Final Report Summary - APPROXNP (Approximation of NP-hard optimization problems)
A major part of this project has been to study maximum constraint satisfaction problems (Max-CSPs). In such a problem we are given a list of constraints where each constraint is the same predicate P applied to a sequence of literals, i.e. variables or negations of variabels. The computational problem is to find an assignment that satisfies the maximal number of contraints. The question is how the choice of the predicate P affects the computational difficulty of this problem. The traditional interpretation of difficult has been approximation resistant which says that if we only care about satisfying P then there is no efficient algorithm that does significantly better than just giving random values to the variables. We have introduced the notion of uselessness which essentially says that the same is true even if we care about any other property.
We feel that uselessness is a more natural concept than (hereditarily) approximation resistance to measure the difficulty of a Max-CSP. The former is now completely characterized (assuming the Unique Games Conjecture (UGC)).
If we do not want to rely on the UGC more open question remain but progress has been substantial. The result of Håstad proving that Max-not-2 is approximation resistant on satisfiable instances completed our understanding of arity three Boolean predicates.
The very recent results by Wenner gave first UGC-free uselessness results for predicates without negation. The results from 2013 by Huang defined the sparest known predicate known to be useless on satisfiable instances. The results by Austrin, Manokaran and Wenner from the same year gave the first UGC-free approximation resistance results for ordering predicates.
Essentially each of these papers introduced new techniques and hence is likely to give a more substantial contribution than the result itself.
Of the other major goals of the project Mömke and Svensson early on made a significant contribution to understanding the approximability of TSP. Inititally we wanted to understand the asymmetric variant while the progress was on the symmetric version in the case of graph-distances. It is interesting to note that Svensson continued to work on the problem and managed to get results for the graph-distance case also in the asymmetric case. This is very recent so this result was obtained after he left the project and joined EPFL as an assistant professor.
We feel that uselessness is a more natural concept than (hereditarily) approximation resistance to measure the difficulty of a Max-CSP. The former is now completely characterized (assuming the Unique Games Conjecture (UGC)).
If we do not want to rely on the UGC more open question remain but progress has been substantial. The result of Håstad proving that Max-not-2 is approximation resistant on satisfiable instances completed our understanding of arity three Boolean predicates.
The very recent results by Wenner gave first UGC-free uselessness results for predicates without negation. The results from 2013 by Huang defined the sparest known predicate known to be useless on satisfiable instances. The results by Austrin, Manokaran and Wenner from the same year gave the first UGC-free approximation resistance results for ordering predicates.
Essentially each of these papers introduced new techniques and hence is likely to give a more substantial contribution than the result itself.
Of the other major goals of the project Mömke and Svensson early on made a significant contribution to understanding the approximability of TSP. Inititally we wanted to understand the asymmetric variant while the progress was on the symmetric version in the case of graph-distances. It is interesting to note that Svensson continued to work on the problem and managed to get results for the graph-distance case also in the asymmetric case. This is very recent so this result was obtained after he left the project and joined EPFL as an assistant professor.