[Paper Review] On the Complexity of the Minimum Cost Homomorphism Problem for Reflexive Multipartite Tournaments

Gregory Gutin, Eun Jung Kim|ArXiv.org|Aug 19, 2007

Advanced Graph Theory Research18 references3 citations

TL;DR

This paper provides a dichotomy classification for the minimum cost homomorphism problem (MinHOMP) on reflexive multipartite tournaments, proving that the problem is polynomial-time solvable if and only if the digraph is a reflexive acyclic tournament or a specific class of 4-vertex acyclic multipartite tournaments with loops. The authors use bipartite representations and structural analysis to show NP-hardness for all other cases, resolving a key open problem in the complexity of MinHOMP for reflexive multipartite tournaments with possible loops.

ABSTRACT

For digraphs $D$ and $H$, a mapping $f: V(D)\dom V(H)$ is a homomorphism of $D$ to $H$ if $uv\in A(D)$ implies $f(u)f(v)\in A(H).$ For a fixed digraph $H$, the homomorphism problem is to decide whether an input digraph $D$ admits a homomorphism to $H$ or not, and is denoted as HOMP($H$). Digraphs are allowed to have loops, but not allowed to have parallel arcs. A natural optimization version of the homomorphism problem is defined as follows. If each vertex $u \in V(D)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of the homomorphism $f$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. For each fixed digraph $H$, we have the {\em minimum cost homomorphism problem for} $H$ and denote it as MinHOMP($H$). The problem is to decide, for an input graph $D$ with costs $c_i(u),$ $u \in V(D), i\in V(H)$, whether there exists a homomorphism of $D$ to $H$ and, if one exists, to find one of minimum cost. In a recent paper, we posed a problem of characterizing polynomial time solvable and NP-hard cases of the minimum cost homomorphism problem for acyclic multipartite tournaments with possible loops (w.p.l.). In this paper, we solve the problem for reflexive multipartite tournaments and demonstrate a considerate difficulty of the problem for the whole class of multipartite tournaments w.p.l. using, as an example, acyclic 3-partite tournaments of order 4 w.p.l.\footnote{This paper was submitted to Discrete Mathematics on April 6, 2007}

Motivation & Objective

• To characterize the polynomial-time solvable and NP-hard cases of the minimum cost homomorphism problem (MinHOMP) for reflexive multipartite tournaments with possible loops (w.p.l.).
• To resolve an open problem posed in prior work on the complexity of MinHOMP for acyclic multipartite tournaments w.p.l.
• To establish a dichotomy classification using structural analysis and bipartite representation techniques.
• To demonstrate the inherent complexity of MinHOMP for the broader class of multipartite tournaments w.p.l. by analyzing challenging cases such as acyclic 3-partite tournaments of order 4 w.p.l.

Proposed method

• Use of bipartite representation of digraphs to analyze MinHOMP complexity, leveraging known results on undirected graphs.
• Application of Theorem 3.6 on proper interval bigraphs to detect NP-hardness via induced subgraphs (e.g., bipartite nets, tents, cycles of length six).
• Structural case analysis based on partite set sizes and dominance relations between vertices in different partite sets.
• Reduction techniques via induced subdigraphs: if a subdigraph's bipartite representation is not a proper interval bigraph, MinHOMP(H) is NP-hard.
• Leveraging prior dichotomy results for reflexive tournaments and specific 4-vertex digraphs to classify polynomial cases.
• Use of Lemmas 3.1, 3.4, 4.4, 4.5, 4.6, 4.7, and 3.7 to systematically prove NP-hardness or polynomial solvability based on structural conditions.

Experimental results

Research questions

• RQ1 For which reflexive multipartite tournaments is the minimum cost homomorphism problem polynomial-time solvable?
• RQ2 What structural properties of reflexive multipartite tournaments lead to NP-hardness in MinHOMP?
• RQ3 Can the complexity of MinHOMP be fully classified for reflexive multipartite tournaments using bipartite representations?
• RQ4 Why is the general case of acyclic multipartite tournaments w.p.l. significantly more complex than the reflexive case?
• RQ5 What are the precise conditions under which a 4-vertex acyclic multipartite tournament with loops yields a polynomial-time MinHOMP?

Key findings

• MinHOMP(H) is polynomial-time solvable if H is a reflexive acyclic tournament.
• MinHOMP(H) is NP-hard if H contains an induced cycle or a bipartite net/tent, as confirmed by non-proper interval bigraph representations.
• When H has two or more partite sets of size greater than one, MinHOMP(H) is NP-hard.
• If any partite set has three or more vertices, MinHOMP(H) is NP-hard, as shown by Lemmas 4.4 and 4.5.
• For a 4-vertex acyclic multipartite tournament H with A(H) = {12,23,34,14,24} ∪ B and B ⊆ {11,22,33}, MinHOMP(H) is polynomial-time solvable if {33} ⊆ B ⊆ {11,22,33}; otherwise, it is NP-hard.
• The case H' with vertex set {1,2,3,4}, arc set {12,23,24} ∪ B (B ⊆ {11,22,33,44}) remains unresolved, particularly for B = {33,44}, indicating complexity beyond current classification.