On the Complexity of Binary Polynomial Optimization Over Acyclic Hypergraphs

Abstract

In this work, we advance the understanding of the fundamental limits of computation for binary polynomial optimization (BPO), which is the problem of maximizing a given polynomial function over all binary points. In our main result we provide a novel class of BPO that can be solved efficiently both from a theoretical and computational perspective. In fact, we give a strongly polynomial-time algorithm for instances whose corresponding hypergraph is $$\beta $$ β -acyclic. We note that the $$\beta $$ β -acyclicity assumption is natural in several applications including relational database schemes and the lifted multicut problem on trees. Due to the novelty of our proving technique, we obtain an algorithm which is interesting also from a practical viewpoint. This is because our algorithm is very simple to implement and the running time is a polynomial of very low degree in the number of nodes and edges of the hypergraph. Our result completely settles the computational complexity of BPO over acyclic hypergraphs, since the problem is NP-hard on $$\alpha $$ α -acyclic instances. Our algorithm can also be applied to any general BPO problem that contains $$\beta $$ β -cycles. For these problems, the algorithm returns a smaller instance together with a rule to extend any optimal solution of the smaller instance to an optimal solution of the original instance.

Simple odd $$\beta $$-cycle inequalities for binary polynomial optimization

“…P(x) is the nonlinear part, consisting of a polynomial of degree four, which incentivizes smoothness by penalizing 2 × 2 sub-images of the output image, the more they look like a checkerboard. We considered seven different sizes ( , h) ∈ {(10, 10), (10,15), (15,15), (15,20), (20,20), (20,25), (25,25)}. For each size, we considered each of the three different base types and each of the three perturbations, where for "all 5%" and "0 s 50%" we created two random instances, resulting in a total of 3 • 5 = 15 instances for each size.…”

“…, 7 8 N , N }, respectively. The third set of instances that we considered are generated randomly, as it is commonly done in the literature [5,7,9,10,15]. We chose a setting similar to the one used in [7,9,10], where we fix the number of nodes |V | and of edges |E| of the hypergraph representing the instance, and we also fix the minimum cardinality of an edge of the hypergraph, which we denote by d. For every edge, the probability that its cardinality is equal to c, for c ∈ {d, .…”

“…We also make sure that there are no parallel edges in the produced hypergraph. We ) in our probabilities is not present in [7,9,10], but is added here because the cardinality c of the edge can be at most |V |. Another difference with the setting examined in [7,9,10] is that in these papers only the case d = 2 is considered.…”

A polynomial-size extended formulation for the multilinear polytope of beta-acyclic hypergraphs

“…As we mentioned before, α-acyclic hypergraphs are the most general type of acyclic hypergraphs. In [14], the authors prove that Problem BMO is strongly NP-hard over α-acyclic hypergraphs. This result implies that, unless P = NP, one cannot construct, in polynomial time, a polynomial-size extended formulation for the multilinear polytope of α-acyclic hypergraphs.…”