106 research outputs found
Monotone Projection Lower Bounds from Extended Formulation Lower Bounds
In this short note, we reduce lower bounds on monotone projections of
polynomials to lower bounds on extended formulations of polytopes. Applying our
reduction to the seminal extended formulation lower bounds of Fiorini, Massar,
Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014;
J. ACM, 2017), we obtain the following interesting consequences.
1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size
projection of the permanent; this both rules out a natural attempt at a
monotone lower bound on the Boolean permanent, and shows that the permanent is
not complete for non-negative polynomials in VNP under monotone
p-projections.
2. The cut polynomials and the perfect matching polynomial (or "unsigned
Pfaffian") are not monotone p-projections of the permanent. The latter, over
the Boolean and-or semi-ring, rules out monotone reductions in one of the
natural approaches to reducing perfect matchings in general graphs to perfect
matchings in bipartite graphs.
As the permanent is universal for monotone formulas, these results also imply
exponential lower bounds on the monotone formula size and monotone circuit size
of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18;
Received: November 10, 2015, Revised: July 27, 2016, Published: December 22,
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Zero-one IP problems: Polyhedral descriptions & cutting plane procedures
A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
A Note on the Approximability of Deepest-Descent Circuit Steps
Linear programs (LPs) can be solved by polynomially many moves along the
circuit direction improving the objective the most, so-called deepest-descent
steps (dd-steps). Computing these steps is NP-hard (De Loera et al., arXiv,
2019), a consequence of the hardness of deciding the existence of an optimal
circuit-neighbor (OCNP) on LPs with non-unique optima.
We prove OCNP is easy under the promise of unique optima, but already
-approximating dd-steps remains hard even for totally
unimodular -dimensional 0/1-LPs with a unique optimum. We provide a matching
-approximation
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Combinatorial Optimization
Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations
Nash equilibria, gale strings, and perfect matchings
This thesis concerns the problem 2-NASH of ļ¬nding a Nash equilibrium of a bimatrix
game, for the special class of so-called āhard-to-solveā bimatrix games. The term āhardto-solveā relates to the exponential running time of the famous and often used Lemkeā
Howson algorithm for this class of games. The games are constructed with the help of
dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled
bitstrings deļ¬ned by the āGale evenness conditionā that characterise the vertices of these
polytopes.
We deļ¬ne the combinatorial problem āAnother completely labeled Gale stringā whose
solutions deļ¬ne the Nash equilibria of any game deļ¬ned by cyclic polytopes, including
the games where the LemkeāHowson algorithm takes exponential time. We show that
āAnother completely labeled Gale stringā is solvable in polynomial time by a reduction to
the āPerfect matchingā problem in Euler graphs. We adapt the LemkeāHowson algorithm
to pivot from one perfect matching to another and show that again for a certain class
of graphs this leads to exponential behaviour. Furthermore, we prove that completely
labeled Gale strings and perfect matchings in Euler graphs come in pairs and that the
LemkeāHowson algorithm connects two strings or matchings of opposite signs.
The equivalence between Nash Equilibria of bimatrix games derived from cyclic polytopes, completely labeled Gale strings, and perfect matchings in Euler Graphs implies that
counting Nash equilibria is #P-complete. Although one Nash equilibrium can be computed in polynomial time, we have not succeeded in ļ¬nding an algorithm that computes
a Nash equilibrium of opposite sign. However, we solve this problem for certain special cases, for example planar graphs. We illustrate the difļ¬culties concerning a general
polynomial-time algorithm for this problem by means of negative results that demonstrate
why a number of approaches towards such an algorithm are unlikely to be successful
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