Equal Sum Sequences and Imbalance Sets of Tournaments

Reid conjectured that any finite set of non-negative integers is the score set of some tournament and Yao gave a non-constructive proof of Reid's conjecture using arithmetic arguments. No constructive proof has been found since. In this paper, we investigate a related problem, namely, which sets of integers are imbalance sets of tournaments. We completely solve the tournament imbalance set problem (TIS) and also estimate the minimal order of a tournament realizing an imbalance set. Our proofs are constructive and provide a pseudo-polynomial time algorithm to realize any imbalance set. Along the way, we generalize the well-known equal sum subsets problem (ESS) to define the equal sum sequences problem (ESSeq) and show it to be NP-complete. We then prove that ESSeq reduces to TIS and so, due to the pseudo-polynomial time complexity, TIS is weakly NP-complete.Comment: Presented at the Retrospective Workshop on Discrete Geometry, Optimization and Symmetry, 25-29 Nov 2013, The Fields Institute, Toronto, Canad

Multiple Subset Problem as an encryption scheme for communication

Using well-known mathematical problems for encryption is a widely used technique because they are computationally hard and provide security against potential attacks on the encryption method. The subset sum problem (SSP) can be defined as finding a subset of integers from a given set, whose sum is equal to a specified integer. The classic SSP has various variants, one of which is the multiple-subset problem (MSSP). In the MSSP, the goal is to select items from a given set and distribute them among multiple bins, en-suring that the capacity of each bin is not exceeded while maximizing the total weight of the selected items. This approach addresses a related problem with a different perspective. Here a related different kind of problem is approached: given a set of sets A={A1, A2..., An}, find an integer s for which every subset of the given sets is summed up to, if such an integer exists. The problem is NP-complete when considering it as a variant of SSP. However, there exists an algorithm that is relatively efficient for known pri-vate keys. This algorithm is based on dispensing non-relevant values of the potential sums. In this paper we present the encryption scheme based on MSSP and present its novel usage and implementation in communication

Subset sum problems with digraph constraints

We introduce and study optimization problems which are related to the well-known Subset Sum problem. In each new problem, a node-weighted digraph is given and one has to select a subset of vertices whose total weight does not exceed a given budget. Some additional constraints called digraph constraints and maximality need to be satisfied. The digraph constraint imposes that a node must belong to the solution if at least one of its predecessors is in the solution. An alternative of this constraint says that a node must belong to the solution if all its predecessors are in the solution. The maximality constraint ensures that no superset of a feasible solution is also feasible. The combination of these constraints provides four problems. We study their complexity and present some approximation results according to the type of input digraph, such as directed acyclic graphs and oriented trees

Changing the focus: worker-centric optimization in human-in-the-loop computations

A myriad of emerging applications from simple to complex ones involve human cognizance in the computation loop. Using the wisdom of human workers, researchers have solved a variety of problems, termed as “micro-tasks” such as, captcha recognition, sentiment analysis, image categorization, query processing, as well as “complex tasks” that are often collaborative, such as, classifying craters on planetary surfaces, discovering new galaxies (Galaxyzoo), performing text translation. The current view of “humans-in-the-loop” tends to see humans as machines, robots, or low-level agents used or exploited in the service of broader computation goals. This dissertation is developed to shift the focus back to humans, and study different data analytics problems, by recognizing characteristics of the human workers, and how to incorporate those in a principled fashion inside the computation loop. The first contribution of this dissertation is to propose an optimization framework and a real world system to personalize worker’s behavior by developing a worker model and using that to better understand and estimate task completion time. The framework judiciously frames questions and solicits worker feedback on those to update the worker model. Next, improving workers skills through peer interaction during collaborative task completion is studied. A suite of optimization problems are identified in that context considering collaborativeness between the members as it plays a major role in peer learning. Finally, “diversified” sequence of work sessions for human workers is designed to improve worker satisfaction and engagement while completing tasks

ON THE COMPLEXITY OF VARIATIONS OF EQUAL SUM SUBSETS

Abstract. 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 (1) 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, (2) 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, (3) 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), (4) 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 subsetsum related problems. Algorithms and Problem Complexity] ACM CCS Categories and Subject Descriptors: F.2 [Analysis o