In-Place Algorithms for Computing a Largest Clique in Geometric Intersection Graphs

Maximum weight independent sets and cliques in intersection graphs of filaments

In this paper, we give a comprehensive overview of the development of clique algorithms and their use for drug design based on the search for cliques in protein product graphs. The maximum clique problem is a computational problem of finding largest subsets of vertices in a graph that are all pairwise adjacent. A related problem is the maximum weight clique problem and the highest weight k-clique problem, which both extend the algorithm to weighted graphs. The review covers our developed algorithms, starting with our improved branch-and-bound algorithm for finding maximum cliques in undirected graphs from 2007 up to the recent developments of algorithms for weighted graphs in 2024. We show the application of these algorithms to early stages of drug discovery, in particular to protein binding site detection based on protein similarity search in large protein databases and to protein-ligand molecular docking.

The intent of maximum clique problem is to discover a clique of largest size or a complete subgraph in a graph; for a graph clique is a subgraph which is complete i.e., every vertex of that subgraph are adjacent to each other. It belongs to the category of the NP-hard problems, for which many solutions exist and many more are continuously being proposed. We are recapitulating the core idea and search strategies of few of the branch and bound based recently proposed algorithms. Intent of this work is to give a comparative study of newly proposed solutions to maximum clique problem which will be helpful to decide which algorithm is efficient for a given graph. It also instigates to design more efficient solutions for maximum clique problem.

The clique operator on graphs with few [formula omitted]'s

In the 90’s Clark, Colbourn and Johnson wrote a seminal paper, where they proved that maximum clique can be solved in polynomial time in unit disk graphs. Since then, the complexity of maximum clique in intersection graphs of (unit) d-dimensional balls has been investigated. For ball graphs, the problem is NP-hard, as shown by Bonamy et al. (FOCS ’18). They also gave an efficient polynomial time approximation scheme (EPTAS) for disk graphs, however the complexity of maximum clique in this setting remains unknown. In this paper, we show the existence of a polynomial time algorithm for solving maximum clique in a geometric superclass of unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS for intersection graphs of convex pseudo-disks.

Whitney triangulations, local girth and iterated clique graphs

The complexity of Dominating Set in geometric intersection graphs

Incremental maximal clique enumeration (IMCE), which maintains maximal cliques in dynamic graphs, is a fundamental problem in graph analysis. A maximal clique has a solid descriptive power of dense structures in graphs. Real-world graph data is often large and dynamic. Studies on IMCE face significant challenges in the efficiency of incremental batch computation and hybrid edge changes. Moreover, with growing graph sizes, new requirements occur on indexing global maximal cliques and obtaining maximal cliques under specific vertex scope constraints. This work presents a new data structure SOMEi to maintain intermediate maximal cliques during construction. SOMEi serves as a space-efficient index to retrieve scope-constrained maximal cliques on the fly. Based on SOMEi, we design a procedure-oriented IMCE algorithm to deal with hybrid edge changes within a unified algorithm framework. In particular, the algorithm is able to process a large batch of edge changes and significantly improve the average processing time of a single edge change through an efficient pruning strategy. Experimental results on real and synthetic graph data demonstrate that the proposed algorithm outperforms all the baselines and achieves good efficiency through pruning.

Shah, Shin and Tetali [25] initiated the study of queuing systems on the medium access channel with arbitrary interference graph. The graph models dependencies between the channel members -- any two connected by an edge are dependent. This problem could be also re-stated as a query system with dependent elements or local scheduling of dependent packets/jobs. In short, if two dependent units, also called stations, want to transmit (or be in the same query, or do jobs in parallel), there is a conflict and none of them succeeds. The problem is particularly challenging, if the units need to make their decisions locally, without any coordination. Prior the paper by Shah, Shin and Tetali [FOCS 2011], the main focus was on the clique graph -- even in this simple topology, many problems related to stability remain open. While the solution by Shah, Shin and Tetali [FOCS 2011] is semi-local, as nodes make use of information about interfering neighbors in the graph, we provide the first purely local stable algorithms. In particular, in our algorithms stations make their decision whether to transmit or not based only on looking at their local queues. Based only on this feature, and without using a priori knowledge of topology, we design an algorithm that allows all stations for implicit transferring information to and from all other stations. We use it as a tool for developing universally stable algorithms for the problem of queuing messages -- i.e., guaranteeing bounded queues when, on average, less than one independent set is injected per round. The first one is stable in adversarial (worst-case) sense, the second one -- in stochastic sense (average-case). We also prove optimality of our algorithm in adversarial sense: no algorithm can be stable against injections with rate ρ=1 (when, on average, one independent set is injected per round). Moreover, for the class of non-adaptive protocols, we show that it is not possible to achieve stability for any injection rate Ω(1/log m), where m is the size of the largest clique in the interference graph. We extend the result by showing that such protocols are not stable on some interference graphs with no cliques of size at least 3 even for injection rates Ω(√[3](log n)/n). This result separates the model with general interference graphs from the classical model of a multiple access channel (in which the interference graph is a clique), in which there exist non-adaptive algorithms stable for adversarial injection rates O(1/log2 n).

