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Chap29/29.2/index.html

@@ -9961,7 +9961,7 @@ <h2 id="292-1">29.2-1<a class="headerlink" href="#292-1" title="Permanent link">
 <blockquote>
 <p>Put the single-pair shortest-path linear program from $\text{(29.44)}$–$\text{(29.46)}$ into standard form.</p>
 </blockquote>
-<p>The objective is already in normal form. However, some of the constraints are equality constraints instead of $\le$ constraints. This means that we need to rewrite them as a pair of inequality constraints, the overlap of whose solutions is just the case where we have equality. we also need to deal with the fact that most of the variables can be negative. To do that, we will introduce variables for the negative part and positive part, each of which need be positive, and we'll just be sure to subtract the negative part. $d_s$ need not be changed in this way since it can never be negative since we are not assuming the existence of negative weight cycles.</p>
+<p>The objective is already in normal form. However, some of the constraints are equality constraints instead of $\le$ constraints. This means that we need to rewrite them as a pair of inequality constraints, the overlap of whose solutions is just the case where we have equality. We also need to deal with the fact that most of the variables can be negative. To do that, we will introduce variables for the negative part and positive part, each of which need be positive, and we'll just be sure to subtract the negative part. $d_s$ need not be changed in this way since it can never be negative since we are not assuming the existence of negative weight cycles.</p>
 <p>$$
 \begin{aligned}
 d_v^+ - d_v^- - d_u^+ + d_u^- \le w(u, v) \text{ for every edge } (u, v) \\
@@ -10059,11 +10059,11 @@ <h2 id="292-7">29.2-7<a class="headerlink" href="#292-7" title="Permanent link">
 <blockquote>
 <p>In the <strong><em>minimum-cost multicommodity-flow problem</em></strong>, we are given directed graph $G = (V, E)$ in which each edge $(u, v) \in E$ has a nonnegative capacity $c(u, v) \ge 0$ and a cost $a(u, v)$. As in the multicommodity-flow problem, we are given $k$ different commodities, $K_1, K_2, \ldots, K_k$, where we specify commodity $i$ by the triple $K_i = (s_i, t_i, d_i)$. We define the flow $f_i$ for commodity $i$ and the aggregate flow $f_{uv}$ on edge $(u, v)$ as in the multicommodity-flow problem. A feasible flow is one in which the aggregate flow on each edge $(u, v)$ is no more than the capacity of edge $(u, v)$. The cost of a flow is $\sum_{u, v \in V} a(u, v)f_{uv}$, and the goal is to find the feasible flow of minimum cost. Express this problem as a linear program.</p>
 </blockquote>
-<p>As in the minimum cost flow problem, we have constraints for the edge capacities, for the conservation of flow, and nonegativity. The difference is that the restraint that before we required exactly $d$ units to flow, now, we require that for each commodity, the right amount of that commodity flows. the conservation equalities will be applied to each different type of commodity independently. If we super script $f$ that will denote the type of commodity the flow is describing, if we do not superscript it, it will denote the aggregate flow</p>
+<p>As in the minimum cost flow problem, we have constraints for the edge capacities, for the conservation of flow, and nonnegativity. The difference is that the restraint that before we required exactly $d$ units to flow, now, we require that for each commodity, the right amount of that commodity flows. the conservation equalities will be applied to each different type of commodity independently. If we superscript $f$, it will denote the type of commodity the flow is describing; if we do not superscript it, it will denote the aggregate flow.</p>
 <p>We want to minimize</p>
 <p>$$\sum_{u, v \in V} a(u, v) f_{uv}.$$</p>
 <p>The capacity constraints are that</p>
-<p>$$\sum_{i \in [k]} \sum_{u, v \in V} f_{uv}^i \le c(u, v).$$</p>
+<p>$$\sum_{i \in [k]} f_{uv}^i \le c(u, v) \text{ for each edge } (u, v).$$</p>
 <p>The conservation constraints are that for every $i \in [k]$, for every $u \in V \backslash \{s_i, t_i\}$.</p>
 <p>$$\sum_{v \in V} f_{uv}^i = \sum_{v \in V} f_{vu}^i.$$</p>
 <p>Now, the constraints that correspond to requiring a certain amount of flow are that for every $i \in [k]$.</p>