Is It Really NP-Hard to Survive in Big-Data?

Prof. Vadim E. Levit
Israel, Department of Computer Science and Mathematics, Ariel University
In order to survive, even in Big Data, one needs a hope. For example,
in computability theory the Church thesis serves us for railing, when it claims informally that a function is computable if and only if it may by programmed in a Turing all-in-one computer. Our thesis reads as follows: “Every intractable problem has its tractable counterpart.” Being more specific, we hint that a reasonable variation of an objective function makes it efficiently computable. On the other hand, we may consider to keep the same objective function and slightly change the data.
In what follows, the protagonist is a maximum independent set of a graph. It may equivalently appear, in the context of our research, as a maximum clique or even a biclique. See, for instance, the maximum vertex biclique problem versus the maximum edge biclique problem. The former can be solved in polynomial time, while the latter is known to be NP-hard [6, 11].
As a primary example of the objective functions exchange we choose the case, when critical independent sets come instead of maximum independent sets.
A set S ⊆ V (G) is independent if no two vertices from S are adjacent,
and by Ind(G) we mean the family of all the independent sets of G. An independent set of maximum size is a maximum independent set of G, and ∝(G) = max{|S| : S ∈ Ind(G)}.
For X⊆V(G), the number|X|—|N(X)|is the difference of X, denoted d(X). The critical difference d (G ) is max {d (X ) : X ⊆ V (G )}. The number max{d (I ) : I ∈ Ind(G )} is the critical independence difference of G, denoted id(G). It is shown in [13] that d(G) and id(G) coincide for every graph G. If A is an independent set in G with d(X) = id(G), then A is a critical independent set [13].
It is known that computing ∝(G) is an NP-hard problem [3]. Each critical independent set is included in some maximum independent set [1]. A critical independent set of maximum size can be found in polynomial time [5].
When our purpose is to modify the data, we invest heavily in König- Egerváry graphs.
A matching is a set M of pairwise non-incident edges of G. A matching
of maximum size, denoted μ(G), is a maximum matching. Recall that if ∝(G ) + μ (G ) =|V (G )|, then G is a König-Egerváry graph [2, 12]. In accord- ance with the celebrated König-Egerváry theorem, each bipartite graph
is a König-Egerváry graph as well. Various properties of König-Egerváry graphs can be found in [4, 8, 9, 10]. Let us also mention that the equality d(G) =∝(G)−μ(G) holds for each König-Egerváry graphG [7].
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