On the representation of polyhedra by polynomial inequalities

A beautiful result of Br\"ocker and Scheiderer on the stability index of basic closed semi-algebraic sets implies, as a very special case, that every $d$-dimensional polyhedron admits a representation as the set of solutions of at most $d(d+1)/2$ polynomial inequalities. Even in this polyhedral case, however, no constructive proof is known, even if the quadratic upper bound is replaced by any bound depending only on the dimension. Here we give, for simple polytopes, an explicit construction of polynomials describing such a polytope. The number of used polynomials is exponential in the dimension, but in the 2- and 3-dimensional case we get the expected number $d(d+1)/2$.Comment: 19 pages, 4 figures; revised version with minor changes proposed by the referee

Flinders Petrie, the Travelling Salesman Problem, and the Beginning of Mathematical Modeling in Archaeology

Abstract. This article describes one of the first attempts to use mathematical modeling and optimization in archaeology. William Matthew Flinders Petrie (1853-1942, eminent British archaeologist, excavating a large graveyard at Naqada in Upper Egypt suggested in his article "Sequences in Prehistoric Remains" [17] to employ a "distance function" to describe the "closeness of graves in time". Petrie's grave distance is known today as Hamming metric, based on which he proposed to establish the chronology of the graves, i.e., the correct sequence of points in time when the graves were built (briefly called seriation). He achieved this by solving a graph theoretic problem which is called weighted Hamiltonian path problem today and is, of course, equivalent to the symmetric travelling salesman problem. This paper briefly sketches a few aspects of Petrie's biographical background and evaluates the significance of seriation. 2010 Mathematics Subject Classification: 01A55, 05-03, 90-03, 90C2

Variable and value elimination in binary constraint satisfaction via forbidden patterns

Variable or value elimination in a constraint satisfaction problem (CSP) can be used in preprocessing or during search to reduce search space size. A variable elimination rule (value elimination rule) allows the polynomial-time identification of certain variables (domain elements) whose elimination, without the introduction of extra compensatory constraints, does not affect the satisfiability of an instance. We show that there are essentially just four variable elimination rules and three value elimination rules defined by forbidding generic sub-instances, known as irreducible existential patterns, in arc-consistent CSP instances. One of the variable elimination rules is the already-known Broken Triangle Property, whereas the other three are novel. The three value elimination rules can all be seen as strict generalisations of neighbourhood substitution.Comment: A full version of an IJCAI'13 paper to appear in Journal of Computer and System Sciences (JCSS

Extended formulations from communication protocols in output-efficient time

Deterministic protocols are well-known tools to obtain extended formulations, with many applications to polytopes arising in combinatorial optimization. Although constructive, those tools are not output-efficient, since the time needed to produce the extended formulation also depends on the number of rows of the slack matrix (hence, on the exact description in the original space). We give general sufficient conditions under which those tools can be implemented as to be output-efficient, showing applications to e.g.~Yannakakis' extended formulation for the stable set polytope of perfect graphs, for which, to the best of our knowledge, an efficient construction was previously not known. For specific classes of polytopes, we give also a direct, efficient construction of extended formulations arising from protocols. Finally, we deal with extended formulations coming from unambiguous non-deterministic protocols

Empfehlungen zur Zukunft des wissenschaftlichen Publikationssystems

Ash M, Carrier M, Dössel O, et al. Empfehlungen zur Zukunft des wissenschaftlichen Publikationssystems. Berlin: Berlin-Brandenburgische Akademie der Wissenschaften; 2015

Tractability in Constraint Satisfaction Problems: A Survey

International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP