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Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials [article]

Bart M.P. Jansen, Astrid Pieterse
2019 arXiv   pre-print
For the Not-All-Equal SAT problem, a compression to size Õ(n^d-1) exists. We put these results in a common framework by analyzing the compressibility of binary CSPs.  ...  We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several  ...  Acknowledgements We are grateful to Fedor Petrov for suggesting Lemma 13.  ... 
arXiv:1606.03233v3 fatcat:mrmshtaddnexziewb6yooz6x7i

Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials

Bart M. P. Jansen, Astrid Pieterse
2019 ACM Transactions on Computation Theory  
The goal of this paper is to understand how the optimal compression size for a binary CSP depends on the type of legal constraints, with the aim of obtaining matching upper and lower bounds.  ...  A problem for which nontrivial polynomial-time sparsification is possible was recently discovered by the current authors [20] .  ...  Since E is a rbds of instance X 1, 2 , for each j ∈ [m B ] at least one vertex from set {v 2 i,j | i ∈ [k]} has a neighbor 71:10 Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials  ... 
doi:10.1145/3349618 fatcat:p7f56ucksrbczgjnlz63fdf67m

Best-Case and Worst-Case Sparsifiability of Boolean CSPs

Hubie Chen, Bart M. P. Jansen, Astrid Pieterse
2020 Algorithmica  
The combination of these algorithmic results allows us to prove two characterizations that capture the optimal sparsification sizes for a range of Boolean CSPs.  ...  We continue the investigation of polynomial-time sparsification for NP-complete Boolean Constraint Satisfaction Problems (CSPs).  ...  It is therefore apparent that finding a low-degree polynomial to capture the constraints of a CSP is a powerful tool to obtain sparsification algorithms for it.  ... 
doi:10.1007/s00453-019-00660-y fatcat:lzg3diwsyfdhzl3jn6rowbolaa

Best-case and Worst-case Sparsifiability of Boolean CSPs [article]

Hubie Chen, Bart M. P. Jansen, Astrid Pieterse
2018 arXiv   pre-print
The combination of these algorithmic results allows us to prove two characterizations that capture the optimal sparsification sizes for a range of Boolean CSPs.  ...  We continue the investigation of polynomial-time sparsification for NP-complete Boolean Constraint Satisfaction Problems (CSPs).  ...  On sparsification for computing treewidth. Bart M. P. Jansen and Astrid Pieterse. Optimal sparsification for some binary CSPs using low-degree polynomials.  ... 
arXiv:1809.06171v1 fatcat:al6bgjkptrfp5gjpaerc5crv6i

The limited blessing of low dimensionality: when 1-1/d is the best possible exponent for d-dimensional geometric problems [article]

Dániel Marx, Anastasios Sidiropoulos
2016 arXiv   pre-print
Our main result is showing that for some of these problems the dependence on 1-1/d is best possible under a standard complexity assumption.  ...  We state the complexity results on CSPs in a way to make them convenient starting points for problem-specific reductions to particular d-dimensional geometric problems and to be reusable in the future  ...  We can prove the same lower bound for cycle-TSP, using a simple modification of the above reduction.  ... 
arXiv:1612.01171v1 fatcat:iugy4xyj6relvcbcxptruu3jbu

Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses [article]

Parinya Chalermsook, Bundit Laekhanukit, Danupon Nanongkai
2013 arXiv   pre-print
It also improves some hardness results in the domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et al., 2013) - For any k larger than some constant, there is no polynomial time  ...  We note an interesting fact that, in contrast to n^1/2-ϵ hardness for polynomial-time algorithms, the k-hypergraph pricing problem admits n^δ approximation for any δ >0 in quasi-polynomial time.  ...  Consider an optimal price function p * for (C, I). Fix some optimal assignment of items to customers with respect to p * .  ... 
arXiv:1308.2617v2 fatcat:4xrgaznlkffobhdducjuamwf6i

Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses

Parinya Chalermsook, Bundit Laekhanukit, Danupon Nanongkai
2013 2013 IEEE 54th Annual Symposium on Foundations of Computer Science  
We present a series of almost settled inapproximability results for three fundamental problems.  ...  The second is the hardness of approximating the maximum induced matching problem on bounded-degree bipartite graphs.  ...  Consider an optimal price function p * for (C, I). Fix some optimal assignment of items to customers with respect to p * .  ... 
doi:10.1109/focs.2013.47 dblp:conf/focs/ChalermsookLN13 fatcat:wvu3lxve3ba7vey7wa6t67mka4

Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs [article]

