An Optimization-Based Sum-of-Squares Approach to Vizing's Conjecture

Vizing's conjecture (open since 1968) relates the sizes of dominating sets in two graphs to the size of a dominating set in their Cartesian product graph. In this paper, we formulate Vizing's conjecture itself as a Positivstellensatz existence question. In particular, we encode the conjecture as an ideal/polynomial pair such that the polynomial is nonnegative if and only if the conjecture is true. We demonstrate how to use semidefinite optimization techniques to computationally obtain numeric sum-of-squares certificates, and then show how to transform these numeric certificates into symbolic certificates approving nonnegativity of our polynomial. After outlining the theoretical structure of this computer-based proof of Vizing's conjecture, we present computational and theoretical results. In particular, we present exact low-degree sparse sum-of-squares certificates for particular families of graphs.

[1]  Shmuel Onn.  Nowhere-zero flow polynomials , 2004, J. Comb. Theory, Ser. A.

[2]  Rekha R. Thomas,et al.  Semidefinite Optimization and Convex Algebraic Geometry , 2012 .

[3]  Illya V. Hicks,et al.  An Algebraic Exploration of Dominating Sets and Vizing's Conjecture , 2012, Electron. J. Comb..

[4]  Klaus G. Fischer,et al.  Symmetric polynomials and Hall's theorem , 1988, Discret. Math..

[5]  R. Saigal,et al.  Handbook of semidefinite programming : theory, algorithms, and applications , 2000 .

[6]  Pravesh Kothari,et al.  A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[7]  Klaudia Frankfurter.  Computers And Intractability A Guide To The Theory Of Np Completeness , 2016 .

[8]  László Lovász,et al.  Stable sets and polynomials , 1994, Discret. Math..

[9]  V. G. Vizing.  SOME UNSOLVED PROBLEMS IN GRAPH THEORY , 1968 .

[10]  S. Onn,et al.  Expressing Combinatorial Optimization Problems by Systems of Polynomial Equations and the Nullstellensatz , 2007, 0706.0578.

[11]  Jesús A. De Loera,et al.  Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz , 2009, Combinatorics, Probability and Computing.

[12]  Donal O'Shea,et al.  Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.) , 1997, Undergraduate texts in mathematics.

[13]  Henry Wolkowicz,et al.  Handbook of Semidefinite Programming , 2000 .

[14]  Wayne Goddard,et al.  Vizing's conjecture: a survey and recent results , 2012, J. Graph Theory.

[15]  Andrew Packard,et al.  Control Applications of Sum of Squares Programming , 2005 .

[16]  Liang Sun.  A result on Vizing's conjecture , 2004, Discret. Math..

[17]  Martin Kreuzer,et al.  Computational Commutative Algebra 1 , 2000 .

[18]  W. Edwin Clark,et al.  Inequality Related to Vizing's Conjecture , 2000, Electron. J. Comb..

[19]  Rekha R. Thomas,et al.  Theta Bodies for Polynomial Ideals , 2008, SIAM J. Optim..

[20]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[21]  Manu Basavaraju,et al.  Acyclic edge coloring of 2-degenerate graphs , 2012, J. Graph Theory.

[22]  Noga Alon,et al.  Colorings and orientations of graphs , 1992, Comb..

[23]  Christopher J. Hillar,et al.  Algebraic characterization of uniquely vertex colorable graphs , 2008, J. Comb. Theory, Ser. B.

[24]  C. D. Boor,et al.  Polynomial interpolation in several variables , 1994 .

[25]  Dima Grigoriev,et al.  Complexity of Null-and Positivstellensatz proofs , 2001, Ann. Pure Appl. Log..

[26]  Pravesh Kothari,et al.  Quantum entanglement, sum of squares, and the log rank conjecture , 2017, Electron. Colloquium Comput. Complex..

[27]  M. Laurent.  Sums of Squares, Moment Matrices and Optimization Over Polynomials , 2009 .