13 research outputs found
The Satisfiability Threshold for a Seemingly Intractable Random Constraint Satisfaction Problem
We determine the exact threshold of satisfiability for random instances of a
particular NP-complete constraint satisfaction problem (CSP). This is the first
random CSP model for which we have determined a precise linear satisfiability
threshold, and for which random instances with density near that threshold
appear to be computationally difficult. More formally, it is the first random
CSP model for which the satisfiability threshold is known and which shares the
following characteristics with random k-SAT for k >= 3. The problem is
NP-complete, the satisfiability threshold occurs when there is a linear number
of clauses, and a uniformly random instance with a linear number of clauses
asymptotically almost surely has exponential resolution complexity.Comment: This is the long version of a paper that will be published in the
SIAM Journal on Discrete Mathematics. This long version includes an appendix
and a computer program. The contents of the paper are unchanged in the latest
version. The format of the arxiv submission was changed so that the computer
program will appear as an ancillary file. Some comments in the computer
program were update
On the uniformity of the approximation for -associated Stirling numbers of the second Kind
The -associated Stirling numbers of the second kind are a natural extension of Stirling numbers of the second kind. A combinatorial interpretation of -associated Stirling numbers of the second kind is the number of ways to partition elements into subsets such that each subset contains at least elements. Calculating the associated Stirling numbers is typically done with a recurrence relation or a generating function that are computationally expensive or alternatively with a closed-form that is practical for only a limited parameter range. In 1994 Hennecart proposed an approximation for the -associated Stirling numbers that is fast to compute, is amenable to analysis over a wide range of parameters, and is conjectured to be asymptotically tight. There are a few other approximations for the associated Stirling numbers, but none of them are as general as Hennecart's. However, until this work, Hennecart's approximation had been utilized without a proper justification due to the absence of a rigorous proof. This work provides a proof of the uniformity of the Hennecart approximation
Phase Transition in Matched Formulas and a Heuristic for Biclique Satisfiability
A matched formula is a CNF formula whose incidence graph admits a matching
which matches a distinct variable to every clause. We study phase transition in
a context of matched formulas and their generalization of biclique satisfiable
formulas. We have performed experiments to find a phase transition of property
"being matched" with respect to the ratio where is the number of
clauses and is the number of variables of the input formula . We
compare the results of experiments to a theoretical lower bound which was shown
by Franco and Gelder (2003). Any matched formula is satisfiable, moreover, it
remains satisfiable even if we change polarities of any literal occurrences.
Szeider (2005) generalized matched formulas into two classes having the same
property -- var-satisfiable and biclique satisfiable formulas. A formula is
biclique satisfiable if its incidence graph admits covering by pairwise
disjoint bounded bicliques. Recognizing if a formula is biclique satisfiable is
NP-complete. In this paper we describe a heuristic algorithm for recognizing
whether a formula is biclique satisfiable and we evaluate it by experiments on
random formulas. We also describe an encoding of the problem of checking
whether a formula is biclique satisfiable into SAT and we use it to evaluate
the performance of our heuristicComment: Conference version submitted to SOFSEM 2018
(https://beda.dcs.fmph.uniba.sk/sofsem2019/) 18 pages(17 without refernces),
3 figures, 8 tables, an algorithm pseudocod
Toward Competency-Based Professional Accreditation in Computing
Program accreditation in medical or religious professions has existedsince the 1800s while accreditation of business and engineeringprograms started in the early twentieth century. With this long history,these disciplines have focused on ensuring the competence oftheir graduates, as modern society demands appropriate expertisefrom doctors and engineers before letting them practice their profession.In computing, however, professional accreditation startedin the last decades of the twentieth century only after computerscience, informatics, and information systems programs becamewidespread. At the same time, although competency-based learninghas existed for centuries, its growth in computing is relativelynew, resulting from recent curricular reports such as ComputingCurricula 2020, which have defined competency comprising knowledge,skills, and dispositions. In addition, demands are being placedon university programs to ensure their graduates are ready forentering and sustaining employment in the computing profession.This work explores the role of accreditation in the formationand development of professional competency in non-computingdisciplines worldwide, building on this understanding to see howcomputing accreditation bodies could play a similar role in computing.This work explores the role of accreditation in the formationand development of professional competency in non-computingdisciplines worldwide, building on this understanding to see howcomputing accreditation bodies could play a similar role in computing.Its recommendations are to incorporate competencies inall computing programs and future curricular guidelines; create competency-based models for computing programs; involve industryin identifying workplace competencies, and ensure accreditationbodies include competencies and their assessment in their standards
Threshold Phenomena in Random Constraint Satisfaction Problems
Despite much work over the previous decade, the Satisfiability Threshold
Conjecture remains open. Random k-SAT, for constant k >= 3,
is just one family of a large number
of constraint satisfaction problems that are conjectured to have exact
satisfiability thresholds, but for which the existence and location of these
thresholds has yet to be proven.
Of those problems for which we are able to prove
an exact satisfiability threshold, each seems to be fundamentally different
than random 3-SAT.
This thesis defines a new family of
constraint satisfaction problems with constant size
constraints and domains and which
contains problems that are NP-complete and a.s.\ have exponential
resolution complexity. All four of these properties hold for k-SAT, k >= 3,
and the
exact satisfiability threshold is not known for any constraint
satisfaction problem
that has all of these properties. For each problem in the
family defined in this
thesis, we determine
a value c such that c is an exact satisfiability threshold if a certain
multi-variable function has a unique maximum at a given point
in a bounded domain. We
also give numerical evidence that this latter condition holds.
In addition to studying the satisfiability threshold, this thesis
finds exact
thresholds for the efficient behavior of DPLL using the unit clause heuristic
and a variation of the generalized unit clause heuristic,
and this thesis proves an analog
of a conjecture on the satisfiability of (2+p)-SAT.
Besides having similar properties as k-SAT, this new family of
constraint satisfaction problems
is interesting to study in its own right because it generalizes the
XOR-SAT problem and it has close ties
to quasigroups.Ph
On the uniformity of the approximation for -associated Stirling numbers of the second Kind
The -associated Stirling numbers of the second kind are a natural extension of Stirling numbers of the second kind. A combinatorial interpretation of -associated Stirling numbers of the second kind is the number of ways to partition elements into subsets such that each subset contains at least elements. Calculating the associated Stirling numbers is typically done with a recurrence relation or a generating function that are computationally expensive or alternatively with a closed-form that is practical for only a limited parameter range. In 1994 Hennecart proposed an approximation for the -associated Stirling numbers that is fast to compute, is amenable to analysis over a wide range of parameters, and is conjectured to be asymptotically tight. There are a few other approximations for the associated Stirling numbers, but none of them are as general as Hennecart's. However, until this work, Hennecart's approximation had been utilized without a proper justification due to the absence of a rigorous proof. This work provides a proof of the uniformity of the Hennecart approximation
Abstract
We investigate multi-source spanning tree problems where, given a graph with edge weights and a subset of the nodes defined as sources, the object is to find a spanning tree of the graph that minimizes some distance related cost metric. This problem can be used to model multicasting in a network where messages are sent from a fixed collection of senders and communication takes place along the edges of a single spanning tree. For a limited set of possible cost metrics of such a spanning tree, we either prove the problem is NP-hard or we demonstrate the existence of an efficient algorithm to find an optimal tree.