Finding Maximum Weight 2‐Packing Sets on Arbitrary Graphs

On the complexity of the independent set problem in triangle graphs

ABSTRACT
 The Ministry of Maritime Affairs and Fisheries (KKP) through the Directorate General of Capture Fisheries (DJPT) provides assistance in the form of fishing vessels to fishermen. However, the aid vessels intended for Cilacap had an empty weight that was heavier when compared to other vessels that had the same main dimensions of the vessel. The vessel that have heavier empty weights will cause poor motion when operating and reduced capacity to accommodate the vessel. The purpose of this research are to calculate the difference in the weights of aid vessel and fishing vessel, formulate the maximum weight of cargo that can be accommodated by aid vessel and fishing vessel through TPC calculations, and estimate the level of income of fishermen per trip with the maximum weight set. The research was conducted on aid vessel and 3 GT sized fishermen vessel made of fiberglass which have different vessel weights. Based on the results of the research, it was found that the empty weight of the aid vessel was 277,1847 kg and the fishermen vessel was 95,4165 kg with the difference between the two vessels 181,7682 kg. The maximum cargo weight that can be accommodated by the aid vessel at the highest draft is 3.98 tons. The maximum cargo weight that can be accommodated by fishing vessel at the highest draft is 5.98 tons. The estimated revenue of the aid vessel in the condition of a maximum load of Rp. 29,371,620 and the estimated income of fishing vessel in the condition of a maximum load of Rp. 46,081,620.
 
 Keywords: draft, fiberglass, income, main dimensions, weight

Maximum weight independent set for [formula omitted]claw-free graphs in polynomial time

A linear time algorithm to compute a maximum weighted independent set on cocomparability graphs

The maximum weight independent set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. In 1982, Alekseev (Comb Algebraic Methods Appl Math 132:3---13, 1982) showed that the M(W)IS problem remains NP-complete on H-free graphs, whenever H is connected, but neither a path nor a subdivision of the claw. We will focus on graphs without a subdivision of a claw. For integers $$i, j, k \ge 1$$i,j,kź1, let $$S_{i, j, k}$$Si,j,k denote a tree with exactly three vertices of degree one, being at distance i, j and k from the unique vertex of degree three. Note that $$S_{i,j, k}$$Si,j,k is a subdivision of a claw. The computational complexity of the MWIS problem for the class of $$S_{1, 2, 2}$$S1,2,2-free graphs, and for the class of $$S_{1, 1, 3}$$S1,1,3-free graphs are open. In this paper, we show that the MWIS problem can be solved in polynomial time for ($$S_{1, 2, 2}, S_{1, 1, 3}$$S1,2,2,S1,1,3, co-chair)-free graphs, by analyzing the structure of the subclasses of this class of graphs. This also extends some known results in the literature.

In this paper an O(kn√logc+γ) time algorithm is presented to solve the maximum weight k-independent set problem on an interval graph with n vertices and non-negative integer weights, where c is the weight of the longest path of the interval graph and γ is the total size of all maximal cliques, given its interval representation. If the intervals are not sorted then considering the time for sorting the time complexity is of O(nlogn+kn √logc+γ). Using this algorithm the maximum weight 2-independent set problem for an interval graph with n vertices can be solved in O(n √logc + γ) time. The best known previous algorithm for 2-independent set problem requires O(n 2) time.

In this paper, we consider the task of computing an independent set of maximum weight in a given d-claw free graph G = (V,E) equipped with a positive weight function w:V → ℝ^+. Thereby, d ≥ 2 is considered a constant. The previously best known approximation algorithm for this problem is the local improvement algorithm SquareImp proposed by Berman [Berman, 2000]. It achieves a performance ratio of d/2+e in time 𝒪(|V(G)|^(d+1)⋅(|V(G)|+|E(G)|)⋅(d-1)²⋅ (d/(2e)+1)²) for any e > 0, which has remained unimproved for the last twenty years. By considering a broader class of local improvements, we obtain an approximation ratio of d/2-(1/63,700,992)+e for any e > 0 at the cost of an additional factor of 𝒪(|V(G)|^(d-1)²) in the running time. In particular, our result implies a polynomial time d/2-approximation algorithm. Furthermore, the well-known reduction from the weighted k-Set Packing Problem to the Maximum Weight Independent Set Problem in k+1-claw free graphs provides a (k+1)/2 -(1/63,700,992)+e-approximation algorithm for the weighted k-Set Packing Problem for any e > 0. This improves on the previously best known approximation guarantee of (k+1)/2 + e originating from the result of Berman [Berman, 2000].

