Vasek Chvátal

A Greedy Heuristic for the Set-Covering Problem

Let A be a binary matrix of size m × n, let cT be a positive row vector of length n and let e be the column vector, all of whose m components are ones. The set-covering problem is to minimize cTx subject to Ax ≥ e and x binary. We compare the value of the objective function at a feasible solution found by a simple greedy heuristic to the true optimum. It turns out that the ratio between the two grows at most logarithmically in the largest column sum of A. When all the components of cT are the same, our result reduces to a theorem established previously by Johnson and Lovasz.

A note on Hamiltonian circuits

Proof. Let G satisfy the hypothesis of Theorem 1. Clearly, G contains a circuit ; let C be the longest one . If G has no Hamiltonian circuit, there is a vertex x with x ~ C . Since G is s-connected, there are s paths starting at x and terminating in C which are pairwise disjoint apart from x and share with C just their terminal vertices x l, X2, . . ., x s (see [ 11, Theorem 1) . For each i = 1, 2, . . ., s, let y i be the successor of x i in a

Many hard examples for resolution

For every choice of positive integers <italic>c</italic> and <italic>k</italic> such that <italic>k</italic> ≥ 3 and <italic>c</italic>2<supscrpt>-<italic>k</italic></supscrpt> ≥ 0.7, there is a positive number ε such that, with probability tending to 1 as <italic>n</italic> tends to ∞, a randomly chosen family of <italic>cn</italic> clauses of size <italic>k</italic> over <italic>n</italic> variables is unsatisfiable, but every resolution proof of its unsatisfiability must generate at least (1 + ε)<supscrpt><italic>n</italic></supscrpt> clauses.

Crossing-Free Subgraphs

If m⩾4 then every planar drawing of a graph with n vertices and m edges contains more than m 3 /100 n 2 edge-crossings and fewer than 10 13n crossing-free subgraphs. The first result settles a conjecture of Erdos and Guy and the second result settles a conjecture of Newborn and Moser.

On Hamilton's ideals

Abstract A theorem is proved that is, in a sense to be made precise, the best possible generalization of the theorems of Dirac, Posa, and Bondy that give successively weaker sufficient conditions for a graph to be Hamiltonian. Some simple corollaries are deduced concerning Hamiltonian paths, n -Hamiltonian graphs, and Hamiltonian bipartite graphs.

Star-cutsets and perfect graphs

Abstract We first establish a certain property of minimal imperfect graphs and then use it to generate large classes of perfect graphs.

Perfectly Ordered Graphs

This note presents a good characterization of a class of strongly perfect graphs which includes all comparability graphs, all triangulated graphs and all complements of triangulated graphs.

Four classes of perfectly orderable graphs

A graph is called “perfectly orderable” if its vertices can be ordered in such a way that, for each induced subgraph F, a certain “greedy” coloring heuristic delivers an optimal coloring of F. No polynomial-time algorithm to recognize these graphs is known. We present four classes of perfectly orderable graphs: Welsh–Powell perfect graphs, Matula perfect graphs, graphs of Dilworth number at most three, and unions of two threshold graphs. Graphs in each of the first three classes are recognizable in a polynomial time. In every graph that belongs to one of the first two classes, we can find a largest clique and an optimal coloring in a linear time.

Tree‐complete graph ramsey numbers

The ramsey number of any tree of order m and the complete graph of order n is 1 + (m − 1)(n − 1).

Hard Knapsack Problems

We consider a class of algorithms which use the combined powers of branch-and-bound, dynamic programming and rudimentary divisibility arguments for solving the zero-one knapsack problem. Our main result identifies a class of instances of the problem which are difficult to solve by such algorithms. More precisely, if reading the data takes t units of time, then the time required to solve the problem grows exponentially with the square root of t.