In the Equal-Subset-Sum problem, we are given a set @math of @math integers and the problem is to decide if there exist two disjoint nonempty subsets @math , whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in @math time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give @math worst case Monte Carlo algorithm. This answers the open problem from Woeginger's inspirational survey. Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in @math time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in @math time and polynomial space.
has some connections to the study of the structure of DNA molecules @cite_15 @cite_19 @cite_26 . considered @math - , in which we need to find @math disjoint subsets of a given set with the same sum. They obtained several algorithms that depend on certain restrictions of the sets (e.g., small cardinality of a solution). In the following work, considered other variants of and proved their NP-hardness.
@cite_19 @cite_15 @cite_26
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The EQUAL SUM SUBSETS problem, where we are given a set of positive integers and we ask for two nonempty disjoint subsets such that their elements add up to the same total, is known to be NP-hard. In this paper we give (pseudo-)polynomial algorithms and or (strong) NP-hardness proofs for several natural variations of EQUAL SUM SUBSETS. Among others we present (i) a framework for obtaining NP-hardness proofs and pseudopolynomial time algorithms for EQUAL SUM SUBSETS variations, which we apply to variants of the problem with additional selection restrictions, (ii) a proof of NP-hardness and a pseudo-polynomial time algorithm for the case where we ask for two subsets such that the ratio of their sums is some fixed rational r > 0, (iii) a pseudo-polynomial time algorithm for finding k subsets of equal sum, with k = O(1), and a proof of strong NP-hardness for the same problem with k = Ω(n), (iv) algorithms and hardness results for finding k equal sum subsets under the additional requirement that the subsets should be of equal cardinality. Our results are a step towards determining the dividing lines between polynomial time solvability, pseudo-polynomial time solvability, and strong NP-completeness of subset-sum related problems. Motivated by the error behavior in the DNA storage channel, in this paper, we extend the previously studied sequence reconstruction problem by Levenshtein. The reconstruction problem studies the model in which the information is read through multiple noisy channels, and the decoder, which receives all channel estimations, is required to decode the information. For the combinatorial setup, the assumption is that all the channels cause at most some @math errors. Levenshtein considered the case in which all the channels have the same behavior, and we generalize this model and assume that the channels are not identical. Thus, different channels may cause different maximum numbers of errors. For example, we assume that there are @math channels, which cause at most @math or @math errors, where @math t_ 1 @math pN @math 0 . If the information codeword belongs to a code with minimum distance @math , the problem is then to find the minimum number of channels that guarantees successful decoding in the worst case. A different problem we study in this paper is where the number of channels is fixed, and the question is finding the minimum distance @math that provides exact reconstruction. We study these problems and show how to apply them for the cases of substitutions and transpositions. The molecule problem is that of determining the coordinates of a set of points in space from a (usually sparse) set of pairwise distance measurements. As its name implies, it has applications in the determination of molecular conformation. Unfortunately, the molecule problem is NP-hard. We present an approach to the molecule problem that uses a very specialized divide-and-conquer technique. Instead of solving a single large problem we try to solve a sequence of smaller, presumably easier ones. These smaller problems consist of subsets of points whose relative locations can be determined uniquely. Once such a subset is positioned, its points can collectively be treated as a rigid body. This can greatly reduce the number of degrees of freedom in the problem. Identifying subsets of points whose relative locations can be uniquely determined requires exploiting some very special structure inherent in the molecule problem. We reduce this identification to a purely combinatoric characterization that ignores the actual distances. We develop necessary graph theoretic conditions for a set of points to have a unique solution, along with efficient algorithms to find subgraphs with these properties. These characterizations and algorithms combine ideas from matching theory, differential topology and matrix computations. These ideas have been implemented in ABBIE, a program to solve three-dimensional instances of the molecule problem. ABBIE combines the recursive decomposition described above with a nonlinear global optimizer to perform the coordinate determinations. Details of this implementation are described, and numerical results of simulated chemical data are presented. In a packing integer program, we are given a matrix @math and column vectors @math with nonnegative entries. We seek a vector @math of nonnegative integers, which maximizes @math subject to @math The edge and vertex-disjoint path problems together with their unsplittable flow generalization are NP-hard problems with a multitude of applications in areas such as routing, scheduling and bin packing. These two categories of problems are known to be conceptually related, but this connection has largely been ignored in terms of approximation algorithms. We explore the topic of approximating disjoint-path problems using polynomial-size packing integer programs. Motivated by the disjoint paths applications, we introduce the study of a class of packing integer programs, called column-restricted. We develop improved approximation algorithms for column-restricted programs, a result that we believe is of independent interest. Additional approximation algorithms for disjoint-paths are presented that are simple to implement and achieve good performance when the input has a special structure.