اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSAT-Encodings for Treecut Width and Treedepth
90
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper we propose, implement, and test the first practical decomposition algorithms for the width parameters treecut width and treedepth. These two parameters have recently gained a lot of attention in the theoretical research community as they offer the algorithmic advantage over treewidth by supporting so-called fixed-parameter algorithms for certain problems that are not fixed-parameter tractable with respect to treewidth. However, the existing research has mostly been theoretical. A main obstacle for any practical or experimental use of these two width parameters is the lack of any practical or implemented algorithm for actually computing the associated decompositions. We address this obstacle by providing the first practical decomposition algorithms. Our approach for computing treecut width and treedepth decompositions is based on efficient encodings of these decomposition methods to the propositional satisfiability problem (SAT). Once an encoding is generated, any satisfiability solver can be used to find the decomposition. Moreover, we propose new characterisations for treecut width and treedepth that are based on sequences of partitions of the vertex set, a method that was pioneered for clique-width. We implemented and systematically tested our encodings on various benchmark instances, including famous named graphs and random graphs of various density. It turned out that for the considered width parameters, our partition-based SAT encoding even outperforms the best existing SAT encoding for treewidth.
قيم البحث
اقرأ أيضاً
The at-most-k constraint is ubiquitous in combinatorial problems, and numerous SAT encodings are available for the constraint. Prior experiments have shown the competitiveness of the sequential-counter encoding for k $>$ 1, and have excluded the para
llel-counter encoding, which is more compact that the binary-adder encoding, from consideration due to its incapability of enforcing arc consistency through unit propagation. This paper presents an experiment that shows astounding performance of the binary-adder encoding for the at-most-k constraint.
One of the most studied models of SAT is random SAT. In this model, instances are composed from clauses chosen uniformly randomly and independently of each other. This model may be unsatisfactory in that it fails to describe various features of SAT i
nstances, arising in real-world applications. Various modifications have been suggested to define models of industrial SAT. Here, we focus mainly on the aspect of community structure. Namely, here the set of variables consists of a number of disjoint communities, and clauses tend to consist of variables from the same community. Thus, we suggest a model of random industrial SAT, in which the central generalization with respect to random SAT is the additional community structure. There has been a lot of work on the satisfiability threshold of random $k$-SAT, starting with the calculation of the threshold of $2$-SAT, up to the recent result that the threshold exists for sufficiently large $k$. In this paper, we endeavor to study the satisfiability threshold for the proposed model of random industrial SAT. Our main result is that the threshold in this model tends to be smaller than its counterpart for random SAT. Moreover, under some conditions, this threshold even vanishes.
We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for $k$-Dominating Set on graphs of t
win-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for Connected $k$-Dominating Set and Total $k$-Dominating Set (albeit with a worse upper bound on the twin-width). The $k$-Independent Set problem admits the same lower bound by a much simpler argument, previously observed [ICALP 21], which extends to $k$-Independent Dominating Set, $k$-Path, $k$-Induced Path, $k$-Induced Matching, etc. On the positive side, we obtain a simple quadratic vertex kernel for Connected $k$-Vertex Cover and Capacitated $k$-Vertex Cover on graphs of bounded twin-width. Interestingly the kernel applies to graphs of Vapnik-Chervonenkis density 1, and does not require a witness sequence. We also present a more intricate $O(k^{1.5})$ vertex kernel for Connected $k$-Vertex Cover. Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most optimization/decision graph problems can be solved in polynomial time on graphs of twin-width at most 1.
Constraint satisfaction problems (CSPs) models many important intractable NP-hard problems such as propositional satisfiability problem (SAT). Algorithms with non-trivial upper bounds on running time for restricted SAT with bounded clause length k (k
-SAT) can be classified into three styles: DPLL-like, PPSZ-like and Local Search, with local search algorithms having already been generalized to CSP with bounded constraint arity k (k-CSP). We generalize a DPLL-like algorithm in its simplest form and a PPSZ-like algorithm from k-SAT to k-CSP. As far as we know, this is the first attempt to use PPSZ-like strategy to solve k-CSP, and before little work has been focused on the DPLL-like or PPSZ-like strategies for k-CSP.
We present a constructive SAT-based algorithm to determine the multiplicative complexity of a Boolean function, i.e., the smallest number of AND gates in any logic network that consists of 2-input AND gates, 2-input XOR gates, and inverters. In order
to speed-up solving time, we make use of several symmetry breaking constraints; these exploit properties of XAGs that may be useful beyond the proposed SAT-based algorithm. We further propose a heuristic post-optimization algorithm to reduce the number of XOR gates once the optimum number of AND gates has been obtained, which also makes use of SAT solvers. Our algorithm is capable to find all optimum XAGs for representatives of all 5-input affine-equivalent classes, and for a set of frequently occurring 6-input functions.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