We study the Open image in new window (shortly, Open image in new window ) problem on geometric intersection graphs such as interval, circular-arc, permutation, unit-disk, outer-string graphs, etc. Given a Open image in new window graph \(G=(V,E)\), where each vertex in V is colored with either “ Open image in new window ” or “ Open image in new window ”, the BCS problem seeks a maximum cardinality induced connected subgraph H of G such that H is Open image in new window , i.e., H contains an equal number of red and blue vertices. We study the computational complexity landscape of the BCS problem while considering geometric intersection graphs. On one hand, we prove that the BCS problem is NP-hard on the unit disk, outer-string, complete grid, and unit square graphs. On the other hand, we design polynomial-time algorithms for the BCS problem on interval, circular-arc and permutation graphs. In particular, we give algorithms for the Open image in new window problem on both interval and circular-arc graphs, and those algorithms are used as subroutines for solving the BCS problem on the same classes of graphs. Finally, we present a FPT algorithm for the BCS problem on general graphs.

Due to the dynamic topology and non infrastructure, network participants cooperate with their neighbors to route packets. The lack of centralized services allows mobile ad hoc networks to be easily and swiftly deployed, but make it difficult to check others' identities on the other hand. Cryptographic tools have been introduced to secure group communications, such as Private and Public Key Infrastructure. The autonomous and distributed nature of mobile ad hoc network demands a decentralized authentication service, where Public Key Infrastructure is considered a better solution. Public Key Infrastructure can ensure both confidentiality and authenticity, but it is impractical to provide an online trusted third party as Certificate Authority (CA) for mobile ad hoc network. In this paper, we proposed a new key management protocol which utilizes certificate graphs and distributed Certificate Authorities. Certificate graph maintained by each user represents the trust among his neighbors, then the maximum clique of certificate graph is selected to be CAs. Based on the assumption that initial certificate graph building is secure [11], good users have more friends while bad ones have less, thus a reliable group can be constructed. The most trustful subset of these good users - the maximum clique - is elected as the governor of this group, which takes the responsibility of certificate authentication.

Finding cliques in graphs is a classical problem which is in general NP-hard and parameterized intractable. However, in typical applications like social networks or protein-protein interaction networks, the considered graphs are scale-free, i.e., their degree sequence follows a power law. Their specific structure can be algorithmically exploited and makes it possible to solve clique much more efficiently. We prove that on inhomogeneous random graphs with n nodes and power law exponent γ, cliques of size k can be found in time \(\mathcal{O}(n^2)\) for γ ≥ 3 and in time \(\mathcal{O}(n\, \exp(k^4))\) for 2 < γ < 3.

If G is a graph, its clique graph K(G) is the intersection graph of all its (maximal) cliques. The complex G<FONT FACE=Symbol></FONT> of a graph G is the simplicial complex whose simplexes are the vertex sets of the complete subgraphs of G. Here we study a sufficient condition for G<FONT FACE=Symbol></FONT> and K(G)<FONT FACE=Symbol></FONT> to be homotopic. Applying this result to Whitney triangulations of surfaces, we construct an infinite family of examples which solve in the affirmative Prisner's open problem 1 in Graph Dynamics (Longman, Harlow, 1995): Are there finite connected graphs G that are periodic under K and where the second modulo 2 Betti number is greater than 0?

On cliques and bicliques

A graph-based approach to finding near-optimal output phase assignments for PLA minimization is presented. A distinctive feature of the approach is that it exploits the necessary and sufficient conditions to reduce the number of product terms needed for PLA implementation and permits the use of existing graph algorithms to solve the PLA output phase optimization problem. The work is based on the transformation of a PLA into a graph whose vertices are the set of conditions required to reduce each product term of the PLA, and whose edges represent the relations between these conditions. Cliques (completely connected subgraphs) in the graph correspond to the output phase assignments required to reduce the product terms represented by the vertices in the cliques. The optimal PLA output phase assignment problem is then formulated as the well-studied problem in graph theory: finding maximum cliques in graphs. Using an existing algorithm for locating cliques in graphs, a modified greedy algorithm is proposed to compute output phase assignments for logic functions in polynomial time. Experimental results using a number of benchmark functions show that the graph-based approach can achieve optimal or near-optimal output phase assignment for PLA minimization and can lead to PLA's with fewer product terms than achieved using existing approaches.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>