Clique Problem Research Articles

Solving maximum clique problem using chemical reaction optimization

Solving maximum clique problem using chemical reaction optimization

Robust quantum circuit for clique problem with intermediate qudits

Clique problem has a wide range of applications due to its pattern matching ability. There are various formulations of clique problem like k-clique problem, maximum clique problem, etc. The k-Clique problem determines whether an arbitrary network has a clique or not whereas maximum clique problem finds the largest clique in a graph. It is already exhibited in the literature that the k-clique or maximum clique problem (NP-problem) can be solved asymptotically faster by using quantum algorithms compared to conventional computing. Quantum computing with higher dimensions is gaining popularity due to its large storage capacity and computation power. In this article, we have shown an improved quantum circuit implementation for the k-clique problem and maximum clique problem (MCP) with the help of higher-dimensional intermediate temporary qudits for the first time to the best of our knowledge. The cost of the state-of-the-art quantum circuit for the k-clique problem is colossal due to a huge number of n-qubit Toffoli gates. We have exhibited an improved cost and depth over the circuit by applying a generalized n-qubit Toffoli gate decomposition with intermediate ququarts (4-dimensional qudits).

Robust Loop Closure Selection Based on Inter-Robot and Intra-Robot Consistency for Multi-Robot Map Fusion

In multi-robot simultaneous localization and mapping (SLAM) systems, the system must create a consistent global map with multiple local maps and loop closures between robot poses. However, false-positive loop closures caused by perceptual aliasing can severely distort the global map, especially in GNSS-denied areas, where a good prior of relative poses between robots is unavailable. In addition, the performance of the consistency metric in existing map fusion methods relies on accurate odometry from each robot. However, in practice, cumulative noise is inevitably present in robot trajectories, which leads to poor map fusion with existing methods. Thus, in this paper, we propose a robust consistency-based inter-robot and intra-robot loop closure selection algorithm for map fusion. We consider both pairwise-loop consistency and loop-odometry consistency to improve robustness against false-positive loop closures and accumulative noise in the odometry. Specifically, we select a reliable inter-robot loop closure measurement with a consistency-based strategy to provide an initial prior of relative pose between two robot trajectories and update the pose variables of the robot trajectories. The loop closure selection problem is formulated as a maximum edge weight clique problem in graph theory. A performance evaluation of the proposed method was conducted on the ManhattanOlson3500, modified CSAIL and Bicocca datasets, and the experimental results demonstrate that the proposed method outperforms the pairwise consistency measurement set maximization method (PCM) under severe accumulative noise and can be integrated with M-estimation methods.

Batch Scheduling with Time Restriction and Clique Search

We consider a graph whose vertices are legally colored using k colors and ask if the graph contains a k-clique. As it turns out this very special type of k-clique problem is in an intimate connection with constructing schedules. The practicality this clique search based construction of schedules is checked by carrying out numerical experiments.

Solving larger maximum clique problems using parallel quantum annealing

Quantum annealing has the potential to find low energy solutions of NP-hard problems that can be expressed as quadratic unconstrained binary optimization problems. However, the hardware of the quantum annealer manufactured by D-Wave Systems, which we consider in this work, is sparsely connected and moderately sized (on the order of thousands of qubits), thus necessitating a minor-embedding of a logical problem onto the physical qubit hardware. The combination of relatively small hardware sizes and the necessity of a minor-embedding can mean that solving large optimization problems is not possible on current quantum annealers. In this research, we show that a hybrid approach combining parallel quantum annealing with graph decomposition allows one to solve larger optimization problem accurately. We apply the approach to the Maximum Clique problem on graphs with up to 120 nodes and 6395 edges.

Logical qubit implementation for quantum annealing: augmented Lagrangian approach

Solving optimization problems on quantum annealers (QA) usually requires each variable of the problem to be represented by a connected set of qubits called a logical qubit or a chain. Chain weights, in the form of ferromagnetic coupling between the chain qubits, are applied so that the physical qubits in a chain favor taking the same value in low energy samples. Assigning a good chain-strength value is crucial for the ability of QA to solve hard problems, but there are no general methods for computing such a value and, even if an optimal value is found, it may still not be suitable by being too large for accurate annealing results. In this paper, we propose an optimization-based approach for producing suitable logical qubits representations that results in smaller chain weights and show that the resulting optimization problem can be successfully solved using the augmented Lagrangian method. Experiments on the D-Wave Advantage system and the maximum clique problem on random graphs show that our approach outperforms both the default D-Wave method for chain-strength assignment as well as the quadratic penalty method.

