The two politicians have been at loggerheads over the new policy for months. Here, the idiom "at loggerheads" means -

Read the following passage. Based on what is stated or implied in the passage, answer questions that follow.To find the job that is right for you, you must set goals. A goal is a purpose you set for yourself. It is what you hope for in the future. You have to identify the skills that you need to get the job you want. Then you need to design a program to help you reach the level of improvement you set for yourself. To reach your goal, you must have a specific goal. Decide what you are going to do and when you are going to do it. Plan ways to achieve your goal. How will you accomplish it? Next, identify problems that will get in the way of accomplishing your goal. Then think of possible ways to solve the problems. Finally, evaluate your progress. Did you reach your gal? Did you get the job you wanted?