Cs 598csc: Combinatorial Optimization
نویسندگان
چکیده
Let f : 2S → R be a submodular set function. We discuss a connection between submodular functions and convexity that was shown by Lovász [3]. Given an arbitrary (not necessarily submodular) set function f : 2S → R, we can view it as assigning values to the integer vectors in the hypercube [0, 1]n where n = |S|. That is, for each U ⊆ S, f(χ(U)) = f(U). We say that a function f̂ : [0, 1]n → R is an extension of f if f̂(χ(U)) = f(U) for all U ⊆ S; that is f̂ assigns a value to each point in the hypercube and agrees with f on the characterstic vectors of the subsets of S. There are several ways to define an extension and we consider one such below. Let S = {1, 2, . . . , n}. Consider a vector c = (c(1), . . . , c(n)) in [0, 1]n and let p1 > p2 > . . . > pk be the distinct values in {c(1), c(2), . . . , c(n)}. Define qk = pk and qj = pj−pj+1 for j = 1, . . . , k−1. For 1 ≤ j ≤ k, we let Uj = {i | c(i) ≥ pj}. Define f̂ as follows:
منابع مشابه
Cs 598csc: Combinatorial Optimization
One of several major contributions of Edmonds to combinatorial optimization is algorithms and polyhedral theorems for matroid intersection, and more generally polymatroid intersection. From an optimization point of view, the matroid intersection problem is the following: Let M1 = (S, I1) and M2 = (S, I2) be two matroids on the same ground set S. Then I1 ∩ I2 is the collection of all sets that a...
متن کاملCs 598csc: Combinatorial Optimization
Now we turn our attention to the proof of Theorem 1. Proof of Theorem 1. Let x be an extreme solution for the polytope (∗). Suppose that M has a loop e. Since rM ({e}) = 0, it follows that x(e) = 0 and we are done. Therefore we may assume that M does not have any loops and thus the polytope (∗) is full dimensional1. Now suppose that x(e) ∈ (0, 1) for all elements e ∈ S. Let n denote the number ...
متن کاملCs 598csc: Combinatorial Optimization 1 Polyhedra and Linear Programming
In this lecture, we will cover some basic material on the structure of polyhedra and linear programming. There is too abundant material on this topic to be covered in a few classes, so pointers will be given for further reading. For algorithmic and computational purposes one needs to work with rational polyhedra. Many basic results, however, are valid for both real and rational polyhedra. There...
متن کاملCs 598csc: Combinatorial Optimization 1 Okamura-seymour Theorem
Theorem 1 Let G = (V, E) be a plane graph and let H = (T, R) be a demand graph where T is the set of vertices of a single face of G. Then if G satisfies the cut condition for H and G + H is eulerian, there is an integral multiflow for H in G. The proof is via induction on 2|E| − |R|. Note that if G satisfies the cut condition for H, then |R| ≤ |E| (why?). There are several " standard " inductio...
متن کاملCs 598csc: Combinatorial Optimization 1 Min Cost Perfect Matching
In this lecture, we will describe a strongly polynomial time algorithm for the minimum cost perfect matching problem in a general graph. Using a simple reduction discussed in lecture 9, one can also obtain an algorithm for the maximum weight matching problem. We also note that when discussing perfect matching, without loss of generality, we can assume that all weights/costs are non-negative (wh...
متن کاملCs 598csc: Combinatorial Optimization Gomory-hu Trees
(The work in this section closely follows [3]) Let G = (V,E) be an undirected graph with non-negative edge capacities defined by c : E → R. We would like to be able to compute the global minimum cut on the graph (i.e., the minimum over all min-cuts between pairs of vertices s and t). Clearly, this can be done by computing the minimum cut for all ( n 2 ) pairs of vertices, but this can take a lo...
متن کامل