Temporal interval cliques and independent sets

Temporal graphs have been recently introduced to model changes in a given network that occur throughout a fixed period of time. The Temporal Δ Clique problem, which generalizes the well known Clique problem to temporal graphs, has been studied in the context of finding nodes of interest in dynamic networks [TCS '16]. We introduce the Temporal Δ Independent Set problem, a temporal generalization of Independent Set. This problem is e.g. motivated in the context of finding conflict-free schedules for maximum subsets of tasks, that have certain (time-varying) constraints within a given time period. We are specifically interested in the case where each task needs to be performed in a certain time-interval on each day and two tasks are in conflict on a certain day if their time-intervals on that day overlap. This leads us to consider both problems on the restricted class of temporal unit interval graphs, i.e., temporal graphs where each layer is a unit interval graph.We present several hardness results as well as positive results. On the algorithmic side, we provide constant-factor approximation algorithms for instances of both problems where τ, the total number of time steps (layers) of the temporal graph, and Δ, a parameter that allows us to model conflict tolerance, are constants. We develop an exact FPT algorithm for Temporal Δ Clique with respect to parameter τ+k. Finally, we use the notion of order preservation for temporal unit interval graphs that, informally, requires the intervals of every layer to obey a common ordering. For both problems, we provide an FPT algorithm parameterized by the size of minimum vertex deletion set to order preservation.

Minimizing the Average Packet Access Time of the Application Layer for Buffered Instantly Decodable Network Coding

In B-IDNC (buffered instantly decodable network coding), each receiver can cache the non-instantly decodable network coded packets (NIDPs) which contain two wanted packets of the network layer for subsequent network decoding, so that more packets can be recovered at the network layer than the traditional instantly decodable network coding (IDNC). By employing B-IDNC, we consider a radio access network wherein a base station (BS) is required to broadcast a block of packets to a set of receivers. After completing network decoding at the network layer, each receiver can deliver its recovered packets from the network layer to the application layer in order. We consider minimizing the average packet access time of the application layer for B-IDNC. For the optimization problem is intractable, we approximate it to reduce the sum minimum access delay of the application layer across all receivers. The approximate problem is shown to be equivalent to a maximum weight encoding clique problem over the B-IDNC graph. We propose a simple heuristic algorithm based on greedy maximum weight vertex search to solve the approximate problem. Simulation results verify the effectiveness of our proposed algorithm as compared with the existing techniques.

A Bio-Inspired Technique for the Maximum Weighted Clique Problem

A Bio-Inspired Technique for the Maximum Weighted Clique Problem

On the binary and Boolean rank of regular matrices

A 0,1 matrix is said to be regular if all of its rows and columns have the same number of ones. We prove that for infinitely many integers k, there exists a square regular 0,1 matrix with binary rank k, such that the Boolean rank of its complement is kΩ˜(log⁡k). This settles, in a strong form, a question of Pullman (1988) [27] and a conjecture of Hefner et al. (1990) [18]. The result can be viewed as a regular analogue of a recent result of Balodis et al. (2021) [2], motivated by the clique vs. independent set problem in communication complexity and by the (disproved) Alon-Saks-Seymour conjecture in graph theory. As an application of the produced regular matrices, we obtain regular counterexamples to the Alon-Saks-Seymour conjecture and prove that for infinitely many integers k, there exists a regular graph with biclique partition number k and chromatic number kΩ˜(log⁡k).

Average Packet Decoding Delay Minimization for Rate-Aware Buffered Instantly Decodable Network Codes

Buffered instantly decodable network coding (B-IDNC) can cache the non-instantly decodable network packets (NIDPs) containing two wanted packets for subsequent network decoding. This letter proposes a cross-layer scheme named as rate-aware B-IDNC (RA-B-IDNC) by jointly considering the B-IDNC decision at the network layer and transmission rate adaption at the physical layer to minimize the average packet decoding delay. Due to the intractability of the considered problem, we approximate it as maximizing the recovery rate which is defined as the ratio of the total number of instantly decoded packets to the transmission time. We design two graphs named as RA-B-IDNC encoding graph which reflects the encoding relations among the vertices of the graph, and RA-B-IDNC decoding graph which indicates the decoding relations among the wanted packets of each receiving node by exploiting its buffered NIDPs. Using these graphs, the approximated problem can be formulated as a maximum weight encoding clique problem. Simulation results demonstrate that RA-B-IDNC can reduce the average packet decoding delay compared with the existing algorithms in the literature.

