63 research outputs found

    Short proofs of the Kneser-LovĂĄsz coloring principle

    Get PDF
    We prove that propositional translations of the Kneser–Lovász 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 neser–Lovász theorem based on the Hilton–Milner theorem; this avoids the topological arguments of prior proofs for all but finitely many base cases. We introduce new “truncated Tucker lemma” principles, which are miniaturizations of the octahedral Tucker lemma. The truncated Tucker lemma implies the Kneser–Lovász 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

    Get PDF
    The orthogonality dimension of a graph G=(V,E)G=(V,E) over a field F\mathbb{F} is the smallest integer tt for which there exists an assignment of a vector uv∈Ftu_v \in \mathbb{F}^t with ⟹uv,uv⟩≠0\langle u_v,u_v \rangle \neq 0 to every vertex v∈Vv \in V, such that ⟹uv,uvâ€Č⟩=0\langle u_v, u_{v'} \rangle = 0 whenever vv and vâ€Čv' are adjacent vertices in GG. 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 cc such that for every sufficiently large integer tt, it is NP\mathsf{NP}-hard to decide whether the orthogonality dimension of an input graph over R\mathbb{R} is at most tt or at least 3t/2−c3t/2-c. 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 nn-vertex graphs whose complement has orthogonality dimension over the binary field at most n1−ήn^{1-\delta} for some constant ÎŽ>0\delta >0. 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

    Kernelization, Proof Complexity and Social Choice

    Get PDF

    Approximating the Orthogonality Dimension of Graphs and Hypergraphs

    Get PDF
    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

    Get PDF
    For integers k≄1k\geq 1 and n≄2k+1n\geq 2k+1, the Kneser graph K(n,k)K(n,k) is the graph whose vertices are the kk-element subsets of {1,
,n}\{1,\ldots,n\} and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form K(2k+1,k)K(2k+1,k) 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 k≄3k\geq 3, the odd graph K(2k+1,k)K(2k+1,k) has a Hamilton cycle. This and a known conditional result due to Johnson imply that all Kneser graphs of the form K(2k+2a,k)K(2k+2^a,k) with k≄3k\geq 3 and a≄0a\geq 0 have a Hamilton cycle. We also prove that K(2k+1,k)K(2k+1,k) has at least 22k−62^{2^{k-6}} distinct Hamilton cycles for k≄6k\geq 6. 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

    Get PDF
    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

    Full text link
    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 GG as a finite vector of homomorphism counts from some fixed finite set of graphs to GG. 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 GG 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 NP\mathsf{NP}-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 GG as additional input, the problem cannot be NP\mathsf{NP}-hard unless P=NP\mathsf{P} = \mathsf{NP}. 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

    On Finding Constrained Independent Sets in Cycles

    Get PDF
    • 

    corecore