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On Completing Latin Squares

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What is a Latin Square and a Partial Latin Square (PLS)? The PLSE problem: Given a PLS, fill the maximum ... Construct a bipartite graph H with. vertex set T T' ... – PowerPoint PPT presentation

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Title: On Completing Latin Squares


1
On Completing Latin Squares
  • Iman Hajirasouliha
  • Joint work with
  • Hossein Jowhari, Ravi Kumar, and Ravi Sundaram

2
Definitions
  • What is a Latin Square and a Partial Latin Square
    (PLS)?
  • The PLSE problem Given a PLS, fill the maximum
    number of empty cells using numbers in n
    without violating the constraints.
  • The k-PLSE problem How many empty cells of a PLS
    can be filled properly using at most k n
    different numbers?

Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
3
Motivations and Applications
  • Interesting object for mathematicians, Evans
    conjecture(1960) says that a PLS with n-1 filled
    cells can be completed. (Proved by Smetaniuk in
    1981)
  • Sudoku puzzles, one of the current fads, are PLSs
    with additional properties.
  • The problem has application in error-correcting
    codes and recently optical networks.

Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
4
Previous and New Results
  • The PLSE problem is NP-Complete (Colbourn 1984)
  • The PLSE problem is APX-hard (This paper)
  • 1-1/ee hardness for the k-PLSE problem. (This
    paper)

Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
5
A problem equivalent to PLSE
  • The 3EDM Problem finding the number of maximum
    edge disjoint triangles in a tripartite simple
    graph.

columns
a
b
c
d
a
a
rows
b
a
b
c
c
b
d
d
c
d
1
3
2
4
numbers
Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
6
Local Search Algorithm for 3EDM
  • Let G be an instance of 3EDM
  • Fix a constant t 7.
  • Start from any arbitrary valid solution.
  • If possible, replace s t triangles in the
    current solution with s1 edge-disjoint triangles
    to get another valid solution.
  • Since the size of solution increases in each step
    by one, the algorithm runs in polynomial time.

Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
7
Local Search Analysis
  • Let TT1,, Tm be the set of edge disjoint
    triangles of OPT and TT1,, Tn be the set
    of triangles found by the heuristic.
  • Construct a bipartite graph H with
  • vertex set T ? T.
  • Connect Ti and Tj in H, iff Ti and Tj share an
    edge in G.

T1
T1
T2
T2
. . .
Tm
Tn
H
Optimal Triangles
Local Search Triangles
Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
8
Hurkens-Schrijver Theorem
  • Let H be a bipartite graph with vertex set X ? Y
    Xn, Ym.
  • Let k 3 and assume
  • For each y in Y, deg (y) k.
  • Every subset of size t of X has a system of
    distinct representatives in Y.
  • Then

Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
9
PLSE and H-S Theorem
  • H satisfies the Hurkens-Schrijver conditions.
  • deg (Tj) 3 for each Tj.
  • Every subset of size t in T has a System of
    Distinct Representation in T (due to local
    search).
  • Setting k3, we get the 2/3-e bound. For t7 we
    beat the previous result

T1
T1
T2
T2
. . .
Tm
Tn
H
Optimal Triangles
Local Search Triangles
Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
10
The k-PLSE problem
  • How many cells of a PLS can be filled using at
    most k n different numbers?
  • A natural greedy algorithm
  • Repeat k times
  • Pick the number c which can fill the most
    cells. Fill those cells with c.
  • The greedy algorithm is a ½ - approximation
    algorithm.

Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
11
Greedy algorithm analysis
  • OPT solution and greedy solution are sets of
    triples (i, j, k). To each triple y in OPT,
    we assign a triple x in greedy solution as
    accountable.
  • Given y(i, j, k) in OPT, we have three cases
  • 1) cell x(i, j, t) is in Greedy. x is
    accountable for y.

Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
12
  • 2) (i, j) is empty in Greedy but k has been used
    in Greedy. We can assign a distinct x(i, j, k)
    in Greedy to y. Consider the iteration where
    Greedy chooses k.
  • Cells with number 1 in OPT
  • Cells with number 1 in Greedy

Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
13
  • 3) cell (i, j) is empty in Greedy and number k
    is missing in Greedy. For each number c in OPT we
    can assign a number c in Greedy which is missing
    in OPT.

OPT
Greedy
Red cells in OPT are mapped to Yellow cells in
Greedy
Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
14
LP relaxation of the problem
  • A way to extend the PLS with a number represents
    a matching.
  • Mc is the set of all matchings that extends the
    PLS with number c.
  • ycM is 1 when Matching M is chosen.

Introduction, 2/3-e Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
15
1-1/e-e approximation
  • 1. Solve the LP program.
  • 2. Multiply the variables by 1-e.
  • 3. For each number pick a matching randomly
    according to the probability associated with the
    matchings.
  • 4. If matchings intersect in a cell, choose one
    of them arbitrarily for the cell.
  • Expectation of the size of solution obtained is
    bigger than (1-1/e-e)LPOPT
  • With a constant probability, at most k numbers
    have been picked.

Introduction, 2/3 Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
16
Conclusion
  • We defined a new and natural variation of the
    PLSE problem and obtained simple approximation
    algorithms for the PLSE and k-PLSE problems.
  • Our results for the PLSE problem is an
    improvement and for the k-PLSE problem is the
    best possible.

Introduction, 2/3 Approx. for PLSE, 1-1/e-e
Approx. for k-PLSE, Conclusion
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