Ron Aharoni

Block-iterative projection methods for parallel computation of solutions to convex feasibility problems

Abstract An iterative method is proposed for solving convex feasibility problems. Each iteration is a convex combination of projections onto the given convex sets where the weights of the combination may vary from step to step. It is shown that any sequence of iterations generated by the algorithm converges if the intersection of the given family of convex sets is nonempty and that the limit point of the sequence belongs to this intersection under mild conditions on the sequence of weight functions. Special cases are block-iterative processes where in each iterative step a certain subfamily of the given family of convex sets is used. In particular, a block-iterative version of the Agmon-Motzkin-Schoenberg relaxation method for solving systems of linear inequalities is derived. Such processes lend themselves to parallel implementation and will be useful in various areas of applications, including image reconstruction from projections, image restoration, and other fully discretized inversion problems.

Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas

Abstract Seymour (Quart. J. Math. Oxford 25 (1974) , 303–312) proved that a minimal non 2-colorable hypergraph on n vertices has at least n edges. A related fact is that a minimal unsatisfiable CNF formula in n variables has at least n + 1 clauses (an unpublished result of M. Tarsi.) The link between the two results is shown; both are given infinite versions and proved using transversal theory (Seymours original proof used linear algebra). For the proof of the first fact we give a strengthening of Konigs duality theorem, both in the finite and infinite cases. The structure of minimal unsatisfiable CNF formulas in n variables containing precisely n + 1 clauses is characterised, and this characterization is given a geometric interpretation.

Hall's theorem for hypergraphs

We prove a hypergraph version of Halls theorem. The proof is topological.

Independent systems of representatives in weighted graphs

The following conjecture may have never been explicitly stated, but seems to have been floating around: if the vertex set of a graph with maximal degree Δ is partitioned into sets Vi of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each Vi at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR’s). En route, we give a survey of some recent developments in the theory of ISR’s.

Ryser's Conjecture for Tripartite 3-Graphs

We prove that in a tripartite 3-graph .

Unfriendly partitions of a graph

Minnesota, Minneapoli.~, Minnesota Communicated by the Managing Editors Received March 8. 1988 It has been conjectured by Cowan and Emerson [3] that every graph has an unfriendly partition; i.e., there is a partition of the vertex set V= V, v V, such that every vertex of V, is joined to at least as many vertices in V, _, as to vertices in V,. It is easily seen that every rinite graph has such a partition, and hence by compact- ness so does any locally finite graph. We show that the conjecture is also true for graphs which satisfy one of the following two conditions: (i) there are only finitely many vertices having infinite degrees; (ii) there are a finite number of infinite cardinals “to < ntI < cm, such that m, is regular for 1 < i 6 X-, there are fewer than m,, vertices having finite degrees, and every vertex having infinite degree has degree m, for some i < k.

Menger's theorem for countable graphs

Abstract The countable case of a conjecture of Erdos is settled: let G = (V, E) be a directed or undirected graph, where V is countable, and let A, B⊆V. There exists then a set P of disjoint A − B paths and an A − B separating set S of vertices so that S consists of the choice of precisely one vertex from each path in P .

Perfect matchings in r-partite r-graphs

Let H be an r-partite r-graph, all of whose sides have the same size n. Suppose that there exist two sides of H, each satisfying the following condition: the degree of each legal r-1-tuple contained in the complement of this side is strictly larger than n2. We prove that under this condition H must have a perfect matching. This answers a question of Kuhn and Osthus.

On a lemma of Scarf

The aim of this note is to point out some combinatorial applications of a lemma of Scarf, proved first in the context of game theory. The usefulness of the lemma in combinatorics has already been demonstrated in a paper by the first author and R. Holzman (J. Combin Theory Ser. B 73 (1) (1998) 1) where it was used to prove the existence of fractional kernels in digraphs not containing cyclic triangles. We indicate some links of the lemma to other combinatorial results, both in terms of its statement (being a relative of the Gale-Shapley theorem) and its proof (in which respect it is a kin of Sperners lemma). We use the lemma to prove a fractional version of the Gale-Shapley theorem for hypergraphs, which in turn directly implies an extension of this theorem to general (not necessarily bipartite) graphs due to Tan (J. Algorithms 12 (1) (1991) 154). We also prove the following result, related to a theorem of Sands et al. (J. Combin. Theory Ser. B 33 (3) (1982) 271): given a family of partial orders on the same ground set, there exists a system of weights on the vertices, which is (fractionally) independent in all orders, and each vertex is dominated by them in one of the orders.

On the strength of Ko¨nig's duality theorem for infinite bipartite graphs

Abstract We prove that Konigs duality theorem for infinite graphs (every graph G has a matching F such that there is a selection of one vertex from each edge in F which forms a cover of G ) is inherently of very high complexity in terms of both the methods of proof it requires and the computational complexity of the covers it produces. In particular, we show that there is a recursive bipartite graph such that any cover as required by the theorem is highly non-computable; indeed it must be above (in Turing degree) all the recursive iterations of the Turing jump. This implies that the theorem is proof theoretically at least as strong as the system ATR 0 which is known to be strictly stronger than compactness or Konigs lemma. Thus the theorem cannot be proven by elementary means plus compactness. Transfinite methods are actually necessary. The actual cover given by the proof considered is seen to have an additional maximality property which makes the assertion of its existence imply a stronger system, Π 1 1 -CA 0 . We refine this known proof of Konigs theorem to show that in fact its consequences are equivalent to Π 1 1 -CA 0 .