Triangle Packing Problem Research Articles

A deterministic approximation algorithm for metric triangle packing

Given an edge-weighted metric complete graph with n vertices, the maximum weight metric triangle packing problem is to find a set of n/3 vertex-disjoint triangles with the total weight of all triangles in the packing maximized. Several simple methods can lead to a 2/3-approximation ratio. However, this barrier is not easy to break. Chen et al. proposed a randomized approximation algorithm with an expected ratio of (0.66768−ε) for any constant ε>0. In this paper, we improve the approximation ratio to (0.66835−ε). Furthermore, we can derandomize our algorithm.

A polynomial time algorithm for the triangle packing problem on interval graphs

The triangle packing problem (TPP) is to find the maximum number of pairwise vertex disjoint triangles in a given graph. The TPP is NP-complete in a general graph and even so when a given graph is restricted to a chordal graph. On the other hand, the TPP can be solved in polynomial time for unit interval graphs. For this reason, it is a well-known open problem to prove whether there exists a polynomial time algorithm solving the TPP on interval graphs. In this paper, we give an answer to the problem.

An improved kernel for planar vertex-disjoint triangle packing

Triangle Packing is an important NP-hard problem that has been well studied in exact and parameterized complexity. In this paper, we study kernelization of the Planar Vertex-Disjoint Triangle Packing problem, which is to check whether a given planar graph has k vertex-disjoint triangles. We prove a kernel of 141k vertices, improving the previous bound of 732k.

A (3 + ϵ)k-vertex kernel for edge-disjoint triangle packing

In the Edge-Disjoint Triangle Packing problem (Etp), we are given a graph G and an integer k, and asked to check whether G contains at least k edge-disjoint triangles. This problem is NP-hard and has been well studied in the parameterized complexity. A vertex kernel of size 4k was proved many years ago. Recently, the size of the vertex kernel was improved to 3.5k. In this paper, we further improve the kernel size. We show that for any fixed ϵ>0, there is a polynomial-time algorithm that can reduce the input instance to an equivalent instance of at most (3+ϵ)k vertices.

Branch-and-bound algorithm for the maximum triangle packing problem

This work addresses the problem of finding the maximum number of unweighted vertex-disjoint triangles in an undirected graph G. It is a challenging NP-hard combinatorial problem and it is well-known to be APX-hard. A branch-and-bound algorithm which uses a lower bound based on neighborhood degree is presented. A naive upper bound is proposed as well as another one based on a surrogate relaxation of the related integer linear program which is analogous to a multidimensional knapsack problem. Further, a Greedy Search algorithm and a genetic algorithm are described to improve the lower bound. A computational comparison of lower bounds, branch-and-bound algorithm and CPLEX solver is provided using randomly generated benchmarks and well-known DIMACS implementation challenges. The empirical study shows that the branch-and-bound finds the optimal triangle packing solution for small randomly generated MTP instances (up to 100 vertices and 200 triangles) and some DIMACS graphs. For some larger instances and DIMACS challenges graphs, we remark that our lower bound outperforms CPLEX solver regarding the triangle packing solution and the computation time.

Towards optimal kernel for edge-disjoint triangle packing

We study the kernelization of the Edge-Disjoint Triangle Packing (Etp) problem, in which we are asked to find k edge-disjoint triangles in an undirected graph. Etp is known to be fixed-parameter tractable with a 4k vertex kernel. The kernelization first finds a maximal triangle packing which contains at most 3k vertices, then the reduction rules are used to bound the size of the remaining graph. The constant in the kernel size is so small that a natural question arises: Could 4k be already the optimal vertex kernel size for this problem? In this paper, we answer the question negatively by deriving an improved 3.5k vertex kernel for the problem.

Deterministic approximation algorithms for the maximum traveling salesman and maximum triangle packing problems

We give derandomizations of known randomized approximation algorithms for the maximum traveling salesman problem and the maximum triangle packing problem: we show how to define pessimistic estimators for certain probabilities, based on the analysis of the randomized algorithms, and show that we can multiply the estimators to obtain pessimistic estimators for the expected weight of the solution. The method of pessimistic estimators (Raghavan (1988) [14]) then immediately implies that the randomized algorithms can be derandomized. For the maximum triangle packing problem, this gives deterministic algorithms with better approximation guarantees than what was previously known.The key idea in our analysis is the specification of conditions on pessimistic estimators of two expectations E[Y] and E[Z], under which the product of the pessimistic estimators is a pessimistic estimator of E[YZ], where Y and Z are two random variables. This approach can be useful when derandomizing algorithms for which one needs to bound the probability of some event that can be expressed as an intersection of multiple events; using our method, one can define pessimistic estimators for the probabilities of the individual events, and then multiply them to obtain a pessimistic estimator for the probability of the intersection of the events.

