8 research outputs found
A Linear Weight Transfer Rule for Local Search
The Divide and Distribute Fixed Weights algorithm (ddfw) is a dynamic local
search SAT-solving algorithm that transfers weight from satisfied to falsified
clauses in local minima. ddfw is remarkably effective on several hard
combinatorial instances. Yet, despite its success, it has received little study
since its debut in 2005. In this paper, we propose three modifications to the
base algorithm: a linear weight transfer method that moves a dynamic amount of
weight between clauses in local minima, an adjustment to how satisfied clauses
are chosen in local minima to give weight, and a weighted-random method of
selecting variables to flip. We implemented our modifications to ddfw on top of
the solver yalsat. Our experiments show that our modifications boost the
performance compared to the original ddfw algorithm on multiple benchmarks,
including those from the past three years of SAT competitions. Moreover, our
improved solver exclusively solves hard combinatorial instances that refute a
conjecture on the lower bound of two Van der Waerden numbers set forth by Ahmed
et al. (2014), and it performs well on a hard graph-coloring instance that has
been open for over three decades
On the van der Waerden numbers w(2;3,t)
We present results and conjectures on the van der Waerden numbers w(2;3,t)
and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed
the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39,
where for t <= 30 we conjecture these lower bounds to be exact. The lower
bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we
present an improved conjecture. We also investigate regularities in the good
partitions (certificates) to better understand the lower bounds.
Motivated by such reglarities, we introduce *palindromic van der Waerden
numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers
w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good
partitions), defined as reading the same from both ends. Different from the
situation for ordinary van der Waerden numbers, these "numbers" need actually
to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide
lower bounds, which we conjecture to be exact, for t <= 35.
All computations are based on SAT solving, and we discuss the various
relations between SAT solving and Ramsey theory. Especially we introduce a
novel (open-source) SAT solver, the tawSolver, which performs best on the SAT
instances studied here, and which is actually the original DLL-solver, but with
an efficient implementation and a modern heuristic typical for look-ahead
solvers (applying the theory developed in the SAT handbook article of the
second author).Comment: Second version 25 pages, updates of numerical data, improved
formulations, and extended discussions on SAT. Third version 42 pages, with
SAT solver data (especially for new SAT solver) and improved representation.
Fourth version 47 pages, with updates and added explanation
Some Results in Extremal Combinatorics
Extremal Combinatorics is one of the central and heavily contributed areas in discrete mathematics,
and has seen an outstanding growth during the last few decades. In general, it deals
with problems regarding determination and/or estimation of the maximum or the minimum size
of a combinatorial structure that satisfies a certain combinatorial property. Problems in Extremal
Combinatorics are often related to theoretical computer science, number theory, geometry, and information
theory. In this thesis, we work on some well-known problems (and on their variants) in
Extremal Combinatorics concerning the set of integers as the combinatorial structure.
The van der Waerden number w(k;t_0,t_1,...,t_{k-1}) is the smallest positive
integer n such that every k-colouring of 1, 2, . . . , n contains a monochromatic
arithmetic progression of length t_j for some colour j in {0,1,...,k-1}. We have
determined five new exact values with k=2 and conjectured several van der Waerden numbers
of the form w(2;s,t), based on which we have formulated a polynomial
upper-bound-conjecture of w(2; s, t) with fixed s. We have provided an efficient SAT
encoding for van der Waerden numbers with k>=3 and computed three new van der Waerden
numbers using that encoding. We have also devised an efficient problem-specific
backtracking algorithm and computed twenty-five new van der Waerden numbers with k>=3
using that algorithm.
We have proven some counting properties of arithmetic progressions and some unimodality
properties of sequences regarding arithmetic progressions. We have generalized Szekeres’
conjecture on the size of the largest sub-sequence of 1, 2, . . . , n without an
arithmetic progression of length k for specific k and n; and provided a construction for
the lower bound corresponding to the generalized conjecture.
A Strict Schur number S(h,k) is the smallest positive integer n such that every
2-colouring of 1,2,...,n has either a blue solution to x_1 +x_2 +···+x_{h-1} = x_h
where x_1 < x_2 < ··· < x_h, or a red solution to x_1+x_2+···+x_{k-1} =x_k where
x_1 <x_2 <···<x_k. We have proven the exact formula for S(3, k)
Algorithmic approaches to problems in probabilistic combinatorics
The probabilistic method is one of the most powerful tools in combinatorics; it has been used to show the existence of many hard-to-construct objects with exciting properties. It also attracts broad interests in designing and analyzing algorithms to find and construct these objects in an efficient way. In this dissertation we obtain four results using algorithmic approaches in probabilistic method:
1. We study the structural properties of the triangle-free graphs generated by a semi-random variant of triangle-free process and obtain a packing extension of Kim's famous R(3,t) results. This allows us to resolve a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo, and answer a problem in extremal graph theory by Esperet, Kang, and Thomasse.
2. We determine the order of magnitude of Prague dimension, which concerns efficient encoding and decomposition of graphs, of binomial random graph with high probability. We resolve conjectures by Furedi and Kantor. Along the way, we prove a Pippenger-Spencer type edge coloring result for random hypergraphs with edges of size O(log n).
3. We analyze the number set generated by r-AP free process, which answers a problem raised by Li and has connection with van der Waerden number in additive combinatorics and Ramsey theory.
4. We study a refined alteration approach to construct H-free graphs in binomial random graphs, which has applications in Ramsey games.Ph.D
Proceedings of the 21st Conference on Formal Methods in Computer-Aided Design – FMCAD 2021
The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing
Efficient local search for Pseudo Boolean Optimization
Algorithms and the Foundations of Software technolog