63 research outputs found
Short proofs of the Kneser-LovĂĄsz coloring principle
We prove that propositional translations of the KneserâLovĂĄsz theorem have polynomial size extended Frege proofs and quasi-polynomial size Frege proofs for all fixed values of k.
We present a new counting-based combinatorial proof of the K neserâLovĂĄsz theorem based on the HiltonâMilner theorem; this avoids the topological arguments of prior proofs for all but finitely many base cases. We introduce new âtruncated Tucker lemmaâ principles, which are miniaturizations of the octahedral Tucker lemma. The truncated Tucker lemma implies the KneserâLovĂĄsz theorem. We show that the
k=1 case of the truncated Tucker lemma has polynomial size extended Frege proofs.Peer ReviewedPostprint (author's final draft
The (Generalized) Orthogonality Dimension of (Generalized) Kneser Graphs: Bounds and Applications
The orthogonality dimension of a graph over a field is
the smallest integer for which there exists an assignment of a vector with to every vertex , such that whenever and are
adjacent vertices in . The study of the orthogonality dimension of graphs is
motivated by various application in information theory and in theoretical
computer science. The contribution of the present work is two-folded.
First, we prove that there exists a constant such that for every
sufficiently large integer , it is -hard to decide whether the
orthogonality dimension of an input graph over is at most or
at least . At the heart of the proof lies a geometric result, which
might be of independent interest, on a generalization of the orthogonality
dimension parameter for the family of Kneser graphs, analogously to a
long-standing conjecture of Stahl (J. Comb. Theo. Ser. B, 1976).
Second, we study the smallest possible orthogonality dimension over finite
fields of the complement of graphs that do not contain certain fixed subgraphs.
In particular, we provide an explicit construction of triangle-free -vertex
graphs whose complement has orthogonality dimension over the binary field at
most for some constant . Our results involve
constructions from the family of generalized Kneser graphs and they are
motivated by the rigidity approach to circuit lower bounds. We use them to
answer a couple of questions raised by Codenotti, Pudl\'{a}k, and Resta (Theor.
Comput. Sci., 2000), and in particular, to disprove their Odd Alternating Cycle
Conjecture over every finite field.Comment: 19 page
Approximating the Orthogonality Dimension of Graphs and Hypergraphs
A t-dimensional orthogonal representation of a hypergraph is an assignment of nonzero vectors in R^t to its vertices, such that every hyperedge contains two vertices whose vectors are orthogonal. The orthogonality dimension of a hypergraph H, denoted by overline{xi}(H), is the smallest integer t for which there exists a t-dimensional orthogonal representation of H. In this paper we study computational aspects of the orthogonality dimension of graphs and hypergraphs. We prove that for every k >= 4, it is NP-hard (resp. quasi-NP-hard) to distinguish n-vertex k-uniform hypergraphs H with overline{xi}(H) = Omega(log^delta n) for some constant delta>0 (resp. overline{xi}(H) >= Omega(log^{1-o(1)} n)). For graphs, we relate the NP-hardness of approximating the orthogonality dimension to a variant of a long-standing conjecture of Stahl. We also consider the algorithmic problem in which given a graph G with overline{xi}(G) <= 3 the goal is to find an orthogonal representation of G of as low dimension as possible, and provide a polynomial time approximation algorithm based on semidefinite programming
Sparse Kneser graphs are Hamiltonian
For integers and , the Kneser graph is the
graph whose vertices are the -element subsets of and whose
edges connect pairs of subsets that are disjoint. The Kneser graphs of the form
are also known as the odd graphs. We settle an old problem due to
Meredith, Lloyd, and Biggs from the 1970s, proving that for every ,
the odd graph has a Hamilton cycle. This and a known conditional
result due to Johnson imply that all Kneser graphs of the form
with and have a Hamilton cycle. We also prove that
has at least distinct Hamilton cycles for .
Our proofs are based on a reduction of the Hamiltonicity problem in the odd
graph to the problem of finding a spanning tree in a suitably defined
hypergraph on Dyck words
Fractional coverings, greedy coverings, and rectifier networks
A rectifier network is a directed acyclic graph with distinguished sources and sinks; it is said to compute a Boolean matrix M that has a 1 in the entry (i,j) iff there is a path from the j-th source to the i-th sink. The smallest number of edges in a rectifier network that computes M is a classic complexity measure on matrices, which has been studied for more than half a century. We explore two techniques that have hitherto found little to no applications in this theory. They build upon a basic fact that depth-2 rectifier networks are essentially weighted coverings of Boolean matrices with rectangles. Using fractional and greedy coverings (defined in the standard way), we obtain new results in this area. First, we show that all fractional coverings of the so-called full triangular matrix have cost at least n log n. This provides (a fortiori) a new proof of the tight lower bound on its depth-2 complexity (the exact value has been known since 1965, but previous proofs are based on different arguments). Second, we show that the greedy heuristic is instrumental in tightening the upper bound on the depth-2 complexity of the Kneser-Sierpinski (disjointness) matrix. The previous upper bound is O(n^{1.28}), and we improve it to O(n^{1.17}), while the best known lower bound is Omega(n^{1.16}). Third, using fractional coverings, we obtain a form of direct product theorem that gives a lower bound on unbounded-depth complexity of Kronecker (tensor) products of matrices. In this case, the greedy heuristic shows (by an argument due to LovĂĄsz) that our result is only a logarithmic factor away from the "full" direct product theorem. Our second and third results constitute progress on open problem 7.3 and resolve, up to a logarithmic factor, open problem 7.5 from a recent book by Jukna and Sergeev (in Foundations and Trends in Theoretical Computer Science (2013)
The Complexity of Homomorphism Reconstructibility
Representing graphs by their homomorphism counts has led to the beautiful
theory of homomorphism indistinguishability in recent years. Moreover,
homomorphism counts have promising applications in database theory and machine
learning, where one would like to answer queries or classify graphs solely
based on the representation of a graph as a finite vector of homomorphism
counts from some fixed finite set of graphs to . We study the computational
complexity of the arguably most fundamental computational problem associated to
these representations, the homomorphism reconstructability problem: given a
finite sequence of graphs and a corresponding vector of natural numbers, decide
whether there exists a graph that realises the given vector as the
homomorphism counts from the given graphs.
We show that this problem yields a natural example of an
\mathsf{NP}^{#\mathsf{P}}-hard problem, which still can be -hard
when restricted to a fixed number of input graphs of bounded treewidth and a
fixed input vector of natural numbers, or alternatively, when restricted to a
finite input set of graphs. We further show that, when restricted to a finite
input set of graphs and given an upper bound on the order of the graph as
additional input, the problem cannot be -hard unless . For this regime, we obtain partial positive results. We also
investigate the problem's parameterised complexity and provide fpt-algorithms
for the case that a single graph is given and that multiple graphs of the same
order with subgraph instead of homomorphism counts are given
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