An efficient algorithm for maximum cliques problem on IoT devices

This work describes how the maximum clique problem (MCP) algorithm can be performed on microcontrollers in a dynamic environment. In practice, many problems can be formalised using MCP and graphs where our problem is considered in the context of a dynamic environment. MCP is, however, a tricky problem NP-Complete for which suitable solutions must be designed for microcontrollers. Microcontrollers are built for specific purposes and optimised to meet different constraints, such as timing, nested recursion depth limitation, or no recursion at all due to recursion stack limitation, power, and RAM limitation. On another side, graph representation and all the algorithms mentioned in the literature to solve the MCP problem, which is recursive, consumes memory and is designed specifically for computers rather than a microcontroller.

Algorithms for the clique problem with multiple-choice constraints under a series–parallel dependency graph

The clique problem with multiple-choice constraints (CPMC), i.e. the problem of finding a k -clique in a k -partite graph with known partition, occurs as a substructure in many real-world applications, in particular scheduling and railway timetabling. Although CPMC is NP-complete in general, it is known to be solvable in polynomial time when the so-called dependency graph of G is a forest. In this article, we focus on the special case CPMCSP, where the dependency graph of G is series–parallel. We give a polynomial-time algorithm for CPMCSP using dynamic programming. Further, we provide some facet-inducing inequalities of the CPMCSP polytope, mainly using properties of the stable set polytope of the complement graph of G . Among these, we give a separation algorithm for the so-called embedded odd-clique-cycle inequalities using dynamic programming. If the number of vertices per subset of the k -partition is bounded, then its runtime is polynomial in the size of the dependency graph.

Initial State Encoding via Reverse Quantum Annealing and H-Gain Features.

Quantum annealing is a specialized type of quantum computation that aims to use quantum fluctuations in order to obtain global minimum solutions of combinatorial optimization problems. Programmable D-Wave quantum annealers are available as cloud computing resources, which allow users low-level access to quantum annealing control features. In this article, we are interested in improving the quality of the solutions returned by a quantum annealer by encoding an initial state into the annealing process. We explore twoD-Wave features that allow one toencode such an initialstate: the reverse annealing (RA) and theh-gain(HG)features.RAaimstorefineaknownsolutionfollowinganannealpathstartingwithaclassical state representing a good solution, going backward to a point where a transverse field is present, and then finishing the annealing process with a forward anneal. The HG feature allows one to put a time-dependent weighting scheme on linear () biases of the Hamiltonian, and we demonstrate that this feature likewise can be used to bias the annealing to start from an initial state. We also consider a hybrid method consisting of a backward phase resembling RA and a forward phase using the HG initial state encoding. Importantly, we investigate the idea of iteratively applying RA and HG to a problem, with the goal of monotonically improving on an initial state that is not optimal. The HG encoding technique is evaluated on a variety of input problems including the edge-weighted maximum cut problem and the vertex-weighted maximum clique problem, demonstrating that the HG technique is a viable alternative to RA for some problems. We also investigate how the iterative procedures perform for both RA and HG initial state encodings on random whole-chip spin glasses with the native hardware connectivity of the D-Wave Chimera and Pegasus chips.

On the absolute and relative oriented clique problems’ time complexity

Let ⃗G = (V, A) be an oriented graph. An oriented k-coloring of ⃗G is a partition of V into k color classes, such that there is no pair of adjacent vertices belonging to the same class and all the arcs between a pair of color classes have the same orientation. The smallest k such that ⃗G admits an oriented k-coloring is the oriented chromatic number Xo(⃗G) = k of ⃗G. In an oriented coloring of ⃗G every pair of vertices with oriented distance at most 2 in ⃗G have different colors. In 2004, Klostermeyer and MacGillivray defined the concept of an “analogue of clique” for oriented coloring in which a subgraph ⃗H of ⃗G is an oriented clique if every pair of vertices of ⃗H is in an oriented distance of at most 2 in ⃗H. The authors defined the absolute oriented clique number of ⃗G as the number of vertices |V(H)| = ωao(⃗G) of the largest oriented clique ⃗H of ⃗G and satisfies that ωao(⃗G) ≤ Xo(⃗G). Ever since, for almost 20 years, the time complexity status of this parameter remained unknown. The relative oriented clique number ωao(⃗G) of an oriented graph ⃗G is the size of the largest set of vertices R, such that every pair of vertices of R is at a maximum oriented distance of 2 in R. For every oriented graph ⃗G, ωao(⃗G) ≤ ωro(⃗G) ≤ Xo(⃗G). In this paper we classify Absolute Oriented Clique - the Klostermeyer and Mac Gillivray's decision problem - proving that given an oriented graph ⃗G and a positive integer k it is NP-complete to decide whether ωao(⃗G) ≥ k. We prove that for all ε > 0, there is no polynomial-time approximation for Relative Oriented Clique and for Absolute Oriented Clique within a factor of n1_ε, unless P = NP. Finally, we prove that Relative Oriented Clique is W[1]-complete and that Absolute Oriented Clique belongs to W[2] and is W[1]-hard.

