{
  "id": "1212.2306",
  "version": "v3",
  "published": "2012-12-11T05:46:18.000Z",
  "updated": "2012-12-20T08:12:49.000Z",
  "title": "Agent Arrangement Problem",
  "authors": [
    "Tomoki Nakamigawa",
    "Tadashi Sakuma"
  ],
  "comment": "32 pages, 5 figures",
  "categories": [
    "math.CO",
    "math.OC"
  ],
  "abstract": "An {\\em arrangement} of an ordered pair $(G_A, G_M)$ of graphs is defined as a function $f$ from $V(G_A)$ to $V(G_M)$ such that, for each vertex $c$ of $G_M$, the vertex-set $f^{-1}(c)$ of $G_A$ either is $\\emptyset$ (the case when $c \\not\\in f(V(G_A))$) or induces a connected subgraph of $G_A$ and that the family $\\{f^{-1}(y) : y \\in V(G_M), f^{-1}(y) \\neq \\emptyset\\}$ is a partition of $V(G_A)$. Let $f$ be an arrangement of $(G_A, G_M)$, let $pq$ be an edge of $G_M$ and let $U$ be a subset of $f^{-1}(p)$ such that each of the three graphs $G_A[U]$, $G_A[f^{-1}(p)\\setminus U]$ and $G_A[f^{-1}(q)\\cup U]$ is ether connected or $\\emptyset$ and that $\\big(f^{-1}(p)\\cup f^{-1}(q) \\big) \\setminus U \\neq \\emptyset$. A {\\em transfer} of $U$ from $p$ to $q$ is defined as the modification $f^{\\prime}$ of $f$ such that $f^{\\prime}(x):=f(x)$ for every $ x \\notin U$ and $f^{\\prime}(u):=q$ for every $u \\in U$. Two arrangements $f$ and $g$ of $(G_A, G_M)$ are called {\\em t-equivalent} if they can be transformed into each other by a finite sequence of transfers. An ordered pair $(G_A, G_M)$ of graphs is called {\\em almighty} if every two arrangements of the pair $(G_A, G_M)$ are t-equivalent. In this study, we consider the following two decision problems. [{\\bf (P1)}]{For a given pair of arrangements $f$ and $g$ of a given ordered pair $(G_A,G_M)$ of graphs, decide whether $f$ is t-equivalent to $g$ or not.} [{\\bf (P2)}]{For a given ordered pair $(G_A,G_M)$ of graphs, decide whether the pair $(G_A,G_M)$ is almighty or not.} We show an $\\Od(|E(G_A)|+(|V(G_M)|+|E(G_A)|)|V(G_A)|)$-time algorithm for {\\bf (P1)}, and prove the $\\co\\np$-completeness of {\\bf (P2)}.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-12-20T08:12:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C85",
      "34B45",
      "94C15",
      "68R10",
      "05C57",
      "G.2.2"
    ],
    "keywords": [
      "agent arrangement problem",
      "ordered pair",
      "t-equivalent",
      "finite sequence",
      "decision problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.2306N"
    }
  }
}