Triangle Packing 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.

Kernelization for edge triangle packing and covering via a discharging method

Edge Triangle Packing and Edge Triangle Covering are dual problems extensively studied in the field of parameterized complexity. Given a graph G and an integer k, Edge Triangle Packing seeks to determine whether there exists a set of at least k edge-disjoint triangles in G, while Edge Triangle Covering aims to find out whether there exists a set of at most k edges that intersects all triangles in G. Previous research has shown that Edge Triangle Packing has a kernel of (3+ϵ)k vertices, while Edge Triangle Covering has a kernel of 6k vertices. In this paper, we show that the two problems allow kernels of 3k vertices, improving all previous results. A significant contribution of our work is the utilization of a novel discharging method for analyzing kernel size, which exhibits potential for analyzing other kernel algorithms.

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.

Improved kernels for triangle packing in tournaments

Improved kernels for triangle packing in tournaments

Covering and packing of triangles intersecting a straight line

We study four geometric optimization problems: set cover , hitting set , piercing set , and independent set with right-triangles (a triangle is a right-triangle whose base is parallel to the x -axis, perpendicular is parallel to the y -axis, and the slope of the hypotenuse is − 1 ). The input triangles are constrained to be intersecting a straight line . The straight line can either be a horizontal or an inclined line (a line whose slope is − 1 ). A right-triangle is said to be a λ -right-triangle , if the length of both its base and perpendicular is λ . For 1-right-triangles where the triangles intersect an inclined line, we prove that the set cover and hitting set problems are NP -hard, whereas the piercing set and independent set problems are in P . The same results hold for 1-right-triangles where the triangles are intersecting a horizontal line instead of an inclined line. We prove that the piercing set and independent set problems with right-triangles intersecting an inclined line are NP -hard. Finally, we give an n O ( ⌈ log c ⌉ + 1 ) time exact algorithm for the independent set problem with λ -right-triangles intersecting a straight line such that λ takes more than one value from [ 1 , c ] , for some integer c . We also present O ( n 2 ) -time dynamic programming algorithms for the independent set problem with 1-right-triangles where the triangles intersect a horizontal line and an inclined line.

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.

Triangle packing and covering in dense random graphs

Triangle packing and covering in dense random graphs

Parallel packing of triangles with squares

Suppose T1 (resp. Δ1) is an isosceles triangle with base length 1 and with height 122 (resp. 132). Let S be a square with a side parallel to the base of T1 (resp. Δ1) and let {Sn} be a sequence of the homothetic copies of S. We first determine the bound of sums of areas of squares from the sequence {Sn} that permits a parallel packing of T1 (resp. Δ1). Then we generalize the results about packing T1 (resp. Δ1) with squares to some other triangles. Finally, we consider the parallel packing of the right triangle T2′ (with leg lengths 1 and 2) with squares.

SnakeBinning: Efficient Temporally Coherent Triangle Packing for Shading Streaming

AbstractStreaming rendering, e.g., rendering in the cloud and streaming via a mobile connection, suffers from increased latency and unreliable connections. High quality framerate upsampling can hide these issues, especially when capturing shading into an atlas and transmitting it alongside geometric information. The captured shading information must consider triangle footprints and temporal stability to ensure efficient video encoding. Previous approaches only consider either temporal stability or sample distributions, but none focuses on both. With SnakeBinning, we present an efficient triangle packing approach that adjusts sample distributions and caters for temporal coherence. Using a multi‐dimensional binning approach, we enforce tight packing among triangles while creating optimal sample distributions. Our binning is built on top of hardware supported real‐time rendering where bins are mapped to individual pixels in a virtual framebuffer. Fragment shader interlock and atomic operations enforce global ordering of triangles within each bin, and thus temporal coherence according to the primitive order is achieved. Resampling the bin distribution guarantees high occupancy among all bins and a dense atlas packing. Shading samples are directly captured into the atlas using a rasterization pass, adjusting samples for perspective effects and creating a tight packing. Comparison to previous atlas packing approaches shows that our approach is faster than previous work and achieves the best sample distributions while maintaining temporal coherence. In this way, SnakeBinning achieves the highest rendering quality under equal atlas memory requirements. At the same time, its temporal coherence ensures that we require equal or less bandwidth than previous state‐of‐the‐art. As SnakeBinning outperforms previous approach in all relevant aspects, it is the preferred choice for texture‐based streaming rendering.