On the complexity of redescription mining

Redescription mining is an important data mining task with the main goals of discovering subsets of instances that can be characterized in multiple ways and constructing the appropriate characterizations. Such characterizations, presented in a form of tuples of logical formulae, are understandable to the domain experts and offer unique insights in different interactions between relevant, presented attributes. In this work, we analyse variants of this task where the input data consists of one or more views – disjoint sets of attributes. The goal of this work is to formally define decision variants of redescription mining tasks with several sets of constraints and to provide corresponding proofs that the decision variants of these tasks are NP-complete. Proofs are realized by reductions from well known decision variants of CLIQUE and Set coverNP-complete problems. We also show that the enumeration variant of the task is NP-hard.

SMC-BRB: an algorithm for the maximum clique problem over large and sparse graphs with the upper bound via $$s^+$$-index

SMC-BRB: an algorithm for the maximum clique problem over large and sparse graphs with the upper bound via $$s^+$$-index

Data-driven optimal sensor placement for high-dimensional system using annealing machine

We propose a novel method for solving optimal sensor placement problem for high-dimensional system using an annealing machine. The sensor points are calculated as a maximum clique problem of the graph, the edge weight of which is determined by the proper orthogonal decomposition mode obtained from data based on the fact that a high-dimensional system usually has a low-dimensional representation. Since the maximum clique problem is equivalent to the independent set problem of the complement graph, the independent set problem is solved using Fujitsu Digital Annealer. In contrast to existing greedy methods, which select the optimal point at each step and never reconsider the point selected previously, the proposed method is superior because it is able to find the optimal set of points. As a demonstration of high dimensional system, the pressure distribution measured by the pressure-sensitive paint method, which is an optical flow diagnose method, is reconstructed from the pressure data at the calculated sensor points. The root mean square errors (RMSEs) between the pressures measured by pressure transducers and the pressures reconstructed from the proposed method, an existing greedy method, and random selection method are compared. The similar RMSE is achieved by the proposed method using approximately 1/5 number of sensor points calculated by the existing method. This method is of great importance as a novel approach for optimal sensor placement problem and a new engineering application of an annealing machine.

A stepped tabu search method for the clique partitioning problem

Given an undirected graph, a clique is a subset of vertices in which the induced subgraph is complete; that is, all pairs of vertices of this subset are adjacent. Clique problems in graphs are very important due to their numerous applications. One of these problems is the clique partitioning problem (CPP), which consists of dividing the set of vertices of a graph into the smallest number of cliques possible. The CPP is an NP-hard problem with many application fields (timetabling, manufacturing, scheduling, telecommunications, etc.). Despite its great applicability, few recent studies have focused on proposing specific resolution methods for the CPP. This article presents a resolution method that combines multistart strategies with tabu search. The most novel characteristic of our method is that it allows unfeasible solutions to be visited, which facilitates exploration of the solution space. The computational tests show that our method performs better than previous methods proposed for this problem. In fact, our method strictly improves the results of these methods in most of the instances considered while requiring less computation time.

A new exact algorithm for the maximum-weight clique problem based on a heuristic vertex-coloring and a backtrack search

In this paper we present an exact algorithm for the maximum-weight clique problem on arbitrary undirected graphs. The algorithm based on a fact that vertices from the same independent set couldn’t be included into the same maximum clique. Those independent sets are obtained from a heuristic vertex coloring where each of them is a color class. Color classes and a backtrack search are used for pruning branches of the maximum-weight clique search tree. Those pruning strategies together result in a very effective algorithm for the maximum-weight clique finding. Computational experiments with random graphs show that the new algorithm works faster than earlier published algorithms; especially on dense graphs.