LetG(n,c/n) andGr(n) be ann‐node sparse random graph and a sparse randomr‐regular graph, respectively, and letI(n,r) andI(n,c) be the sizes of the largest independent set inG(n,c/n) andGr(n). The asymptotic value ofI(n,c)/nasn→ ∞, can be computed using the Karp‐Sipser algorithm whenc≤e. For random cubic graphs,r= 3, it is only known that .432 ≤ lim infnI(n,3)/n≤ lim supnI(n,3)/n≤ .4591 with high probability (w.h.p.) asn→ ∞, as shown in Frieze and Suen [Random Structures Algorithms 5 (1994), 649–664] and Bollabas [European J Combin 1 (1980), 311–316], respectively. In this paper we assume in addition that the nodes of the graph are equipped with nonnegative weights, independently generated according to some common distribution, and we consider instead the maximum weight of an independent set. Surprisingly, we discover that for certain weight distributions, the limit limnI(n,c)/ncan be computed exactly even whenc>e, and limnI(n,r)/ncan be computed exactly for somer≥ 1. For example, when the weights are exponentially distributed with parameter 1, limnI(n,2e)/n≈ .5517, and limnI(n,3)/n≈ .6077. Our results are established using the recently developedlocal weak convergencemethod further reduced to a certainlocal optimalityproperty exhibited by the models we consider. We extend our results to maximum weight matchings inG(n,c/n) andGr(n). For the case of exponential distributions, we compute the corresponding limits for everyc> 0 and everyr≥ 2. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

We present improved results for approximating maximum-weight independent set (MaxIS) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let n and Δ be the number of nodes and maximum degree, respectively, and let MIS(n,Δ) be the running time of finding a maximal independent set (MIS) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a Δ-approximation for MaxIS in O(MIS(n,Δ)log W) rounds, where W is the maximum weight of a node in the graph, which can be as large as poly (n). Whether their algorithm is deterministic or randomized that succeeds with high probability depends on the MIS algorithm that is used as a black-box. Our results: 1) A deterministic O(MIS(n,Δ)/e)-round algorithm that finds a (1+e)Δ-approximation for MaxIS in the CONGEST model. 2) A randomized (poly(log log n)/e)-round algorithm that finds, with high probability, a (1+e)Δ-approximation for MaxIS in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an exponential speed-up in the running time over the previous best known result. 3) A randomized O(log n⋅ poly(log log n)/e)-round algorithm that finds, with high probability, a 8(1+e)α-approximation for MaxIS in the CONGEST model, where α is the arboricity of the graph. For graphs of arboricity α < Δ/(8(1+e)), this result improves upon the previous best known result in both the approximation factor and the running time. One may wonder whether it is possible to approximate MaxIS with high probability in fewer than poly(log log n) rounds. Interestingly, a folklore randomized ranking algorithm by Boppana implies a single round algorithm that gives an expected Δ-approximation in the CONGEST model. However, it is unclear how to convert this algorithm to one that succeeds with high probability without sacrificing a large number of rounds. For unweighted graphs of maximum degree Δ ≤ n/log n, we show a new analysis of the randomized ranking algorithm, which we combine with the local-ratio technique, to provide a O(1/e)-round algorithm in the CONGEST model that, with high probability, finds an independent set of size at least n/((1+e)(Δ+1)). This result cannot be extended to very high degree graphs, as we show a lower bound of Ω(log^*n) rounds for any randomized algorithm that with probability at least 1-1/log n finds an independent set of size Ω(n/Δ). This lower bound holds even for the LOCAL model. The hard instances that we use to prove our lower bound are graphs of maximum degree Δ = Ω(n/log^*n).

The Exact Weighted Independent Set Problem in Perfect Graphs and Related Classes

An efficient algorithm for finding a maximum weight 2-independent set on interval graphs

Given a graph G, a non-negative integer k, and a weight function that maps each vertex in G to a positive real number, the Maximum Weighted Budgeted Independent Set (MWBIS) problem is about finding a maximum weighted independent set in G of cardinality at most k. A special case of MWBIS, when the weight assigned to each vertex is equal to its degree in G, is called the Maximum Independent Vertex Coverage (MIVC) problem. In other words, the MIVC problem is about finding an independent set of cardinality at most k with maximum coverage.

