Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

An international cohort study of autosomal dominant tubulointerstitial kidney disease due to REN mutations identifies distinct clinical subtypes

There have been few clinical or scientific reports of autosomal dominant tubulointerstitial kidney disease due to REN mutations (ADTKD-REN), limiting characterization. To further study this, we formed an international cohort characterizing 111 individuals from 30 families with both clinical and laboratory findings. Sixty-nine individuals had a REN mutation in the signal peptide region (signal group), 27 in the prosegment (prosegment group), and 15 in the mature renin peptide (mature group). Signal group patients were most severely affected, presenting at a mean age of 19.7 years, with the prosegment group presenting at 22.4 years, and the mature group at 37 years. Anemia was present in childhood in 91% in the signal group, 69% prosegment, and none of the mature group. REN signal peptide mutations reduced hydrophobicity of the signal peptide, which is necessary for recognition and translocation across the endoplasmic reticulum, leading to aberrant delivery of preprorenin into the cytoplasm. REN mutations in the prosegment led to deposition of prorenin and renin in the endoplasmic reticulum Golgi intermediate compartment and decreased prorenin secretion. Mutations in mature renin led to deposition of the mutant prorenin in the endoplasmic reticulum, similar to patients with ADTKD-UMOD, with a rate of progression to end stage kidney disease (63.6 years) that was significantly slower vs. the signal (53.1 years) and prosegment groups (50.8 years) (significant hazard ratio 0.367). Thus, clinical and laboratory studies revealed subtypes of ADTKD-REN that are pathophysiologically, diagnostically, and clinically distinct

The complexity of approximately counting retractions to square-free graphs

A retraction is a homomorphism from a graph G to an induced subgraph H of G that is the identity on H. In a long line of research, retractions have been studied under various algorithmic settings. Recently, the problem of approximately counting retractions was considered. We give a complete trichotomy for the complexity of approximately counting retractions to all square-free graphs (graphs that do not contain a cycle of length 4). It turns out there is a rich and interesting class of graphs for which this problem is complete in the class #BIS. As retractions generalise homomorphisms, our easiness results extend to the important problem of approximately counting homomorphisms. By giving new #BIS-easiness results, we now settle the complexity of approximately counting homomorphisms for a whole class of non-trivial graphs that were previously unresolved

The complexity of approximately counting retractions

Let G be a graph that contains an induced subgraph H. A retraction from G to H is a homomorphism from G to H that is the identity function on H. Retractions are very well studied: Given H, the complexity of deciding whether there is a retraction from an input graph G to H is completely classified, in the sense that it is known for which H this problem is tractable (assuming P ≠ NP). Similarly, the complexity of (exactly) counting retractions from G to H is classified (assuming FP ≠ #P). However, almost nothing is known about approximately counting retractions. Our first contribution is to give a complete trichotomy for approximately counting retractions to graphs without short cycles. The result is as follows: (1) Approximately counting retractions to a graph H of girth at least 5 is in FP if every connected component of H is a star, a single looped vertex, or an edge with two loops. (2) Otherwise, if every component is an irreflexive caterpillar or a partially bristled reflexive path, then approximately counting retractions to H is equivalent to approximately counting the independent sets of a bipartite graph—a problem that is complete in the approximate counting complexity class RH Π 1. (3) Finally, if none of these hold, then approximately counting retractions to H is equivalent to approximately counting the satisfying assignments of a Boolean formula. Our second contribution is to locate the retraction counting problem for each H in the complexity landscape of related approximate counting problems. Interestingly, our results are in contrast to the situation in the exact counting context. We show that the problem of approximately counting retractions is separated both from the problem of approximately counting homomorphisms and from the problem of approximately counting list homomorphisms—whereas for exact counting all three of these problems are interreducible. We also show that the number of retractions is at least as hard to approximate as both the number of surjective homomorphisms and the number of compactions. In contrast, exactly counting compactions is the hardest of all of these exact counting problems

The power of the combined basic linear programming and affine relaxation for promise constraint satisfaction problems

In the field of constraint satisfaction problems (CSPs), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: “strict” and “weak,” and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not being able to satisfy all the weak constraints. The most commonly cited example of a promise CSP is the approximate graph coloring problem-which has recently seen exciting progress [Bulín, Krokhin, and Oprs̆al, Proceedings of the Symposium on Theory of Computing, 2019, pp. 602--613 and Wrochna and Živný, Proceedings of the Symposium on Discrete Algorithms, 2020, pp. 1426--1435] benefiting from a systematic algebraic approach to promise CSPs based on “polymorphisms,” operations that map tuples in the strict form of each constraint to tuples in the corresponding weak form. In this work, we present a simple algorithm which in polynomial time solves the decision problem for all promise CSPs that admit infinitely many symmetric polymorphisms, which are invariant under arbitrary coordinate permutations. This generalizes previous work of the first two authors [Brakensiek and Guruswami, Proceedings of the Symposium on Discrete Algorithms, 2019, pp. 436--455]. We also extend this algorithm to a more general class of block-symmetric polymorphisms. As a corollary, this single algorithm solves all polynomial-time tractable Boolean CSPs simultaneously. These results give a new perspective on Schaefer's classic dichotomy theorem and shed further light on how symmetries of polymorphisms enable algorithms. Finally, we show that block symmetric polymorphisms are not only sufficient but also necessary for this algorithm to work, thus establishing its precise power