We study how to allocate a common budget across agents facing heterogeneous finite-horizon budgeted Markov decision processes so as to optimize the resulting value vector according to the leximin criterion. Boutilier and Lu (2016) show that, under deterministic policies, each agent’s attainable value is represented by finitely many budget-value pairs. Using this finite representation, we establish an exact two-stage decomposition of the leximin allocation problem. First, the leximin criterion requires maximizing the number of agents attaining positive value. Second, conditional on this maximum number, the planner must jointly decide which agents receive positive budgets and which budget levels are assigned to them. This second-stage finite problem can be formulated as a leximin multiple-choice knapsack problem with a cardinality constraint and solved exactly by dynamic programming.
For a graph G, let c1(G) be the largest distortion necessary to embed any shortest-path metric on G into ℓ1, and for any natural number n,m∈N, denote Kn,m as the complete bipartite graph. In this note, we calculate the value of c1(K2,n), more precisely we prove c1(K2,n)=3k−22k−1 where k=⌈n2⌉.
In recent years, research on the spectral extremal problems of fractional [a, b]-factors has attracted the attention of many scholars. The main content of our research is the generalization of fractional [a, b]-factors, namely fractional (a, b, l)-critical graph. Let a and b be two positive integers. A graph.. is called a fractional (a, b, l)-critical graph if after deleting any l vertices of G the remaining graph of G has a fractional [a, b]-factor. In this paper, we present spectral radius and size conditions for a graph to be fractional (a, b, l)-critical.
The atom-bond sum-connectivity (ABS) index of a graph is a variant of several well-known chemical topological indices, such as the randić index, the sum-connectivity index and the atom-bond connectivity index. For a graph G=(V(G),E(G)), the ABS index of G is defined as ABS(G)=∑uv∈E(G)1−2dG(u)+dG(v), where dG(u) denotes the degree of the vertex u in G. A cactus is a connected graph in which each block is either an edge or a cycle. For two integers n≥2 and k≥0, let G(n,k) be the set of cacti of order n and with k cycles. Obviously, G(n,0) is the set of all trees of order n and G(n,1) is the set of all unicyclic graphs of order n. In this paper, we determine the maximum ABS index of graphs among G(n,k), and also characterize the corresponding extremal graphs.
The independence number a of a graph is the cardinality of a largest set of vertices that are pairwise independent. Calculating this number is NP-hard. Various attempts have been made to better understand the structure of a graph in the hopes of yielding, for instance, computational improvements (more efficient algorithms). In this paper we show that two of these, a classical linear programming approach and a more recent purely graph-theoretic approach, are in fact the same. We conclude with a new characterization of the well-studied class of K & odblac;nig-Egerv & aacute;ry graphs.
