134 research outputs found
On the Hardness of Almost All Subset Sum Problems by Ordinary Branch-and-Bound
Given positive integers , and a positive integer right
hand side , we consider the feasibility version of the subset sum
problem which is the problem of determining whether a subset of
adds up to . We show that if the right hand side
is chosen as for a constant and if the 's are independentand identically distributed from a
discrete uniform distribution taking values , then the probability that the instance of the subset sum problem
generated requires the creation of an exponential number of branch-and-bound
nodes when one branches on the individual variables in any order goes to as
goes to infinity.Comment: 5 page
Compressing Branch-and-Bound Trees
A branch-and-bound (BB) tree certifies a dual bound on the value of an
integer program. In this work, we introduce the tree compression problem (TCP):
Given a BB tree T that certifies a dual bound, can we obtain a smaller tree
with the same (or stronger) bound by either (1) applying a different
disjunction at some node in T or (2) removing leaves from T? We believe such
post-hoc analysis of BB trees may assist in identifying helpful general
disjunctions in BB algorithms. We initiate our study by considering
computational complexity and limitations of TCP. We then conduct experiments to
evaluate the compressibility of realistic branch-and-bound trees generated by
commonly-used branching strategies, using both an exact and a heuristic
compression algorithm
Improved Online Algorithm for Fractional Knapsack in the Random Order Model
The fractional knapsack problem is one of the classical problems in combinatorial optimization, which is well understood in the offline setting. However, the corresponding online setting has been handled only briefly in the theoretical computer science literature so far, although it appears in several applications. Even the previously best known guarantee for the competitive ratio was worse than the best known for the integral problem in the popular random order model. We show that there is an algorithm for the online fractional knapsack problem that admits a competitive ratio of 4.39. Our result significantly improves over the previously best known competitive ratio of 9.37 and surpasses the current best 6.65-competitive algorithm for the integral case. Moreover, our algorithm is deterministic in contrast to the randomized algorithms achieving the results mentioned above
Parallel Approximation and Integer Programming Reformulation
We show that in a knapsack feasibility problem an integral vector , which
is short, and near parallel to the constraint vector gives a branching
direction with small integer width.
We use this result to analyze two computationally efficient reformulation
techniques on low density knapsack problems. Both reformulations have a
constraint matrix with columns reduced in the sense of Lenstra, Lenstra, and
Lov\'asz. We prove an upper bound on the integer width along the last variable,
which becomes 1, when the density is sufficiently small.
In the proof we extract from the transformation matrices a vector which is
near parallel to the constraint vector The near parallel vector is a good
branching direction in the original knapsack problem, and this transfers to the
last variable in the reformulations
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