Finding the largest independent set in a graph is a notoriously difficult N P-complete combinatorial optimization problem. Moreover, even for graphs with largest degree 3, no polynomial time approximation algorithm exists with a 1.0071-factor approximation guarantee, unless P = N P [BK98].We consider the related problem of finding the maximum weight independent set in a bounded degree graph, when the node weights are generated i.i.d. from a common distribution. Surprisingly, we discover that the problem becomes tractable for certain distributions. Specifically, we construct a randomized PTAS (Polynomial-Time Approximation Scheme) for the case of exponentially distributed weights and arbitrary graphs with degree at most 3. We extend our result to graphs with larger constant degrees but for distributions which are mixtures of exponential distributions. At the same time, we prove that no PTAS exists for computing the expected size of the maximum weight independent set in the case of exponentially distributed weights for graphs with sufficiently large constant degree, unless P=NP. Our algorithm, cavity expansion, is new and is based on the combination of several powerful ideas, including recent deterministic approximation algorithms for counting on graphs and local weak convergence/correlation decay methods.

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)PTAS for maximum weight independent set problem with random weights in bounded degree graphsDavid Gamarnik, David Goldberg, and Theophane WeberDavid GamarnikOperations Research Center and Sloan School of Management, MIT, Cambridge, MAOperations Research Center and Sloan School of Management, MIT, Cambridge, MAOperations Research Center, MIT, Cambridge, MASearch for more papers by this author, David GoldbergOperations Research Center and Sloan School of Management, MIT, Cambridge, MAOperations Research Center and Sloan School of Management, MIT, Cambridge, MAOperations Research Center, MIT, Cambridge, MASearch for more papers by this author, and Theophane WeberOperations Research Center and Sloan School of Management, MIT, Cambridge, MAOperations Research Center and Sloan School of Management, MIT, Cambridge, MAOperations Research Center, MIT, Cambridge, MASearch for more papers by this authorpp.268 - 278Chapter DOI:https://doi.org/10.1137/1.9781611973075.23PDFBibTexSections ToolsAdd to favoritesDownload CitationsTrack CitationsEmail SectionsAboutAbstract Finding the largest independent set in a graph is a notoriously difficult NP-complete combinatorial optimization problem. Moreover, even for graphs with largest degree 3, no polynomial time approximation algorithm exists with a 1.0071-factor approximation guarantee, unless P = NP [BK98]. We consider the related problem of finding the maximum weight independent set in a bounded degree graph, when the node weights are generated i.i.d. from a common distribution. Surprisingly, we discover that the problem becomes tractable for certain distributions. Specifically, we construct a randomized PTAS (Polynomial-Time Approximation Scheme) for the case of exponentially distributed weights and arbitrary graphs with degree at most 3. We extend our result to graphs with larger constant degrees but for distributions which are mixtures of exponential distributions. At the same time, we prove that no PTAS exists for computing the expected size of the maximum weight independent set in the case of exponentially distributed weights for graphs with sufficiently large constant degree, unless P=NP. Our algorithm, cavity expansion, is new and is based on the combination of several powerful ideas, including recent deterministic approximation algorithms for counting on graphs and local weak convergence/correlation decay methods. Previous chapter Next chapter RelatedDetails Published:2010ISBN:978-0-89871-701-3eISBN:978-1-61197-307-5 https://doi.org/10.1137/1.9781611973075Book Series Name:ProceedingsBook Code:PR135Book Pages:xviii + 1667

The maximum independent set problem is one of the most important problems in graph algorithms and has been extensively studied in the line of research on the worst-case analysis of exact algorithms for NP-hard problems. In the weighted version, each vertex in the graph is associated with a weight and we are going to find an independent set of maximum total vertex weight. In this paper, we design several reduction rules and a fast exact algorithm for the maximum weighted independent set problem, and use the measure-and-conquer technique to analyze the running time bound of the algorithm. Our algorithm works on general weighted graphs and it has a good running time bound on sparse graphs. If the graph has an average degree at most 3, our algorithm runs in \(O^*(1.1443^n)\) time and polynomial space, improving previous running time bounds for the problem in cubic graphs using polynomial space.