs and finding the weighted DDS on weighted directed graphs, and we develop efficient non-trivial algorithms to solve these two problems by extending our DDS algorithms. We have performed an extensive evaluation of our approaches on 15 real large datasets. The results show that our proposed solutions are up to six orders of magnitude faster than the state-of-the-art.

Finding densest [formula omitted]-connected subgraphs

Dense subgraph discovery is an important graph-mining primitive with a variety of real-world applications. One of the most well-studied optimization problems for dense subgraph discovery is the densest subgraph problem, where given an edge-weighted undirected graph G=(V,E,w), we are asked to find S⊆V that maximizes the density d(S), i.e., half the weighted average degree of the induced subgraph G[S]. This problem can be solved exactly in polynomial time and well-approximately in almost linear time. However, a densest subgraph has a structural drawback, namely, the subgraph may not be robust to vertex/edge failure. Indeed, a densest subgraph may not be well-connected, which implies that the subgraph may be disconnected by removing only a few vertices/edges within it. In this paper, we provide an algorithmic framework to find a dense subgraph that is well-connected in terms of vertex/edge connectivity. Specifically, we introduce the following problems: given a graph G=(V,E,w) and a positive integer/real k, we are asked to find S⊆V that maximizes the density d(S) under the constraint that G[S] is k-vertex/edge-connected. For both problems, we propose polynomial-time (bicriteria and ordinary) approximation algorithms, using classic Mader’s theorem in graph theory and its extensions.

A matheuristic approach for the maximum balanced subgraph of a signed graph

A graph G = (V, E) with its edges labeled in the set {+,-} is called a signed graph. It is balanced if its set of vertices V can be partitioned into two sets V1 and V2, such that all positive edges connect nodes within V1 or V2, and all negative edges connect nodes between V1 and V2. The maximum balanced subgraph problem (MBSP) for a signed graph is the problem of finding a balanced subgraph with the maximum number of vertices. In this work, we present the first polynomial integer linear programming formulation for this problem and a matheuristic to obtain good quality solutions in a short time. The results obtained for different sets of instances show the effectiveness of the matheuristic, optimally solving several instances and finding better results than the exact method in a much shorter computational time.

Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs

We consider a well-studied generalization of the maximum clique problem which is defined as follows. Given a graph G on n vertices and a fixed parameter $$d\ge 1$$ , in the maximum diameter-bounded subgraph problem (MaxDBS for short) the goal is to find a (vertex) maximum subgraph of G of diameter at most d. For $$d=1$$ , this problem is equivalent to the maximum clique problem and thus it is NP-hard to approximate it within a factor $$n^{1-\epsilon }$$ , for any $$\epsilon >0$$ . Moreover, it is known that, for any $$d\ge 2$$ , it is NP-hard to approximate MaxDBS within a factor $$n^{1/2-\epsilon }$$ , for any $$\epsilon >0$$ . In this paper we focus on MaxDBS for the class of unit disk graphs. We provide a polynomial-time constant-factor approximation algorithm for the problem. The approximation ratio of our algorithm does not depend on the diameter d. Even though the algorithm itself is simple, its analysis is rather involved. We combine tools from the theory of hypergraphs with bounded VC-dimension, k-quasi planar graphs, fractional Helly theorems, and several geometric properties of unit disk graphs.

The Maximum k-Colorable Subgraph Problem and Related Problems

The maximum k-colorable subgraph (MkCS) problem is to find an induced k-colorable subgraph with maximum cardinality in a given graph. This paper is an in-depth analysis of the MkCS problem that considers various semidefinite programming relaxations, including their theoretical and numerical comparisons. To simplify these relaxations, we exploit the symmetry arising from permuting the colors, as well as the symmetry of the given graphs when applicable. We also show how to exploit invariance under permutations of the subsets for other partition problems and how to use the MkCS problem to derive bounds on the chromatic number of a graph. Our numerical results verify that the proposed relaxations provide strong bounds for the MkCS problem and that those outperform existing bounds for most of the test instances. Summary of Contribution: The maximum k-colorable subgraph (MkCS) problem is to find an induced k-colorable subgraph with maximum cardinality in a given graph. The MkCS problem has a number of applications, such as channel assignment in spectrum sharing networks (e.g., Wi-Fi or cellular), very-large-scale integration design, human genetic research, and so on. The MkCS problem is also related to several other optimization problems, including the graph partition problem and the max-k-cut problem. The two mentioned problems have applications in parallel computing, network partitioning, floor planning, and so on. This paper is an in-depth analysis of the MkCS problem that considers various semidefinite programming relaxations, including their theoretical and numerical comparisons. Further, our analysis relates the MkCS results with the stable set and the chromatic number problems. We provide extended numerical results that verify that the proposed bounding approaches provide strong bounds for the MkCS problem and that those outperform existing bounds for most of the test instances. Moreover, our lower bounds on the chromatic number of a graph are competitive with existing bounds in the literature.

