Ok, It is clear you don't understand real estate law and fence and property line issues, or at least you are not likeing what you are being told.
You can cut what ever is on your side of the property line, yo cannot make them remove their vines.
Since you don't know what they are and have not bothered to take a leaf and flower to the nusery to have them identified, I can't help you specifically, but since they don't spread by roots you are lucky! Simply replace your fence they don't last forever anyway. Dig a trench down 2-3 feet up to the property line removibg all the soil and roots, so frost isn't a problem, fill this with "con Mix" this will take approximately 5-6 cubic yards or one standard dump truck of con mix to fill the treach to make a footing that allows drainage and for soil movement during your harsh MN winters, don't worry, you can easily move this yourself, I moved a similar amount when I was 9 mos pregnant in 100 degree weather. Then install a cinderblock fence up to the property line 6-8 feet high, install a 2" copper barrier on the top, this will solve your problem. I would have a contractor install the fence, be sure to get building permits. More than likely, this will kill off the vines.
Look up the infamous Kudzu vine that covers the south, it won't grow in MN, now that's a vine to contend with! It can grow 60' in a year ans 1' a day in the summer!
Kudzu's History:
Up and Down the Power Pole
Kudzu was introduced to the United States in 1876 at the Centennial Exposition in Philadelphia, Pennsylvania. Countries were invited to build exhibits to celebrate the 100th birthday of the U.S. The Japanese government constructed a beautiful garden filled with plants from their country. The large leaves and sweet-smelling blooms of kudzu captured the imagination of American gardeners who used the plant for ornamental purposes.
Florida nursery operators, Charles and Lillie Pleas, discovered that animals would eat the plant and promoted its use for forage in the 1920s. Their Glen Arden Nursery in Chipley sold kudzu plants through the mail. A historical marker there proudly proclaims "Kudzu Developed Here."
During the Great Depression of the 1930s, the Soil Conservation Service promoted kudzu for erosion control. Hundreds of young men were given work planting kudzu through the Civilian Conservation Corps. Farmers were paid as much as eight dollars an acre as incentive to plant fields of the vines in the 1940s.
http://www.cptr.ua.edu/kudzu/
"Cotton isn't king in the South anymore.
Kudzu is king!"
Channing Cope
Kudzu's most vocal advocate was Channing Cope of Covington, Georgia who promoted use of the vine to control erosion. Cope wrote about kudzu in articles for the Atlanta Journal-Constitution and talked about its virtues frequently on his daily WSB-AM radio program broadcast from his front porch. During the 1940s, he traveled across the southeast starting Kudzu Clubs to honor what he called "the miracle vine."
Cope was very disappointed when the U.S. government stopped advocating the use of kudzu in 1953.
Kudzu Control:
An Impossible Dream?
The problem is that it just grows too well! The climate of the Southeastern U.S. is perfect for kudzu. The vines grow as much as a foot per day during summer months, climbing trees, power poles, and anything else they contact. Under ideal conditions kudzu vines can grow sixty feet each year.
While they help prevent erosion, the vines can also destroy valuable forests by preventing trees from getting sunlight. This problem led Dr. James H. Miller of the U.S. Forest Service in Auburn, Alabama to research methods for killing kudzu. In eighteen years of research, he has found that one herbicide actually makes kudzu grow better while many have little effect. Miller recommends repeated herbicide treatments for at least four years, but some kudzu plants may take as long as ten years to kill, even with the most effective herbicides.
The USDA declared kudzu to be a weed in 1972!
Dr. Errol G. Rhoden, along with other researchers at Tuskegee University, has successfully raised Angora goats in fields of kudzu which would otherwise be considered wasted land. The goats keep the kudzu from spreading further while producing profitable milk and wool products. Rhoden says constant grazing will eventually eradicate kudzu. If kudzu is to provide a continuing food source, animals must be removed from the fields occasionally to allow the vines time to grow.
Uses for Kudzu:
It's here. It's free... Why not?
Basket makers have found that the rubber-like vines are excellent for decorative and functional creations. Ruth Duncan of Greenville, Alabama makes over 200 kudzu baskets each year and says she doesn't mind that people call her the "Queen of Kudzu."
Regina Hines of Ball Ground, Georgia, has developed unique basket styles which incorporate curled kudzu vines. She weaves with other vines as well, but says that kudzu is the most versatile.
Nancy Basket of Walhalla, South Carolina, makes paper from kudzu which she uses in colorful collages. Her designs vary from geometric shapes to images of rural life and Native American themes.
Diane Hoots of Dahlonega, Georgia has developed a company to market her kudzu products which include kudzu blossom jelly and syrup, kudzu baskets, and books. Her book, Kudzu: The Vine to Love or Hate, co-written with Juanita Baldwin, is an in-depth study of the South's love/hate relationship with the vine. The book includes recipes and basket making instructions.
Henry and Edith Edwards of Rutherfordton, North Carolina have found many uses for kudzu over the past 30 years. Henry produces over 1,000 bales of kudzu hay each year on his Kudzu Cow Farm. The hay is high in nutritive value, but many people have found kudzu difficult to cut and bale. Henry says the secret is to "cut it low and bale it high."
Edith Edwards makes deep-fried kudzu leaves, kudzu quiche, and many other kudzu dishes. She found recipes in The Book of Kudzu: A Culinary and Healing Guide by William Shurtleff and Akiko Aoyagi, and thought this was a good use for a plentiful resource. She has demonstrated kudzu cooking for clubs, schools, and visitors to the Knoxville World's Fair.
Common names for kudzu include:
mile-a-minute vine,
foot-a-night vine,
and the vine that ate the South.
Current research may lead to new medicines made from kudzu, but for now only hamsters and mice can benefit from these drugs. Research with laboratory animals at Harvard Medical School has revealed that a drug extracted from kudzu root may help in the treatment of alcoholism. The drug is based on a 2,000 year old Chinese herbal medicine. Several years of testing may be required before the drug can be made available for human consumption.
In China and Japan, ground kudzu root (called kuzu) has been a common ingredient in foods and medications for centuries. Kudzu is respected and enjoyed there. It's far more versatile than say, turnips. But kudzu grows better in the South than it does in its native lands. Its natural insect enemies were not brought to the U.S. with it.
That's why visitors to the South are sometimes awe-struck by scenic vistas which reveal miles and miles of seemingly endless vines.
Southerners just close their windows at night to keep the kudzu out.