Limit theory of combinatorial optimization for random geometric graphs

In the random geometric graph G(n,rn), n vertices are placed randomly in Euclidean d-space and edges are added between any pair of vertices distant at most rn from each other. We establish strong laws of large numbers (LLNs) for a large class of graph parameters, evaluated for G(n,rn) in the thermodynamic limit with nrnd= const., and also in the dense limit with nrnd→∞, rn→0. Examples include domination number, independence number, clique-covering number, eternal domination number and triangle packing number. The general theory is based on certain subadditivity and superadditivity properties, and also yields LLNs for other functionals such as the minimum weight for the traveling salesman, spanning tree, matching, bipartite matching and bipartite traveling salesman problems, for a general class of weight functions with at most polynomial growth of order d−ε, under thermodynamic scaling of the distance parameter.

Packing Arc-Disjoint Cycles in Tournaments

A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k. We refer to these problems as Arc-disjoint Cycles in Tournaments (ACT) and Arc-disjoint Triangles in Tournaments (ATT), respectively. Although the maximization version of ACT can be seen as the dual of the well-studied problem of finding a minimum feedback arc set (a set of arcs whose deletion results in an acyclic graph) in tournaments, surprisingly no algorithmic results seem to exist for ACT. We first show that ACT and ATT are both NP-complete. Then, we show that the problem of determining if a tournament has a cycle packing and a feedback arc set of the same size is NP-complete. Next, we prove that ACT is fixed-parameter tractable via a $$2^{\mathcal {O}(k \log k)} n^{\mathcal {O}(1)}$$ -time algorithm and admits a kernel with $$\mathcal {O}(k)$$ vertices. Then, we show that ATT too has a kernel with $$\mathcal {O}(k)$$ vertices and can be solved in $$2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}$$ time. Afterwards, we describe polynomial-time algorithms for ACT and ATT when the input tournament has a feedback arc set that is a matching. We also prove that ACT and ATT cannot be solved in $$2^{o(\sqrt{n})} n^{\mathcal {O}(1)}$$ time under the exponential-time hypothesis.

Closing the Random Graph Gap in Tuza's Conjecture through the Online Triangle Packing Process

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 10 July 2020Accepted: 16 June 2021Published online: 20 September 2021KeywordsTuza' conjecture, random graphs, triangle packingAMS Subject Headings05B40, 05C80, 05D40Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

A randomized approximation algorithm for metric triangle packing

A randomized approximation algorithm for metric triangle packing

Large triangle packings and Tuza’s conjecture in sparse random graphs

AbstractThe triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.

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.

Covering Triangles in Edge-Weighted Graphs

Let G = (V, E) be a simple graph and \(\mathbf {w}\in \mathbb {Z}^{E}_{>0}\) assign each edge e ∈ E a positive integer weight w(e). A subset of E that intersects every triangle of G is called a triangle cover of (G, w), and its weight is the total weight of its edges. A collection of triangles in G (repetition allowed) is called a triangle packing of (G, w) if each edge e ∈ E appears in at most w(e) members of the collection. Let τ t (G, w) and ν t (G, w) denote the minimum weight of a triangle cover and the maximum cardinality of a triangle packing of (G, w), respectively. Generalizing Tuza’s conjecture for unit weight, Chapuy et al. conjectured that τ t (G, w)/ν t (G, w) ≤ 2 holds for every simple graph G and every \(\mathbf {w}\in \mathbb {Z}^{E}_{>0}\). In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding triangle covers of small weights. These algorithms imply new sufficient conditions for the conjecture of Chapuy et al. More precisely, given (G, w), suppose that all edges of G are covered by the set \({\mathscr {T}}_{G}\) consisting of edge sets of triangles in G. Let \(|E|_{w}={\sum }_{e\in E}w(e)\) and \(|{\mathscr {T}}_{G}|_{w}={\sum }_{\{e,f,g\}\in {\mathscr {T}}_{G}}w(e)w(f)w(g)\) denote the weighted numbers of edges and triangles in (G, w), respectively. We show that a triangle cover of (G, w) of weight at most 2ν t (G, w) can be found in strongly polynomial time if one of the following conditions is satisfied: (i) \(\nu _{t}(G,\mathbf {w})/|{\mathscr {T}}_{G}|_{w}\ge \frac {1}{3}\), (ii) \(\nu _{t}(G,\mathbf {w})/|E|_{w}\ge \frac {1}{4}\), (iii) \(|E|_{w}/|{\mathscr {T}}_{G}|_{w}\ge 2\).

Tuza's Conjecture is Asymptotically Tight for Dense Graphs

An old conjecture of Z. Tuza says that for any graphG, the ratio of the minimum size, τ3(G), of a set of edges meeting all triangles to the maximum size, ν3(G), of an edge-disjoint triangle packing is at most 2. Here, disproving a conjecture of R. Yuster, we show that for any fixed, positive α there are arbitrarily large graphsGof positive density satisfying τ3(G) > (1 −o(1))|G|/2 and ν3(G) < (1 + α)|G|/4.

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.