Efficient constructions of convex combinations for 2-edge-connected subgraphs on fundamental classes

We present coloring-based algorithms for tree augmentation and use them to construct convex combinations of 2-edge-connected subgraphs. This classic tool has been applied previously to the problem, but our algorithms illustrate its flexibility, which – in coordination with the choice of spanning tree – can be used to obtain various properties (e.g., 2-vertex connectivity) that are useful in our applications.We use these coloring algorithms to design approximation algorithms for the 2-edge-connected multigraph problem (2ECM) and the 2-edge-connected spanning subgraph problem (2ECS) on two well-studied types of LP solutions. The first type of points, half-integer square points, belong to a class of fundamental extreme points, which exhibit the same integrality gap as the general case. For half-integer square points, the integrality gap for 2ECM is known to be between 65 and 43. We improve the upper bound to 97. The second type of points we study are uniform points whose support is a 3-edge-connected graph and each entry is 23. Although the best-known upper bound on the integrality gap of 2ECS for these points is less than 43, previous results do not yield an efficient algorithm. We give the first approximation algorithm for 2ECS with ratio below 43 for this class of points.

A 4 + ϵ approximation for k-connected subgraphs

We obtain approximation ratio 4+2ℓ≈4+4lg⁡klg⁡n−lg⁡k for the (undirected) k-Connected Subgraph problem, where ℓ=⌊lg⁡n−lg⁡k+12lg⁡k+1⌋ is the largest integer such that 2ℓ−1k2ℓ+1≤n. For large values of n this improves the ratio 6 of Cheriyan and Végh [4] when n≥k3 (the case ℓ=1). Our result implies an fpt-approximation ratio 4+ϵ that matches (up to the “+ϵ” term) the best known ratio 4 for k=6,7 for both the general and the easier augmentation versions of the problem. Similar results are shown for the problem of covering an arbitrary symmetric crossing supermodular biset function.

Complexity and inapproximability results for balanced connected subgraph problem

This work is devoted to the study of the Balanced Connected Subgraph Problem (BCS) from a complexity, inapproximability and approximation point of view. The input is a graph G=(V,E), with each vertex having been colored, “red” or “blue”; the goal is to find a maximum connected subgraph G′=(V′,E′) from G that is color-balanced (having exactly |V′|/2 red vertices and |V′|/2 blue vertices). This problem is known to be NP-complete in general but polynomial in paths and trees. We propose a polynomial-time algorithm for block graph. We propose some complexity results for bounded-degree or bounded-diameter graphs, and also for bipartite graphs. We also propose inapproximability results for some graph classes, including chordal, planar, or subcubic graphs.

Minimum 2-vertex strongly biconnected spanning directed subgraph problem

A directed graph $G=(V,E)$ is strongly biconnected if $G$ is strongly connected and its underlying graph is biconnected. A strongly biconnected directed graph $G=(V,E)$ is called $2$-vertex-strongly biconnected if $|V|\geq 3$ and the induced subgraph on $V\setminus\left\lbrace w\right\rbrace $ is strongly biconnected for every vertex $w\in V$. In this paper we study the following problem. Given a $2$-vertex-strongly biconnected directed graph $G=(V,E)$, compute an edge subset $E^{2sb} \subseteq E$ of minimum size such that the subgraph $(V,E^{2sb})$ is $2$-vertex-strongly biconnected.

Efficient Directed Densest Subgraph Discovery

Given a directed graph G, the directed densest subgraph (DDS) problem refers to the finding of a subgraph from G, whose density is the highest among all the subgraphs of G. The DDS problem is fundamental to a wide range of applications, such as fraud detection, community mining, and graph compression. However, existing DDS solutions suffer from efficiency and scalability problems: on a threethousand- edge graph, it takes three days for one of the best exact algorithms to complete. In this paper, we develop an efficient and scalable DDS solution. We introduce the notion of [x, y]-core, which is a dense subgraph for G, and show that the densest subgraph can be accurately located through the [x, y]-core with theoretical guarantees. Based on the [x, y]-core, we develop both exact and approximation algorithms. We have performed an extensive evaluation of our approaches on eight real large datasets. The results show that our proposed solutions are up to six orders of magnitude faster than the state-of-the-art.