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

Body indices and basic vital signs in Helicobacter pylori positive and negative persons

It has been hypothesized that Helicobacter pylori (Hp) infection may contribute to reduced stature, risk of hypertension or obesity. The aim was to evaluate body indices in Hp positive and negative persons. A total of 2436 subjects (4–100 years old) were tested for Hp status by 13Curea breath test. Data on height and weight were collected for 84%, and blood pressure for 80% of the study subjects. The prevalence of Hp infection was 41.6%. The odds ratio for a 10-year increase in age was 1.21 (95% CI 1.17–1.25, p-value <0.001). Statistically significant negative association of Hp positivity with body height was most pronounced in the younger age groups, while a positive association of Hp positivity with body mass index was only seen in those aged 15+ years. There was a negative effect of Hp positivity on systolic and diastolic blood pressure in subjects below 25 and a relatively strong positive effect on blood pressure in subjects over 65 years. Residual confounding by social characteristics as a possible explanation for the associations of Hp positivity with height and blood pressure cannot be excluded. Unmeasured factors related to social and family environment may cause the apparent association between Hp positivity and children’s growth and blood pressure

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

Beyond PCSP(1-in-3,NAE)

The promise constraint satisfaction problem (PCSP) is a recently introduced vast generalisation of the constraint satisfaction problem (CSP) that captures approximability of satisfiable instances. A PCSP instance comes with two forms of each constraint: a strict one and a weak one. Given the promise that a solution exists using the strict constraints, the task is to find a solution using the weak constraints. While there are by now several dichotomy results for fragments of PCSPs, they all consider (in some way) symmetric PCSPs. 1-in-3-SAT and Not-All-Equal-3-SAT are classic examples of Boolean symmetric (non-promise) CSPs. While both problems are NP-hard, Brakensiek and Guruswami showed [SODA'18] that given a satisfiable instance of 1-in-3-SAT one can find a solution to the corresponding instance of (weaker) Not-All-Equal-3-SAT. In other words, the PCSP template (-in-,NAE) is tractable. We focus on non-symmetric PCSPs. In particular, we study PCSP templates obtained from the Boolean template (-in-, NAE) by either adding tuples to -in- or removing tuples from NAE. For the former, we classify all templates as either tractable or not solvable by the currently strongest known algorithm for PCSPs, the combined basic LP and affine IP relaxation of Brakensiek and Guruswami [SODA'20]. For the latter, we classify all templates as either tractable or NP-hard.</p

The complexity of Valued CSPs

We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems (VCSPs). We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify general features of the complexity landscape

Approximate counting CSP seen from the other side

In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP(C, −), in which the goal is, given a relational structure A from a class C of structures and an arbitrary structure B, to find the number of homomorphisms from A to B. Flum and Grohe showed that #CSP(C, −) is solvable in polynomial time if C has bounded treewidth [FOCS’02]. Building on the work of Grohe [JACM’07] on decision CSPs, Dalmau and Jonsson then showed that, if C is a recursively enumerable class of relational structures of bounded arity, then assuming FPT , #W[1], there are no other cases of #CSP(C, −) solvable exactly in polynomial time (or even fixed-parameter time) [TCS’04].We show that, assuming FPT , W[1] (under randomised parameterised reductions) and for C satisfying certain general conditions, #CSP(C, −) is not solvable even approximately for C of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP(C, −). In particular, our condition generalises the case when C is closed under taking minors.</p