69,423 research outputs found
Sensitivity analysis of the greedy heuristic for binary knapsack problems
Greedy heuristics are a popular choice of heuristics when we have to solve a large variety of NP -hard combinatorial problems. In particular for binary knapsack problems, these heuristics generate good results. If some uncertainty exists beforehand regarding the value of any one element in the problem data, sensitivity analysis procedures can be used to know the tolerance limits within which the value may vary will not cause changes in the output. In this paper we provide a polynomial time characterization of such limits for greedy heuristics on two classes of binary knapsack problems, namely the 0-1 knapsack problem and the subset sum problem. We also study the relation between algorithms to solve knapsack problems and algorithms to solve their sensitivity analysis problems, the conditions under which the sensitivity analysis of the heuristic generates bounds for the toler-ance limits for the optimal solutions, and the empirical behavior of the greedy output when there is a change in the problem data.
More on quasi-random graphs, subgraph counts and graph limits
We study some properties of graphs (or, rather, graph sequences) defined by
demanding that the number of subgraphs of a given type, with vertices in
subsets of given sizes, approximatively equals the number expected in a random
graph. It has been shown by several authors that several such conditions are
quasi-random, but that there are exceptions. In order to understand this
better, we investigate some new properties of this type. We show that these
properties too are quasi-random, at least in some cases; however, there are
also cases that are left as open problems, and we discuss why the proofs fail
in these cases.
The proofs are based on the theory of graph limits; and on the method and
results developed by Janson (2011), this translates the combinatorial problem
to an analytic problem, which then is translated to an algebraic problem.Comment: 35 page
Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
We establish the existence of free energy limits for several combinatorial
models on Erd\"{o}s-R\'{e}nyi graph and
random -regular graph . For a variety of models, including
independent sets, MAX-CUT, coloring and K-SAT, we prove that the free energy
both at a positive and zero temperature, appropriately rescaled, converges to a
limit as the size of the underlying graph diverges to infinity. In the zero
temperature case, this is interpreted as the existence of the scaling limit for
the corresponding combinatorial optimization problem. For example, as a special
case we prove that the size of a largest independent set in these graphs,
normalized by the number of nodes converges to a limit w.h.p. This resolves an
open problem which was proposed by Aldous (Some open problems) as one of his
six favorite open problems. It was also mentioned as an open problem in several
other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999
(Canterbury) (1999) 239-298 Cambridge Univ. Press]; Bollob\'{a}s and Riordan
[Random Structures Algorithms 39 (2011) 1-38]; Janson and Thomason [Combin.
Probab. Comput. 17 (2008) 259-264] and Aldous and Steele [In Probability on
Discrete Structures (2004) 1-72 Springer].Comment: Published in at http://dx.doi.org/10.1214/12-AOP816 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Quantum SDP-Solvers: Better upper and lower bounds
Brand\~ao and Svore very recently gave quantum algorithms for approximately
solving semidefinite programs, which in some regimes are faster than the
best-possible classical algorithms in terms of the dimension of the problem
and the number of constraints, but worse in terms of various other
parameters. In this paper we improve their algorithms in several ways, getting
better dependence on those other parameters. To this end we develop new
techniques for quantum algorithms, for instance a general way to efficiently
implement smooth functions of sparse Hamiltonians, and a generalized
minimum-finding procedure.
We also show limits on this approach to quantum SDP-solvers, for instance for
combinatorial optimizations problems that have a lot of symmetry. Finally, we
prove some general lower bounds showing that in the worst case, the complexity
of every quantum LP-solver (and hence also SDP-solver) has to scale linearly
with when , which is the same as classical.Comment: v4: 69 pages, small corrections and clarifications. This version will
appear in Quantu
Spectral statistics for unitary transfer matrices of binary graphs
Quantum graphs have recently been introduced as model systems to study the
spectral statistics of linear wave problems with chaotic classical limits. It
is proposed here to generalise this approach by considering arbitrary, directed
graphs with unitary transfer matrices. An exponentially increasing contribution
to the form factor is identified when performing a diagonal summation over
periodic orbit degeneracy classes. A special class of graphs, so-called binary
graphs, is studied in more detail. For these, the conditions for periodic orbit
pairs to be correlated (including correlations due to the unitarity of the
transfer matrix) can be given explicitly. Using combinatorial techniques it is
possible to perform the summation over correlated periodic orbit pair
contributions to the form factor for some low--dimensional cases. Gradual
convergence towards random matrix results is observed when increasing the
number of vertices of the binary graphs.Comment: 18 pages, 8 figure
Infinite combinatorial issues raised by lifting problems in universal algebra
The critical point between varieties A and B of algebras is defined as the
least cardinality of the semilattice of compact congruences of a member of A
but of no member of B, if it exists. The study of critical points gives rise to
a whole array of problems, often involving lifting problems of either diagrams
or objects, with respect to functors. These, in turn, involve problems that
belong to infinite combinatorics. We survey some of the combinatorial problems
and results thus encountered. The corresponding problematic is articulated
around the notion of a k-ladder (for proving that a critical point is large),
large free set theorems and the classical notation (k,r,l){\to}m (for proving
that a critical point is small). In the middle, we find l-lifters of posets and
the relation (k, < l){\to}P, for infinite cardinals k and l and a poset P.Comment: 22 pages. Order, to appea
Limits of Order Types
The notion of limits of dense graphs was invented, among other reasons, to attack problems in extremal graph theory. It is straightforward to define limits of order types in analogy with limits of graphs, and this paper examines how to adapt to this setting two approaches developed to study limits of dense graphs.
We first consider flag algebras, which were used to open various questions on graphs to mechanical solving via semidefinite programming. We define flag algebras of order types, and use them to obtain, via the semidefinite method, new lower bounds on the density of 5- or 6-tuples in convex position in arbitrary point sets, as well as some inequalities expressing the difficulty of sampling order types uniformly.
We next consider graphons, a representation of limits of dense graphs that enable their study by continuous probabilistic or analytic methods. We investigate how planar measures fare as a candidate analogue of graphons for limits of order types. We show that the map sending a measure to its associated limit is continuous and, if restricted to uniform measures on compact convex sets, a homeomorphism. We prove, however, that this map is not surjective. Finally, we examine a limit of order types similar to classical constructions in combinatorial geometry (Erdos-Szekeres, Horton...) and show that it cannot be represented by any somewhere regular measure; we analyze this example via an analogue of Sylvester\u27s problem on the probability that k random points are in convex position
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