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# ps2 - Homework 2 IE418 Discrete Optimization Dr Ralphs Due...

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Homework 2 IE418 – Discrete Optimization Dr. Ralphs Due February 13, 2007 1. Given a graph ( G = ( V, E ) with weights c v for all v V , formulate the following problems as mixed-integer programs. (a) Find a maximum-weight clique. (b) Find a minimum-weight dominating set (a set of nodes U V such that every node of V is adjacent to some node of U ). 2. Recall the classical formulation of the Traveling Salesman Problem (TSP). Given a directed graph G = ( N, A ), feasible solutions consist of x Z A satisfying X j :( i,j ) A x ij = 1 for i N, (1) X i :( i,j ) A x ij = 1 for j N, and (2) X ( i,j ) A : i U,j N \ U x ij 1 for U N with 2 ≤ | U | ≤ | N | - 2 , (3) 1 x 0 (4) where the binary variables x ij for each ( i, j ) A represent whether the salesman travels from i to j in the final tour (see Section 1.3 of the textbook). One drawback of this classical formulation of the TSP is that its size is exponential in the size of the original problem description. It is possible, however, to obtain a polynomially- sized formulation using a more subtle set of constraints. We do this by creating an

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