# The complexity of finite-valued CSPs

## Stanislav Živný (Warwick)

When: June 24, 2pm

Where: room G2.91b/G215

**Abstract**

Let L be a set of rational-valued functions on a fixed finite domain; such a set is called a finite-valued constraint language. We are interested in the problem of minimising a function given explicitly as a sum of functions from L. We establish a dichotomy theorem with respect to exact solvability for all finite-valued languages defined on domains of arbitrary finite size. We present a simple algebraic condition that characterises the tractable cases. Moreover, we show that a single algorithm based on linear programming solves all tractable cases. Furthermore, we show that there is a single reason for intractability; namely, a very specific reduction from Max-Cut.

Based on work published at FOCS'12 and STOC'13, joint work with J. Thapper.