# Max-k-cut with negative edge weights

Let $$G=(V,E,w)$$ be a graph and for edge $$e\in E$$, there is associated weight $$w_e$$. The max-k-cut wants to partition V into k subsets $$P_1,\cdots,P_k$$ and maximize $$\sum_{1\leq r.

If $$k=2$$, or we assume non-negative weights, there is $$O(1)$$-approximation algorithms. I am wondering if we can get $$O(1)$$ or logarithmic approximation for general $$k$$ and possibly negative weights.

Besides, I am more interested in the case where the objective is to maximize $$|\sum_{1\leq r.

• See Approximating the Cut-Norm via Grothendieck’s Inequality'' for k=2, and Improved approximation algorithms for MAX k-CUT and MAX BISECTION'' for general k with non-negative weights. Aug 3, 2021 at 7:44
• Hasn't this question already been asked and answered, in the negative, for $k=2$? See cstheory.stackexchange.com/questions/2312/… Aug 3, 2021 at 13:52
• Yes, it has been answered for k=2 Aug 3, 2021 at 14:56
• But isn't the answer there (by Peter Shor) that it's not possible for $k=2$? So it's also not possible for general $k$? So the answer to your question is no? Aug 3, 2021 at 16:00
• The answer depends on whether we maximize the sum or its absolute value. In the former case, we cannot even determine the sign of the optimal value in polynomial time; in the later case, we can get a constant factor approximation.
– Yury
Aug 3, 2021 at 21:28

There is no approximation algorithm for the problem of maximizing $$\sum_{(i,j)\text{ is cut}} w_{ij}$$, since it's even NP-hard to determine whether the optimal value is positive. However, there is a constant-factor approximation algorithm for the problem of maximizing $$\left|\sum_{(i,j)\text{ is cut}} w_{ij}\right|$$. I will briefly describe how this algorithm works.

I. For a fixed instance of the problem, let $$OPT_k$$ be the optimal value when we partition $$V$$ into $$k$$ parts $$P_1, \dots, P_k$$. We assume that some parts may be empty. We now show that $$OPT_k/2 \leq OPT_2 \leq OPT_k.$$

• Let $$(P_1, P_2)$$ be an optimal solution with 2 parts. Pad this partition with $$k-2$$ empty parts. The value of the obtained solution $$(P_1, P_2, \varnothing, \dots, \varnothing)$$ is $$OPT_2$$. Thus, $$OPT_k \geq OPT_2$$.
• Now consider an optimal solution $$(P_1, \dots, P_k)$$ for $$k$$ parts. Randomly divide all clusters in two groups. Let $$P_1'$$ be the union of parts in the first group; $$P_2'$$ be the union of groups in the second. It's easy to see that $${\mathbb E}\left[\sum_{(i,j)\text{ is cut by }(P_1', P_2')} w_{ij}\right] = \frac12 \sum_{(i,j)\text{ is cut by }(P_1,\dots, P_k)} w_{ij}.$$ Therefore, $${\mathbb E}\left[\left|\sum_{(i,j)\text{ is cut by }(P_1', P_2')} w_{ij}\right|\right] \geq \left|\frac12 \sum_{(i,j)\text{ is cut by }(P_1,\dots, P_k)} w_{ij}\right|.$$ We conclude that $$OPT_k \geq OPT_2/2$$.

Thus, it is sufficient to design an approximation algorithm for $$OPT_2$$.

II. Recall that the Grothendieck inequality states that $$\max_{x,y\in\{-1,1\}^n} \sum_{i,j}w_{ij} x_i y_j \geq \frac{1}{K} \max_{\text{unit vectors }u_1,\dots,u_n,v_1,\dots, v_n} \sum_{i,j}w_{ij} \langle u_i, v_j\rangle$$ where $$K$$ is an absolute constant. Using semidefinite programming and the Grothendieck inequality, we can get a constant factor approximation for the following Quadratic Programming problem [Alon and Naor 2004].

$$\max_{x,y\in\{-1,1\}^n} \sum_{i,j}w_{ij} x_i y_j$$

The same algorithm gives a constant factor approximation for $$\max_{z\in\{-1,1\}^n} \left|\sum_{i,j}w_{ij} z_i z_j\right|$$ (we first find $$x$$ and $$y$$ and then output the better of the two solutions $$z=x$$ and $$z=y$$).

III. Let $$W = \sum_{i and $$QP= \max_{z\in\{-1,1\}^n} \left|\sum_{i Note that there is one-to-one correspondence between solutions to the QP and the partitioning problem; namely, a solution $$z_1,\dots,z_n$$ defines the following partition $$P_1 = \{i:z_i=1\}$$ and $$P_2=\{i:z_i = -1\}$$.

If solution $$(P_1, P_2)$$ has value $$x$$ then the value of the corresponding QP solution is either $$|W-2x|=|2x-W|$$ or $$\left|2x + W\right|$$. It follows that $$|OPT_2- 2QP| \leq |W|$$.

IV. Now we solve the QP approximately, obtain a solution $$z_1',\dots, z_n'$$, and output the better of the following two solution:

• partition $$(P_1', P_2')$$ defined by the QP solution $$z_1', \dots, z_n'$$; the value of this solution is $$\Theta(OPT_2) - O(|W|)$$.
• a random partition; the expected value of this solution is at least $$|W|/2$$.

It's easy to see that this algorithm gives a constant factor approximation for the partitioning problem.

• Thank you very much! Aug 6, 2021 at 3:46