A Heuristic Algorithm for Solving Triangle Packing Problem

The research on the triangle packing problem has important theoretic significance, which has broad application prospects in material processing, network resource optimization, and so forth. Generally speaking, the orientation of the triangle should be limited in advance, since the triangle packing problem is NP-hard and has continuous properties. For example, the polygon is not allowed to rotate; then, the approximate solution can be obtained by optimization method. This paper studies the triangle packing problem by a new kind of method. Such concepts as angle region, corner-occupying action, corner-occupying strategy, and edge-conjoining strategy are presented in this paper. In addition, an edge-conjoining and corner-occupying algorithm is designed, which is to obtain an approximate solution. It is demonstrated that the proposed algorithm is highly efficient, and by the time complexity analysis and the analogue experiment result is found.

Planar graph vertex partition for linear problem kernels

A simple partition of the vertex set of a graph is introduced to analyze kernels for planar graph problems in which vertices and edges not in a solution have small distance to the solution. This method directly leads to improved kernel sizes for several problems, without needing new reduction rules. Moreover, new kernelization algorithms are developed for Connected Vertex Cover, Edge Dominating Set, and Maximum Triangle Packing problems, further improving the kernel sizes for these problems.

On approximating four covering and packing problems

In this paper, we consider approximability issues of the following four problems: triangle packing, full sibling reconstruction, maximum profit coverage and 2 -coverage. All of them are generalized or specialized versions of set-cover and have applications in biology ranging from full-sibling reconstructions in wild populations to biomolecular clusterings; however, as this paper shows, their approximability properties differ considerably. Our inapproximability constant for the triangle packing problem improves upon the previous results in [A. Caprara, R. Rizzi, Packing triangles in bounded degree graphs, Inform. Process. Lett. 84 (4) (2002) 175–180; J. Chlebíková, M. Chlebík, Complexity of approximating bounded variants of optimization problems, Theoret. Comput. Sci. 354 (3) (2006) 320–338]; this is done by directly transforming the inapproximability gap of Håstad for the problem of maximizing the number of satisfied equations for a set of equations over GF(2) [J. Håstad, Some optimal inapproximability results, in: Proc. of the 29th Annual ACM Symp. on Theory of Computing, 1997, pp. 1–10] and is interesting in its own right. Our approximability results on the full siblings reconstruction problems answers questions originally posed by Berger-Wolf et al. [T.Y. Berger-Wolf, B. DasGupta, W. Chaovalitwongse, M.V. Ashley, Combinatorial reconstruction of sibling relationships, in: Proc. of the 6th International Symposium on Computational Biology and Genome Informatics, 2005, pp. 1252–1255; T.Y. Berger-Wolf, S. Sheikh, B. DasGupta, M.V. Ashley, I. Caballero, W. Chaovalitwongse, S.L. Putrevu, Reconstructing sibling relationships in wild populations, Bioinformatics 23 (13) (2007) i49–i56] and our results on the maximum profit coverage problem provides almost matching upper and lower bounds on the approximation ratio, answering a question posed by Hassin and Or [R. Hassin, E. Or, A maximum profit coverage algorithm with application to small molecules cluster identification, in: 5th International Workshop Experimental Algorithms, in: Lecture Notes in Comput. Sci., vol. 4007, Springer-Verlag, 2006, pp. 265–276].

An improved randomized approximation algorithm for maximum triangle packing

This paper deals with the maximum triangle packing problem. For this problem, Hassin and Rubinstein gave a randomized polynomial-time approximation algorithm that achieves an expected ratio of 43 83 ( 1 − ϵ ) ( ≈ 0.518 ( 1 − ϵ ) ) for any constant ϵ > 0 . By modifying their algorithm, we obtain a new randomized polynomial-time approximation algorithm for the problem which achieves an expected ratio of 0.5257 ( 1 − ϵ ) for any constant ϵ > 0 .

An approximation algorithm for maximum triangle packing

We present a randomized ( 89 169 - ε ) -approximation algorithm for the weighted maximum triangle packing problem, for any given ε > 0 . This is the first algorithm for this problem whose performance guarantee is better than 1 2 . The algorithm also improves the best-known approximation bound for the maximum 2-edge path packing problem.

A heuristic method for solving triangle packing problem

Given a set of triangles and a rectangle container, the triangle packing problem is to determine if these triangles can be placed into the container without overlapping. Triangle packing problem is a special case of polygon packing problem and also NP-hard, so it is unlikely that an efficient and exact algorithm can be developed to solve this problem. In this paper, a new concept of rigid placement is proposed, based on which a discrete solution space called rigid solution space is constructed. Each solution in the rigid solution space can be built by continuously applying legal rigid placements one by one until all the triangles are placed into the rectangle container without overlapping. The proposed Least-Destruction-First (LDF) strategy determines which rigid placement has the privilege to go into the rectangle container. Based on this, a heuristic algorithm is proposed to solve the problem. Combining Least-Destruction-First strategy with backtracking, the corresponding backtracking algorithm is proposed. Computational results show that our proposed algorithms are efficient and robust. With slight modification, these techniques can be conveniently used for solving polygon packing problem.

A heuristic method for solving triangle packing problem

A heuristic method for solving triangle packing problem