Refutation of Aslams Proof that NP = P

Aslam presents an algorithm he claims will count the number of perfect matchings in any incomplete bipartite graph with an algorithm in the function-computing version of NC, which is itself a subset of FP. Counting perfect matchings is known to be #P-complete; therefore if Aslam’s algorithm is correct, then NP=P. However, we show that Aslam’s algorithm does not correctly count the number of perfect matchings and offer an incomplete bipartite graph as a concrete counter-example.

💡 Research Summary

The paper under review is a detailed refutation of Aslam’s claim that he has devised an NC‑class algorithm (hence polynomial‑time) for counting perfect matchings in any incomplete bipartite graph, a problem known to be #P‑complete. If correct, Aslam’s algorithm would imply #P = FP and consequently NP = P. The authors demonstrate that Aslam’s method fails to count perfect matchings accurately and provide a concrete counter‑example.

The critique begins by restating Aslam’s framework. He models each perfect matching of a balanced bipartite graph (B G_n) as a permutation (\pi) in the symmetric group (S_n). Each permutation is expressed as a product of transpositions, and each transposition corresponds to a node ((i,k,j_i)) in a directed acyclic “generating graph” (\Gamma(n)). A complete valid multiplication path (CVMP) in (\Gamma(n)) of length (n) uniquely encodes a permutation, and therefore a perfect matching.

Aslam’s algorithm partitions the set of all CVMPs into subsets called VMPSet((a_i,a_j)). Two fundamental operations are defined on these subsets: multiplication (concatenating a VMPSet((a,b)) with VMPSet((b,c)) to obtain VMPSet((a,c))) and addition (union of two VMPSet’s with the same endpoints). The key claim is that after (O(\log n)) rounds of these operations, the algorithm will have constructed a single VMPSet((1,n)) whose cardinality equals the number of CVMPs whose edge‑requirements (ER) are empty, i.e., the number of valid perfect matchings in the given incomplete graph.

The authors identify a critical flaw: the algorithm assumes that the ER of every VMPSet remains uniform throughout the multiplication and addition steps. ER is defined as the set of required edges that are not present in the input graph, after removing surplus edges (SE) that appear only because of the internal structure of a CVMP. However, SE can differ from one CVMP to another, and when two VMPSet’s are multiplied, the resulting paths may inherit different SE patterns, leading to different ERs. Consequently, a VMPSet produced by multiplication may contain some CVMPs with ER = ∅ (valid matchings) and others with ER ≠ ∅ (invalid), violating the uniform‑ER invariant required for the final summation.

To substantiate this theoretical observation, the authors construct a concrete counter‑example. They consider any (n\ge9) and focus on a subgraph (\gamma) of (\Gamma(9)) that contains exactly five CVMPs (denoted (p_{aa}, p_{ab}, p_{ba}, p_{bb}, p_{ad})). Each CVMP corresponds to a distinct permutation, and the authors explicitly list the node labels and the associated edge sets. They then define an incomplete bipartite graph (B G’_n) that omits certain edges required by some of these CVMPs. In this setting, only a subset of the five CVMPs have empty ER; the others have non‑empty ER because they rely on missing edges. Aslam’s algorithm, when applied to (\gamma), would count all five CVMPs as if they were valid, thereby over‑counting the number of perfect matchings in (B G’_n).

Beyond the specific example, the paper presents several lemmas that expose the combinatorial explosion of distinct SE patterns. Lemma 3 shows that at least ((n-1)!) CVMPs have pairwise distinct SE sets, implying that any attempt to keep all CVMPs with the same ER in a single VMPSet would require an exponential number of VMPSet’s. Lemma 4 further proves that the number of VMPSet’s sharing the same SE up to level (i) is bounded below by (\binom{n}{i}), again indicating that the algorithm cannot remain polynomial‑time while preserving the uniform‑ER property.

The authors also critique Aslam’s supporting lemmas (Lemma 5.8 and Lemma 5.9). While Lemma 5.8 (every pair of paths must be concatenable) is necessary, Lemma 5.9 (all nodes covered by a VMPSet must have identical ER) is shown to be unnecessary and, more importantly, insufficient to guarantee correctness. The paper demonstrates scenarios where Lemma 5.9 holds but the multiplication still yields a VMPSet with mixed ER, breaking the algorithm’s logic.

In conclusion, the refutation establishes that Aslam’s algorithm does not correctly count perfect matchings in incomplete bipartite graphs. The failure stems from an incorrect assumption about the preservation of edge‑requirements during the VMPSet multiplication process, which leads to over‑counting in the presence of surplus edges. Consequently, the claimed proof that NP = P collapses, and the algorithm does not provide a polynomial‑time solution to a #P‑complete problem. The paper’s combination of theoretical analysis and a concrete counter‑example convincingly invalidates Aslam’s claim.