Hybrid quantum-classical algorithms such as the quantum approximate optimization algorithm (QAOA) are considered one of the most promising approaches for leveraging near-term quantum computers for practical applications. Such algorithms are often implemented in a variational form, combining classical optimization methods with a quantum machine to find parameters to maximize performance. The quality of the QAOA solution depends heavily on quality of the parameters produced by the classical optimizer. Moreover, multiple local optima in the space of parameters make it harder for the classical optimizer. In this paper we study the use of a multistart optimization approach within a QAOA framework to improve the performance of quantum machines on important graph clustering problems. We also demonstrate that reusing the optimal parameters from similar problems can improve the performance of classical optimization methods, expanding on similar results for MAXCUT.
@cite_38 show that for MAXCUT on 3-regular graphs, the objective function value is concentrated; that is, typical instances have nearly the same value of objective function. They make a case that the same holds for any combinatorial search problem where the number of clauses with a given variable is bounded (e.g., MAXCUT on a bounded-degree graph). They propose reusing optimal parameters between problems that come from the same distribution and refining them using a local optimization heuristic. In this work, we successfully apply this strategy to modularity clustering, a problem where the number of clauses in which a variable can appear grows with @math (see Sec. ).
@cite_38
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In (2015), we proposed a graph-based formulation that links and clusters person hypotheses over time by solving a minimum cost subgraph multicut problem. In this paper, we modify and extend (2015) in three ways: (1) We introduce a novel local pairwise feature based on local appearance matching that is robust to partial occlusion and camera motion. (2) We perform extensive experiments to compare different pairwise potentials and to analyze the robustness of the tracking formulation. (3) We consider a plain multicut problem and remove outlying clusters from its solution. This allows us to employ an efficient primal feasible optimization algorithm that is not applicable to the subgraph multicut problem of (2015). Unlike the branch-and-cut algorithm used there, this efficient algorithm used here is applicable to long videos and many detections. Together with the novel pairwise feature, it eliminates the need for the intermediate tracklet representation of (2015). We demonstrate the effectiveness of our overall approach on the MOT16 benchmark ( 2016), achieving state-of-art performance. We initiate a study of when the value of mathematical relaxations such as linear and semi-definite programs for constraint satisfaction problems (CSPs) is approximately preserved when restricting the instance to a sub-instance induced by a small random subsample of the variables. Let C be a family of CSPs such as 3SAT, Max-Cut, etc., and let Π be a mathematical program that is a relaxation for C, in the sense that for every instance P ∈ C, Π(P) is a number in [0, 1] upper bounding the maximum fraction of satisfiable constraints of P. Loosely speaking, we say that subsampling holds for C and Π if for every sufficiently dense instance P ∈ C and every e > 0, if we let P' be the instance obtained by restricting P to a sufficiently large constant number of variables, then Π(P') ∈ (1 ± e)Π(P). We say that weak subsampling holds if the above guarantee is replaced with Π(P') = 1 − θ(γ) whenever Π(P) = 1 − γ, where θ hides only absolute constants. We obtain both positive and negative results, showing that: 1. Subsampling holds for the BasicLP and BasicSDP programs. BasicSDP is a variant of the semi-definite program considered by Raghavendra (2008), who showed it gives an optimal approximation factor for every constraint-satisfaction problem under the unique games conjecture. BasicLP is the linear programming analog of BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of unique games type. 3. There are non-unique CSPs for which even weak subsampling fails for the above tighter semi-definite programs. Also there are unique CSPs for which (even weak) subsampling fails for the Sherali-Adams linear programming hierarchy. As a corollary of our weak subsampling for strong semi-definite programs, we obtain a polynomial-time algorithm to certify that random geometric graphs (of the type considered by Feige and Schechtman, 2002) of max-cut value 1 − γ have a cut value at most 1 − γ 10. More generally, our results give an approach to obtaining average-case algorithms for CSPs using semi-definite programming hierarchies. We introduce a quantum algorithm that produces approximate solutions for combinatorial optimization problems. The algorithm depends on a positive integer p and the quality of the approximation improves as p is increased. The quantum circuit that implements the algorithm consists of unitary gates whose locality is at most the locality of the objective function whose optimum is sought. The depth of the circuit grows linearly with p times (at worst) the number of constraints. If p is fixed, that is, independent of the input size, the algorithm makes use of efficient classical preprocessing. If p grows with the input size a different strategy is proposed. We study the algorithm as applied to MaxCut on regular graphs and analyze its performance on 2-regular and 3-regular graphs for fixed p. For p = 1, on 3-regular graphs the quantum algorithm always finds a cut that is at least 0.6924 times the size of the optimal cut. We investigate a natural online version of the well-known Maximum Directed Cut problem on DAGs. We propose a deterministic algorithm and show that it achieves a competitive ratio of @math . We then give a lower bound argument to show that no deterministic algorithm can achieve a ratio of @math for any ?>0 thus showing that our algorithm is essentially optimal. Then, we extend our technique to improve upon the analysis of an old result: we show that greedily derandomizing the trivial randomized algorithm for MaxDiCut in general graphs improves the competitive ratio from 4 to 3, and also provide a tight example.