Showing papers by "Richard Cole published in 2007"

A Generalization of Kotzig’s Theorem and Its Application

Abstract: An edge of a graph is light when the sum of the degrees of its end-vertices is at most 13. The well-known Kotzig theorem states that every 3-connected planar graph contains a light edge. Later, Borodin [J. Reine Angew. Math., 394 (1989), pp. 180-185] extended this result to the class of planar graphs of minimum degree at least 3. We deal with generalizations of these results for planar graphs of minimum degree 2. Borodin, Kostochka, and Woodall [J. Combin. Theory Ser. B, 71 (1997), pp. 184-204] showed that each such graph contains a light edge or a member of two infinite sets of configurations, called 2-alternating cycles and 3-alternators. This implies that planar graphs with maximum degree $\Delta \geq 12$ are $\Delta$-edge-choosable. We prove a similar result with 2-alternating cycles and 3-alternators replaced by five fixed bounded-sized configurations called crowns. This gives another proof of $\Delta$-edge-choosability of planar graphs with $\Delta \geq 12$. However, we show efficient choosability; i.e., we describe a linear-time algorithm for $\max\{\Delta,12\}$-edge-list-coloring planar graphs. This extends the result of Chrobak and Yung [J. Algorithms, 10 (1989), pp. 35-51].

Indivisible Markets with Good Approximate Equilibrium Prices

Abstract: This paper considers the tradeoff between divisibility and the hardness of approximating equilibrium prices. Tight bounds are obtained for smooth Fisher markets that obey a relaxed weak gross substitutes property (WGS). A smooth market is one in which small changes in prices cause only proportionately small changes in demand, which we capture by a parameter k. Specifically, assuming that the total wealth is at least r times the total number of goods, this paper gives a polynomial time algorithm to compute prices achieving a (1 O(k/r))approximation and shows that it is NP-hard to do better. A second contribution of this paper is a new consideration of how to measure the quality of an approximation to equilibrium prices. Our approach takes the notion of compensatory payments from welfare economics and applies it to indivisible markets. This allows the dissatisfaction, or discontent, of individual agents to be combined in a natural way. In addition, an important observation is that in the indivisible setting, standard utility functions, such as CES, need not obey the standard WGS property.