Finding a subgraph of cliques with the minimum total sum weight

Question

Consider the following graph problem. For a number $K$ and a set $\mathcal{K} = \{ 1, \ldots,K\}$, we have a set of vertices $V_k^s$ for all $s \subset \mathcal{K} \setminus \{k\}$, $s$ is not empty and for all $k$. For example, if $K =2$, we have $V_1^{\{2\}}$ and $V_2^{\{1\}}$. if $K =3$, we have $V_1^{\{2,3\}}$, $V_1^{\{2\}}$, $V_1^{\{3\}}$, $V_2^{\{1,3\}}, V_2^{\{1\}}, V_2^{\{3\}}, V_3^{\{1,2\}}, V_3^{\{1\}}, V_3^{\{2\}}$. The weight of vertex $V_k^s$ is denoted by $v_k^s$.

There is an edge between two vertices $V_k^s$ and $V_l^t$, if $k \in t$ and $l \in s$. For example, for $K=3$, there is an edge between $V_1^{\{2,3\}}$ and $V_2^{\{1\}}$, but there is no edge between $V_1^{\{2\}}$ and $V_3^{\{1\}}$.

The resulting graphs for $K=2$ and $K=3$ have been shown in the two following figures. Note that the corresponding graph for $K=k$ have $Q_k = k \times (2^{k-1}-1)$ vertices.

For each clique (complete subgraph) of the original graph, the weight is defined as the maximum of the weights of the vertices of the clique. More specifically, if vertices $X_1, \ldots, X_k$ form a clique where $x_i$ is the weight of vertex $X_i$, then the weight of this clique is $\max\{x_1,\ldots,x_k \}$. For example, the weight of the following clique is defined as $\max\{v_1^{\{2,3\}},v_2^{\{1,3\}},v_3^{\{1,2\}} \}$.

Then, the total weight of a subgraph with multiple cliques is the sum of the weights of the cliques. For example, the weight of the following subgraph is $\max\{v_1^{\{2,3\}},v_2^{\{1,3\}},v_3^{\{1,2\}} \}+ \max \{v_1^{\{2\}},v_2^{\{1\}} \} + \max \{v_1^{\{3\}},v_3^{\{1\}} \} +v_2^{\{3\}} + v_3^{\{2\}}$.

For a given number $K$ and its corresponding graph, we want to find the minimum weight subgraph consisting of disjoint cliques which includes all vertices (that is, all vertices are present in the subgraph and each vertex is exactly in one clique). Is the following greedy algorithm optimal?

*Assume we have access to an oracle that can say whether a set of vertices can form a clique or not. Now, we sort the vertices based on their weights in decreasing order (the first one has the highest weight) and if two have the same weight we sort them based on the number of edges of them. For example for $K=3$, if all vertices have the same weight, we sort them as $v_1^{\{2,3\}},v_2^{\{1,3\}},v_3^{\{1,2\}}, v_1^{\{2\}},v_1^{\{3\}}, v_2^{\{1\}}, v_2^{\{3\}}, v_3^{\{1\}}, v_3^{\{2\}}$.

Assume the sorted list is $X_1, \ldots, X_9$ where $X_i$ is one of the vertices. Now, we start from the clique $S = X_1$ and check the list $X_2, \ldots, X_9$ from left to right. For each element, we check whether adding $X_j$ to $S$ results in a clique or not. If yes, we form a clique of $S$ and $X_j$, and remove $X_j$ from the list. We continue this process (now for $S = \{X_1,X_j\}$) until we reach the end of list $X_2, \ldots, X_9$. Now we remove the vertices of set S, from $X_1, \ldots, X_9$, and repeat the above procedure for the first element of $X_1, \ldots, X_9$ minus the set $S$.

Answer 1

Not sure what you mean by saying vertices "can form an acceptable connected component" but no, your greedy algorithm isn't right.

Here are two examples that disprove what you want based on what exactly you're saying (example 2 is probably better).

Example 1: [this might not be a counterexample depending on what you intended your algorithm to do. See example 2 if you aren't happy with this.] As in your example, I'll call the vertices $\{A_1, A_2, A_3, B_1, \ldots, C_3\}$. The graph is the complete graph minus two edges. Namely, we insist $A_1 \not \sim A_2$ and $A_2 \not \sim A_3$. The weights should be $A_1$ has weight 1000, $A_2$ has weight 999, $A_3$ has weight $2$, and all the others have weight 1. If your algorithm picks the heaviest vertex and then grows this by greedily finding the heaviest adjacent vertex (et cetera), then your algorithm with fail on this input.

Example 2: this has 9 vertices (but we only need the four non-isolated vertices). The graph has exactly 3 edges, which are $A_1 \sim B_2 \sim B_1 \sim A_2$. The weights are that $B_1$ and $B_2$ have weight 100, $A_1$ and $A_2$ have weight $99$, and the other vertices have weight 0. The best min-weight decomposition here has weight $200$ but [if yours does anything close to what I think it does] your algorithm will give a weight of $100 + 99 +99$.

Example 3: the greedy algorithm fails for the graph you gave as an example. Let the weights for $A_i$ all be equal to $1.1$, and let all the other weights be $1.$ Greedy algorithm would give $4.1$ but best is $3.3$.

That said! Is there a polynomial-time algorithm? I bet there is by some sort of greedy with augmentation, but I haven't thought too much about that.

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Added after question was edited:

The original question posted was about decomposing the graph into connected components, and it has since changed to be about decomposing the graph into cliques (and notation has been changed). With this new problem, the greedy algorithm still fails [in fact, I believe it's easier to see this]. To see that it fails, take the graph you show and weight it so that $V_{1}^{\{2,3\}}$ and $V_2 ^{\{1\}}$ have weight 1.1 with all others having weight 1.