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Inapproximability of Treewidth, One-Shot Pebbling, and Related Layout Problems [chapter]

Per Austrin, Toniann Pitassi, Yu Wu
2012 Lecture Notes in Computer Science  
We study the approximability of a number of graph problems: treewidth and pathwidth of graphs, one-shot black (and black-white) pebbling costs of directed acyclic graphs, and a variety of different graph  ...  We show that, assuming the recently introduced Small Set Expansion Conjecture, all of these problems are hard to approximate within any constant factor. Research supported by NSERC.  ...  A central open problem is to determine whether or not there exists a polynomial time constant factor approximation algorithm for treewidth (see e.g., [9, 13, 8] ).  ... 
doi:10.1007/978-3-642-32512-0_2 fatcat:cykgo367bfdb5k3slkwvopqtxm

A Note on Multiflows and Treewidth

Chandra Chekuri, Sanjeev Khanna, F. Bruce Shepherd
2007 Algorithmica  
We consider multicommodity flow problems in capacitated graphs where the treewidth of the underlying graph is bounded by r. The parameter r is allowed to be a function of the input size.  ...  We obtain the following results in such graphs. • An O(r log r log n) approximation for EDP and UFP. • The integrality gap of the multicommodity flow relaxation for EDP and UFP is O(min{r log n, √ n}).  ...  Using by now standard ideas [25] an algorithm for EDP based on the flow relaxation can be extended to obtain an algorithm with a comparable performance (to within constant factors) for UFP if d max ≤  ... 
doi:10.1007/s00453-007-9129-z fatcat:addxk77fejhatpd3v5vufqfy24

Relax, Compensate and Then Recover [chapter]

Arthur Choi, Adnan Darwiche
2011 Lecture Notes in Computer Science  
First, our notion of an approximation is based on "relaxing" equality constraints, for the purposes of simplifying a problem so that it can be solved more readily.  ...  We present in this paper a framework of approximate probabilistic inference which is based on three simple concepts.  ...  Discussion The performance of our system in the UAI'10 approximate inference evaluation was clearly a strong demonstration of the practical effectiveness of the relax, compensate and then recover framework  ... 
doi:10.1007/978-3-642-25655-4_16 fatcat:vrsjf5zs2nhdtmg4spleghv4vu

Large-Scale Traffic Signal Offset Optimization [article]

Yi Ouyang, Richard Y. Zhang, Javad Lavaei, Pravin Varaiya
2019 arXiv   pre-print
Specifically, we solve a convex relaxation of the nonconvex problem using a tree decomposition reduction, and use randomized rounding to recover a near-global solution.  ...  In this paper, we present a novel algorithm to solve this new formulation to near-global optimality on a large-scale.  ...  In practice, the resulting algorithm is about a constant factor of 10 times slower than the one proposed in this paper.  ... 
arXiv:1911.08368v1 fatcat:ekkipngoindhrfw4yxi66kvkti

Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs

Chandra Chekuri, Alina Ene, Marcin Pilipczuk
2018 SIAM Journal on Discrete Mathematics  
The main technical contribution is to show that a planar digraph with directed treewidth h contains a constant congestion crossbar of size Ω(h/polylog(h)). paths.  ...  It is known that even the simpler case of ANF and with congestion c allowed is hard to approximate to within a factor of n Ω(1/c) [15] ; moreover this holds in acyclic graphs.  ...  7:13 Does a planar directed graph with treewidth h have a constant congestion crossbar of size Ω(h). This would strengthen our result. In particular, is there a cylindrical grid minor of size Ω(h)?  ... 
doi:10.1137/17m1150694 fatcat:qzb4xwxhbrctjk6wgsxvnxlrlm

Convex Relaxations for Learning Bounded Treewidth Decomposable Graphs [article]

K. S. Sesh Kumar, Francis Bach
2012 arXiv   pre-print
A supergradient method is used to solve the dual problem, with a run-time complexity of O(k^3 n^k+2 n) for each iteration, where n is the number of variables and k is a bound on the treewidth.  ...  We consider the problem of learning the structure of undirected graphical models with bounded treewidth, within the maximum likelihood framework.  ...  We would also like to thank other members of the SIERRA and WILLOW project-teams for helpful discussions.  ... 
arXiv:1212.2573v1 fatcat:s7h5odxm6bd6hncumavdiypat4

Inapproximability of Treewidth and Related Problems

Y. Wu, P. Austrin, T. Pitassi, D. Liu
2014 The Journal of Artificial Intelligence Research  
We show that, assuming the recently introduced Small Set Expansion Conjecture, all of these problems are NP-hard to approximate to within any constant factor in polynomial time.  ...  Therefore, finding or approximating the Treewidth of a graph is a fundamental problem related to inference in graphical models.  ...  It is a longstanding open question whether or not there is a constant factor approximation algorithm for Treewidth.  ... 
doi:10.1613/jair.4030 fatcat:zyqijghvife2zlvgkhnhpjwfcm

Maximizing submodular functions using probabilistic graphical models [article]

