From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz

The next few years will be exciting as prototype universal quantum processors emerge, enabling implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation, and which have the potential to significantly expand the breadth of quantum computing applications. A leading candidate is Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the Quantum Alternating Operator Ansatz, is the consideration of general parametrized families of unitaries rather than only those corresponding to the time-evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.Comment: 51 pages, 2 figures. Revised to match journal pape

On the Complexity of Winner Determination and Strategic Control in Conditional Approval Voting

We focus on a generalization of the classic Minisum approval voting rule, introduced by Barrot and Lang (2016), and referred to as Conditional Minisum (CMS), for multi-issue elections with preferential dependencies. Under this rule, voters are allowed to declare dependencies between different issues, but the price we have to pay for this higher level of expressiveness is that we end up with a computationally hard rule. Motivated by this, we first focus on finding special cases that admit efficient algorithms for CMS. Our main result in this direction is that we identify the condition of bounded treewidth (of an appropriate graph, emerging from the provided ballots) as the necessary and sufficient condition for exact polynomial algorithms, under common complexity assumptions. We then move to the design of approximation algorithms. For the (still hard) case of binary issues, we identify natural restrictions on the voters' ballots, under which we provide the first multiplicative approximation algorithms for the problem. The restrictions involve upper bounds on the number of dependencies an issue can have on the others and on the number of alternatives per issue that a voter can approve. Finally, we also investigate the complexity of problems related to the strategic control of conditional approval elections by adding or deleting either voters or alternatives and we show that in most variants of these problems, CMS is computationally resistant against control. Overall, we conclude that CMS can be viewed as a solution that achieves a satisfactory tradeoff between expressiveness and computational efficiency, when we have a limited number of dependencies among issues, while at the same time exhibiting sufficient resistance to control

Exclusive graph searching vs. pathwidth

International audienceIn Graph Searching, a team of searchers aims at capturing an invisible fugitive moving arbitrarily fast in a graph. Equivalently, the searchers try to clear a contaminated network. The problem is to compute the minimum number of searchers required to accomplish this task. Several variants of Graph Searching have been studied mainly because of their close relationship with the pathwidth of a graph. Blin et al. defined the Exclusive Graph Searching where searchers cannot " jump " and no node can be occupied by more than one searcher. In this paper, we study the complexity of this new variant. We show that the problem is NP-hard in planar graphs with maximum degree 3 and it can be solved in linear-time in the class of cographs. We also show that monotone Exclusive Graph Searching is NP-complete in split graphs where Pathwidth is known to be solvable in polynomial time. Moreover, we prove that monotone Exclusive Graph Searching is in P in a subclass of star-like graphs where Pathwidth is known to be NP-hard. Hence, the computational complexities of monotone Exclusive Graph Searching and Pathwidth cannot be compared. This is the first variant of Graph Searching for which such a difference is proved

On primal-dual schema for the minimum satisfiability problem

Satisfiability problem is the first problem known to be NP-complete [8, 28]. In this thesis, we have studied the minimization version of the satisfiability problem called the MINSAT. Given a set of boolean variables and a set of clauses, such that each clause is a disjunction of variables, the goal is to find the boolean values of the variables so that minimum number of clauses are satisfied. We have used the concept of linear programming and the primal-dual method to study the problem. We have constructed the Linear program of the MINSAT and its restricted version. We have proposed two combinatorial methods to solve the dual of the restricted primal of the MINSAT. Further to this, these two algorithms also obtain an integral solution to the dual of the MINSAT problem. Lastly, we performed a comparison analysis of our proposed algorithms with the simplex method