Abstract

1
Information-theoretic approaches to branching in
search Andrew Gilpin and Tuomas Sandholm, CMU,
CSD This was work was funded by, and conducted
at, CombineNet, Inc., Fifteen 27th St.,
Pittsburgh, PA 15222.

• Abstract
• Deciding what to branch on at each node is a key
  element of search algorithms
• We present four families of methods for selecting
  what question to branch on
• Each is information-theoretically motivated to
  reduce uncertainty in remaining sub-problems
• Experiments demonstrate improvement over
  state-of-the-art

• Motivation
• Integer programming problem
• xargmin cx Ax b, xi integer
• Integer programs are at the center of many
  multi-agent systems (combinatorial market
  clearing, equilibrium computation)
• These problems are usually solved using
  algorithms based on branch-and-bound
• Beginning of search total uncertainty about
  optimal solution
• End of search zero uncertainty about optimal
  solution
• Idea Measure uncertainty and drive the search to
  minimize this uncertainty quickly

• Information theory and entropy
• Let X be a binary random variable with events A
  and B
• Let p be the probability of A and 1-p the
  probability of B
• entropy(X) -p log2 p (1-p) log2 (1-p)
• Entropy is additive for independent random
  variables
• Big assumption treat variables as independent
  random variables

• Strong branching2
• Let w be fractional soln at current node
• w argmin cw Aw b
• Let C c1,,cn be index set for n fractional
  variables (the candidates)
• For each i in C
• xi,l argmin cx Ax b, xci floor(wci)
• xi,r argmin cx Ax b, xci floor(wci)
  1
• Branch on variable
• j mini max cxi,l, cxi,r

• Entropic branching
• Let w be fractional soln at current node
• w argmin cw Aw b
• Let C c1,,cn be index set for n fractional
  variables (the candidates)
• For each i in C
• xi,l argmin cx Ax b, xci floor(wci)
• xi,r argmin cx Ax b, xci floor(wci)
  1
• Branch on variable
• j mini max entropy(xi,l),
  entropy(xi,r)

• Combining strong branching and entropic branching
• Tie-breaking
• Use SB, EB to break ties
• Can consider close ties
• Combinational
• Convex combination
• Combining ranks of each
• Since strong branching and entropic branching are
  such similar computations, there is no overhead
  for combining

• Indicator entropic branching (IEB)
• Combinatorial procurement
• M 1,,m goods to procure
• S 1,,s suppliers
• B 1,,n bids, indicating bundle, price from a
  supplier
• Buyer specifies max number of winning suppliers K
• Want min cost allocation satisfying max supplier
  constraint
• NP-complete, even if bids on single items only3
• Branching strategy
• Integer programming methods have been successful
  in clearing combinatorial markets1, but still
  need to be improved
• Natural formulation of above problem contains an
  indicator variable for each supplier
• For each supplier, compute the entropy on that
  suppliers bids
• Strategy Branch on the indicator variable for
  the supplier having the greatest entropy

IEB experimental results ssuppliers,
rregions, mitems/region, bbids/region, Kmax
suppliers Solution time for complete search
(third row indicates gap after 1 hour)
s r m b k CPLEX CPLEX-IF IEB
20 10 10 100 5 25.63 15.13 11.81
30 15 15 150 8 5755.92 684.83 551.82
40 20 20 200 10 37.05 32.38 30.57

• Branching on multiple variables
• Generalizes one-step look-ahead on individual
  variables
• Given a set X x1,,xn of variables, let k
  floor(x1xn)
• We can generate these branches
• x1xn k and x1xn k1
• This is the only value of k worth consider all
  others lead to one child being the same
• Can skip evaluation of sets where their
  fractional sum is already integral

• Selected bibliography
• Arne Andersson, Mattias Tenhunen, and Fredrik
  Ygge. Integer programming for auctions with bids
  for combinations. In Proc. 4th International
  Conference on Multi-Agent Systems (ICMAS-00),
  2000.
• David Applegate, Robert Bixby, Vasek Chvatal,
  and William Cook. The traveling salesman problem.
  Technical report, DIMACS, 1994.
• Tuomas Sandholm and Subhash Suri. Side
  constraints and non-price attributes in markets.
  In IJCAI-2001 Workshop on Distributed Constraint
  Reasoning, Seattle, WA, 2001.