K. S. Sesh Kumar, Francis Bach
2013 arXiv   pre-print
based on outer approximations of the marginal polytope and maximum likelihood bounded treewidth structures.  ...  In a graphical model, the entropy of the joint distribution decomposes as a sum of marginal entropies of subsets of variables; moreover, for any distribution, the entropy of the closest distribution factorizing  ...  [10] proposed constant factor approximation algorithms for maximizing non-negative submodular functions.  ... 
arXiv:1309.2593v1 fatcat:x4ckgjwbnnehfl7ypcgvmzl4ni

Tree Diet: Reducing the Treewidth to Unlock FPT Algorithms in RNA Bioinformatics [article]

Bertrand Marchand, Yann Ponty, Laurent Bulteau
2021 bioRxiv   pre-print
Our rationale is that the time gained thanks to a smaller treewidth in a parameterized algorithm compensates the extra post-processing needed to take deleted edges into account.  ...  This work paves the way for a wider adoption of tree-decomposition-based algorithms in Bioinformatics.  ...  However it seems unlikely to induce a 175 "practical" exact algorithms.  ... 
doi:10.1101/2021.04.30.442158 fatcat:ovpvupwbfbftzb5radjf2iriri

Improved Approximation Algorithms for Minimum Weight Vertex Separators

Uriel Feige, MohammadTaghi Hajiaghayi, James R. Lee
2008 SIAM journal on computing (Print)  
For graphs excluding a fixed graph as a minor (which includes, e.g. bounded genus graphs), we give a constant-factor approximation for the treewidth; this can be used to obtain the first polynomial-time  ...  Likewise, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k √ log k), whereas previous algorithms yielded  ...  A wellknown open problem is whether treewidth can be approximated within a constant factor.  ... 
doi:10.1137/05064299x fatcat:hedhyzomrjdwpojnnall7heg3y

Improved approximation algorithms for minimum-weight vertex separators

Uriel Feige, MohammadTaghi Hajiaghayi, James R. Lee
2005 Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05  
For graphs excluding a fixed graph as a minor (which includes, e.g. bounded genus graphs), we give a constant-factor approximation for the treewidth; this can be used to obtain the first polynomial-time  ...  Likewise, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k √ log k), whereas previous algorithms yielded  ...  A wellknown open problem is whether treewidth can be approximated within a constant factor.  ... 
doi:10.1145/1060590.1060674 dblp:conf/stoc/FeigeHL05 fatcat:ad2nkwcizzeynmbucegyxvyrdi

Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs [chapter]

Krishnendu Chatterjee, Rasmus Ibsen-Jensen, Andreas Pavlogiannis
2015 Lecture Notes in Computer Science  
First, for constant treewidth graphs we present an algorithm that approximates the mean-payoff value within a multiplicative factor of in time O(n · log(n/ )) and linear space, as compared to the classical  ...  Beyond the mathematical elegance of the treewidth property for graphs, there are many classes of graphs which arise in practice and have constant treewidth.  ...  First, we present a linear-time algorithm that computes c * exactly (if c * ≥ 0) or approximate within a polynomial factor (if c * < 0).  ... 
doi:10.1007/978-3-319-21690-4_9 fatcat:246qqug5l5bolp7jo6o2jorsai

Algorithmic Meta-Theorems for Monotone Submodular Maximization [article]

Masakazu Ishihata, Takanori Maehara, Tomas Rigaux
2018 arXiv   pre-print
Formally, we prove the following three 'algorithmic metatheorems.' (1) If the constraint is specified by a monadic second-order logic on a graph of bounded treewidth, the problem is solved in n^O(1) time  ...  factor of 2. (3) If the constraint is specified by a first-order logic on a graph of bounded expansion, the problem is solved in n^O( k) time with an approximation factor of O( k), where k is the number  ...  Acknowledgment We thank Antoine Amarilli for confirming the structuredness of their construction of DNNF in [1] .  ... 
arXiv:1807.04575v1 fatcat:ye2o43726zfyvfc7vijifqaptm

An Improvement of Reed's Treewidth Approximation [article]

Mahdi Belbasi, Martin Fürer
2020 arXiv   pre-print
Reed simply claims time O(n log n) for bounded k for his constant factor approximation algorithm, but the bound of 2^Ω(k log k) n log n is well known.  ...  We present a new approximation algorithm for the treewidth problem which constructs a corresponding tree decomposition as well. Our algorithm is a faster variation of Reed's classical algorithm.  ...  Then, the running time of our algorithm has another factor of 1/ .  ... 
arXiv:2010.03105v1 fatcat:xhy2egcprzdqrg5sbsrbh32324

Prize-collecting Network Design on Planar Graphs [article]

MohammadHossein Bateni and MohammadTaghi Hajiaghayi and Dániel Marx
2010 arXiv   pre-print
More precisely, we show any α-approximation algorithm for these problems on graphs of bounded treewidth gives an (α + ϵ)-approximation algorithm for these problems on planar graphs (and more generally  ...  is known to be polynomially solvable on series-parallel graphs and admits a PTAS on graphs of bounded treewidth.  ...  For any given constant > 0, an α-approximation algorithm for SPCSF on graphs of bounded treewidth gives a (α + )-approximation algorithm for SPCSF on planar graphs.  ... 
arXiv:1006.4339v1 fatcat:tnfakl2efvdzneuc2hn64wgrje
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