Vincent Cohen-Addad, Éric Colin de Verdière, Daniel Marx, Arnaud de Mesmay
2021 arXiv   pre-print
In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for  ...  The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value g of the genus.  ...  Acknowledgements We are grateful to the anonymous reviewers for a careful reading of our manuscript and helpful remarks.  ... 
arXiv:1903.08603v3 fatcat:uasnm56bvje37mnm2ociswceqa

A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs [article]

Pasin Manurangsi, Prasad Raghavendra
2016 arXiv   pre-print
In particular, for any sufficiently large i and for every k ≥ 2, we show the following results: - We exhibit an O(q^1/i)-approximation algorithm for dense Max k-CSPs with alphabet size q via O_k(i)-level  ...  As an application of our birthday repetition theorem, we obtain new fine-grained hardness of approximation results for dense CSPs.  ...  He also thanks Grigory Yaroslavtsev for providing insights on his algorithm for dense CSPs in [Yar14] and Madhur Tulsiani for bringing [BRS11, GS11] to his attention.  ... 
arXiv:1607.02986v1 fatcat:thsrikpxp5gmzpel6dd3cmjdfm

Weak bases of Boolean co-clones

Victor Lagerkvist
2014 Information Processing Letters  
This algebraic approach allows us to obtain a more nuanced view of the complexity CSP(S) than possible with algebras of total functions, clones.  ...  For MAX-ONES(S) we use partial clone theory and prove that MAX-ONES({R = = = 1/3 }) is the easiest NPcomplete MAX-ONES(S) problem.  ...  If Inv(C) is polynomially closed, then s Inv(C) (κ) ≤ p(κ) for some polynomial p. D .  ... 
doi:10.1016/j.ipl.2014.03.011 fatcat:4rhokivpgnaxrptedshzn2emji

Hamiltonian sparsification and gap-simulations [article]

Dorit Aharonov, Leo Zhou
2018 arXiv   pre-print
We also show a complementary result where degree-reduction is possible when the strength of interactions is allowed to grow polynomially.  ...  We initiate the rigorous study of the physical resources required for such simulations, where we focus on the task of Hamiltonian sparsification.  ...  Another common use of sparsification in classical computer science is degree-reduction (DR), used in the study of local Constraint Satisfaction Problems (CSPs) and PCPs [35] .  ... 
arXiv:1804.11084v2 fatcat:bszhvsm5vzfw3nmsj7ry4d3i6u

Time and Space Bounds for Planning

Christer Bäckström, Peter Jonsson
2017 The Journal of Artificial Intelligence Research  
We provide a number of upper- and lower-bound results (the latter based on various complexity-theoretic assumptions such as the Exponential Time Hypothesis) for both satisficing and optimal planning.  ...  We show that many classes of planning instances exhibit a dichotomy: either they can be solved in polynomial time or they cannot be solved in subexponential time.  ...  instances is very low for other values of the ratio.  ... 
doi:10.1613/jair.5535 fatcat:gkhyikoyo5ep3l27fji5kgnknm

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Andreas Emil Feldmann, Karthik C. S., Euiwoong Lee, Pasin Manurangsi
2020 Algorithms  
After this branching finishes and we are left with low-degree graphs, just run the known polynomial-time approximation algorithms [283] on these graphs.  ...  This algorithm is randomized, but can be derandomized using the sparsification lemma [284] .  ... 
doi:10.3390/a13060146 fatcat:2u2vv3uksfguvj6473t2gsq42a

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms [article]

Andreas Emil Feldmann, Karthik C. S., Euiwoong Lee, Pasin Manurangsi
2020 arXiv   pre-print
After this branching finishes and we are left with low-degree graphs, just run the known polynomial-time approximation algorithms [263] on these graphs.  ...  This algorithm is randomized, but can be derandomized using the sparsification lemma [264] .  ... 
arXiv:2006.04411v1 fatcat:hjgu7f3s7zbydkcnioq3qlzgza

Cardinality Minimization, Constraints, and Regularization: A Survey [article]

Andreas M. Tillmann, Daniel Bienstock, Andrea Lodi, Alexandra Schwartz
2022 arXiv   pre-print
solutions for cardinality optimization problems, even in large-scale real-world settings.  ...  knowledge that may stem from different fields of application and, e.g., shed light on structural properties of a model or its solutions, or lead to the development of efficient heuristics; we also provide some  ...  There are some further interesting methods that, while certainly related to some degree, do not quite fit into the previous categories.  ... 
arXiv:2106.09606v2 fatcat:msfqmu4cc5d3hgi7wwfumvo2b4
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